Free energy analyticity of the disordered XY model and Debye screening in the 2D Coulomb gas

FREE ENER GY ANAL YTICITY OF THE DISORDERED XY MODEL AND DEBYE SCREENING IN THE 2D COULOMB GAS LUCAS D’ALIMONTE AND PIET LAMMERS Abstra ct. W e consider three mo dels of statistical mec hanics: the classical XY mo del in arbitrary dimension, the lattice Coulomb gas in dimension tw o, and the square well model in arbitrary dimension. F or each of these three mo dels, we pro ve that the free energy is analytic in the disordered regime (the square well mo del is disordered at any p ositive temp erature). In order to prov e these results, we pro v e that the Gibbs measures of these mo dels are factors of i.i.d. with information clusters of exponentially deca ying size (volume). In the case of the Coulomb gas, we obtain a strong v ersion of Deb ye screening with an arbitrary n umber of arbitrary local observ ables of the Coulomb gas, and we prov e that the Deb ye phase is equiv alen t to the complement of the Berezinskii–Kosterlitz–Thouless phase. Contents 1. Preface 2 2. Main results 4 3. Pro of ov erview and article organisation 8 P art I. The square w ell mo del 10 4. SWM: Pro of ov erview 10 5. SWM: Glaub er dynamic 11 6. SWM: Space-time mixing 14 7. SWM: Construction of the EFI ID 19 8. SWM: Spatial mixing of the Gibbs measure 21 P art I I. The XY mo del 23 9. XY: Pro of ov erview and monotone represen tation 23 10. XY: Glaub er dynamic 25 11. XY: Space-time mixing 27 12. XY: Construction of the EFI ID 29 13. XY: Spatial mixing of the Gibbs measure 31 14. The Villain mo del and the 2D Coulomb gas 32 A c knowledgemen ts 33 References 33 Date : 27 Marc h 2026. Key wor ds and phr ases. XY mo del, Coulomb gas, Deb ye screening, BKT transition, square w ell mo del, analyticit y of the free energy , factors of i.i.d., coupling from the past, cluster expansion. 1 2 LUCAS D’ALIMONTE AND PIET LAMMERS 1. Pref a ce 1.1. Classifying phase transitions b y regularit y . The idea of classifying phase transi- tions of lattice mo dels according to the regularity of their free energy was in troduced b y P aul Ehrenfest in 1933 [ Ehr33 ]. He predicted that a generic thermo dynamical quan tit y — the fr e e ener gy — considered as a function of the parameters of the system (often the temp er atur e ) should b e analytic, except at sp ecial tr ansition p oints . A t those p oints a phase tr ansition o ccurs, the nature of whic h can b e further inv estigated by studying the regularity of the free energy . In Ehrenfest’s terminology , the transition is said to b e of k -th or der at some point β c if the free energy at that p oint is C k − 1 but not C k . The determination of the order of phase transitions is of deep theoretical interest, and has been thoroughly discussed since the systematic study of statistical mec hanics. 1.2. Probabilistic p ersp ectiv es. Phase transitions can also b e c haracterised in terms of qualitativ e changes in the probabilistic b eha viour of the mo del when the inv erse temp era- ture passes certain threshold v alues (spontaneous magnetisation in the Ising mo del, the app earance of an inﬁnite cluster in percolation theory , etc.). The probabilistic viewp oint pro vides a welcome testing ground for mathematicians for the analysis of phase transitions, and those probabilistic questions can b e inv estigated using tools coming from sev eral ﬁelds (com binatorics, complex analysis, measure theory , algebra, and others). This also giv es rise to an integrated approach: analysing the relation b etw een the regularity of the free energy (or other thermo dynamical quantities) and the probabilistic b ehaviour at and around criticalit y . 1.3. The BKT transition. Berezinskii [ Ber71 ; Ber72 ] and later K osterlitz and Thou- less [ KT73 ] conjectured a new t yp e of phase transition in t w o-dimensional media with a con tin uous symmetry . This phase transition is driven b y the changing b eha viour of so-called top olo gic al defe cts , and a key feature of the prediction in [ KT73 ] is that the free energy is smo oth (but not analytic) at the critical p oin t ( inﬁnite or der in Ehrenfest’s terminology). This transition is no w called the Berezinskii–K osterlitz–Thouless (BKT) transition. While this prediction is widely exp ected to b e correct, a rigorous analysis of the near-critical windo w is still missing due to the lac k of precise to ols that apply in the nonp erturbative regime. The 2D XY mo del is not exp ected to b e integrable, and while progress is b eing made on its probabilistic analysis, the lac k of a hard quantativ e input mak es it diﬃcult to obtain a complete picture at the moment. Let us compare this to recent dev elopments in the context of the 2D random-cluster mo del with cluster w eight q ≥ 1 , and the asso ciated six-vertex mo del. By using the Bethe Ansatz (a manifestation of integrabilit y), it has b een pro v ed that the phase transition is con tin uous for q ∈ [1 , 4] [ DST17 ] and discon tin uous for q > 4 [ Dum+21 ]. This is closely related to the b ehaviour of the free energy of the mo del as a function of q and of the slop e in the six-v ertex mo del. W e now understand the b ehaviour for small slop es for all q > 0 [ Dum+22 ]. This implies that the mass of the model (the inv erse of the correlation length, which is zero for q ≤ 4 ) is analytic on (4 , ∞ ) and drops smo othly to zero near q = 4 . Similarly , the explicit expressions in [ Dum+22 ] rev eal that the free energy of the mo del is smo oth in q and analytic ev erywhere except at q = 4 . These b ehaviours are reminiscent of the BKT transition, but w e cannot at the moment answ er these questions for the XY mo del with similar metho ds b ecause exact integrabilit y (leading to explicit expressions for the free energy) app ears not to b e a v ailable. 1.4. Analyticit y aw ay from criticalit y. In the absence of integrabilit y , this article aims to progress on the probabilistic analysis of the XY mo del in the sub critic al regime (in the op en in terv al [0 , β c ) ). It derives analyticit y of the free energy in this in terv al. As a b y-pro duct, we also obtain analyticity of the free energy of the closely related 2D lattice Coulom b gas on the same sub critical in terv al, and we establish that the interactions b etw een FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 3 the Coulomb particles deca y exp onen tially fast in the distance in the same temp erature regime (Deby e screening). While the question of analyticity throughout the en tire sub critical regime may app ear simple at ﬁrst sight, it is not uncommon that subtle complications arise. In the tradition of deriving analyticity , let us ﬁrst men tion the remark able and celebrated work of Lee and Y ang [ L Y52a ; L Y52b ] regarding the analyticity of the free energy as a function of the magnetic ﬁeld, which is the ﬁrst successful non-p erturbative approac h of this type of question. Georgakopoulos and Panagiotis prov ed in [ GP23 ] that the density of the inﬁnite cluster in Bernoulli p ercolation is analytic in the whole sup ercritical regime, providing a ﬁrst example of a non-p erturbative pro of of analyticity in a p ercolation mo del using some form of renormalisation or coarse-graining argumen t com bined with in v olv ed com binatorial constructions. W e also men tion the work [ PS22 ], that establishes a similar result in the case of a related mo del, the Gaussian fr e e ﬁeld p er c olation . In the Ising mo del, the most complete picture has b een obtained by Ott, who prov ed in [ Ott20b ; Ott20a ] that the free energy of the Ising mo del is analytic in β ev erywhere but at β c . In p erturbative setting, the most complete picture was describ ed in the con text of general Gibbs measures in [ Pra83 ]. Finally , w e men tion that the corresp ondence b et w een Ehrenfest’s notion of phase transi- tion and the probabilistic viewp oin t w as questioned by Griﬃths, who discov ered in [ Gri69 ] the very intriguing phenomenon that now b ears the name of Griﬃths singularities . In the setting of dilute d magnets (Ising mo dels with random Hamiltonians), he sho w ed that the analyticit y of the free energy of the mo del could b e broken away of the critic al p oint , due to the random ﬂuctuations of the Hamiltonian. In the absence of those random defects (on so-called pur e mo dels), this phenomenon should ho w ever not exist, and Ehrenfest’s corresp ondance is exp ected to b e exact. 1.5. Background on the XY mo del and the discrete 2D Coulomb gas. The critical b eha viour of the XY mo del dep ends starkly on the dimension of the underlying lattice. In dimension tw o, it has b een known since the 1960s that the XY mo del do es not undergo a magnetisation transition (like the Ising mo del) due to the celebrated w ork of Mermin and W agner [ MW66 ]. At the same time, n umerical simulations op ened up the idea that the mo del might still undergo a phase transition of a diﬀeren t t yp e [ SK66 ]. As mentioned b efore, such a phase transition was theoretically motiv ated by Berezinskii [ Ber71 ; Ber72 ] and Kosterlitz and Thouless [ KT73 ]. A t and b elow the critical temp erature, it is physically exp ected that the model has a conformally in v ariant scaling limit [ Gin88 ]. While a clear picture of the near-critical b eha viour of the 2D XY mo del remains elusive to mathematicians, the mo del has been the sub ject of in tense research since the 1980s. In their landmark work, F röhlic h and Sp encer [ FS81 ] (cf. [ KP17 ]) prov ed that the lo w-temp erature XY mo del exhibits p olynomial decay of correlations, thus rigorously proving that a phase transition o ccurs (exp onential deca y at high temp erature is classical). The authors obtain this result by viewing the dual of the XY mo del (an inte ger-value d height function ) as a p erturbation around the discr ete Gaussian fr e e ﬁeld (a r e al-value d height function ), and b y using a renormalisation group analysis to control the eﬀect of the p erturbation. The argumen t is robust; for example, it was sho wn that the argument is stable under certain p erturbations [ GS23b ], and that it can b e extended to a larger range of p otentials, ev en at a slop e [ OS25 ]. On the heigh t function side, it was recently pro ved (for a slightly diﬀeren t p oten tial) that the heigh t function conv erges to the Gaussian free ﬁeld in the scaling limit at high temp erature [ BPR24a ; BPR24b ]. W e ﬁnally refer to [ GS23a ] for an analysis of the relations b etw een the Villain (mo diﬁed XY) mo del, the Coulomb gas, and the Gaussian free ﬁeld, and the wa y that correlations in the diﬀerent mo del relate to one another. Later, a second argumen t for a phase transition app eared, based on the non-co existence theorem in percolation (see [ She05 , Theorem 9.3.1] and [ DR T19 , Theorem 1.5]). In [ Lam22 ] these ideas were used to sho w that 2D integer-v alued heigh t functions delo calise at high 4 LUCAS D’ALIMONTE AND PIET LAMMERS temp erature, and [ EL23 ; Aiz+22 ] deriv es from this the p olynomial deca y of correlations in the low-temperature 2D XY mo del. F urther results w ere obtained in [ Lam23a ; Lam23b ], where it w as prov ed that the BKT transition in the XY mo del transition is equiv alent to the delo calisation transition in the dual heigh t function. The dualit y transform was further explored in [ EL25 ] to obtain GFF-t yp e estimates and monotonicit y in parameters in large generalit y . The Z 2 Coulom b gas alluded to ab ov e has a con tin uum counterpart, in whic h the Coulomb particles live in R 2 rather than Z 2 . This complicates certain asp ects of the mo del (for example, it is problematic when particles of opp osing charge come v ery close to each other). Boursier and Serfaty [ BS24 ; BS25 ] recen tly obtained impro v ed electrostatic estimates for the contin uum Coulom b gas (cf. [ FS81 ]), whic h allows for the deriv ation of a sequence of m ultip ole transitions at temp eratures accum ulating ab ov e the BKT temperature. These m ultip oles are connected to transitions in the co eﬃcients of a May er series arising in the free energy expansion. In dimension d ≥ 3 , the XY mo del is kno wn to exhibit sp ontaneous magnetisation (or con tin uous symmetry breaking) at lo w temp erature [ FSS76 ] (cf. [ GS22 ] for a new pro of based on [ Abb+18 ]). Moreov er, it is kno wn that in dimension d ≥ 5 , the mo del exhibits trivial (Gaussian) behaviour at the critical p oint [ F rö82 ]. F or the Ising mo del, such metho ds w ere recen tly extended to co v er the dimension d = 4 [ AD21 ] (cf. [ Aiz82 ]). Our mathematical kno wledge on the higher dimensional XY mo del remains limited, with k ey problems (suc h as contin uit y and sharpness of the phase transition) remaining op en in an y dimension d ≥ 3 . 1.6. Bac kground on the square well mo del. Finally , we apply our techniques to the square well mo del. This mo del has real-v alued spins and a quadratic in teraction term, but the v alues are conditioned to lie in the interv al [ − 1 , 1] . It is known that this conditioning destro ys long-range in teractions; the tw o-p oin t function exhibits exp onential decay of correlations at all temp eratures thanks to the w ork of McBry an and Sp encer [ MS77 ]. F urthermore, the study of this mo del w as pursued in [ BEF86 ] in which precise asymptotics of the mass of the mo del are derived when the cutoﬀ [ ± 1] is replaced by [ ± L ] and L tends to inﬁnity . The mo del is included in the analysis as it show cases the nov el asp ects of our argumen t in a slightly simpler setting than the one of the XY mo del. W e also include a new pro of of the McBryan–Spencer result of exponential decay of this mo del, relying on the F ortuin–Kasteleyn–Ginibre inequality [ FKG71 ] and “absolute-v alue-FKG inequalit y” [ LO24 ]. 2. Main resul ts F or an y d ∈ Z ≥ 1 , let Z d denote the d -dimensional square lattice graph endow ed with nearest-neigh b our connectivit y (denoted ∼ ). A domain is a ﬁnite subset of Z d . Let Λ n := ( − n, n ) d ∩ Z d . 2.1. F ree energy of the XY mo del. Let S 1 ⊂ C denote the unit circle in the complex plane. Deﬁnition 2.1 (XY mo del) . Fix d ∈ Z ≥ 1 and β ∈ R ≥ 0 . F or an y domain Λ and ζ : Z d → S 1 , the XY mo del on Λ with b oundary condition ζ at inv erse temp erature β is the probability measure on σ ∈ ( S 1 ) Λ giv en by d µ ζ XY , Λ ,β ( σ ) = 1 Z ζ XY , Λ ,β e − β H ζ XY , Λ ( σ ) d σ, where Z ζ XY , Λ ,β is the normalising constant, d σ the Haar measure, and H ζ XY , Λ ( σ ) := X { x,y }⊂ Λ x ∼ y | σ x − σ y | 2 / 2 + X x ∈ Λ , y ∈ Λ c x ∼ y | σ x − ζ y | 2 / 2 . FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 5 The critic al inverse temp er atur e is deﬁned as the largest β c = β c ( d ) ∈ (0 , ∞ ) suc h that µ +1 XY , Λ n ,β [ σ 0 ] decays exponentially fast in n for any ﬁxed β ∈ [0 , β c ) . Classical argumen ts based on correlations inequalities imply the existence of an inﬁnite-v olume Gibbs measure with +1 b oundary conditions at in v erse temp erature β ≥ 0 , that we refer to as µ +1 XY ,β . The lab el XY is omitted from notations when it is clear from the context. Theorem 2.2 (Analyticit y of the free energy of the XY mo del) . Fix d ∈ Z ≥ 1 . Then f XY : R ≥ 0 → R , β 7→ lim n →∞ 1 | Λ n | log Z +1 XY , Λ n ,β , the fr e e ener gy of the mo del, is analytic on [0 , β c ( d )) . 2.2. F ree energy of the discrete t w o-dimensional Coulom b gas. Now consider d = 2 . Recall that Λ n = ( − n, n ) 2 ∩ Z 2 is the base graph for the XY mo del. W e also use it as the base graph for the Villain mo del, whic h is a sligh t mo diﬁcation of the XY mo del (describ ed formally in Section 14 ). The Coulomb gas is deﬁned on the faces F (Λ n ) of Λ n , that is, the unit squares with vertices in [ − n, n ] 2 ∩ Z 2 . W e think of F (Λ n ) as a graph; tw o faces are adjacen t if they share an edge (faces in the bulk hav e four neigh b ors; edges on the boundary t w o or three). W e let ∆ denote the Laplacian on this graph of faces, and ∆ − 1 its inv erse. Deﬁnition 2.3 (T wo-dimensional Coulom b gas) . Fix d = 2 and β ∈ R ≥ 0 . The two- dimensional lattic e Coulomb gas on Λ n with free b oundary conditions at inv erse temp erature β is the probability measure on q ∈ Z F (Λ n ) giv en by µ free Coulomb , Λ n ,β [ { q = Q } ] = 1 Z free Coulomb , Λ n ,β 1 h X f ∈ F (Λ n ) Q ( f ) = 0 i · e − β (2 π ) 2 2 ⟨ Q, − ∆ − 1 Q ⟩ . Notice that, con trary to the XY mo del, the Hamiltonian of the Coulomb gas is non-lo cal due to the presence of the inv erse Laplacian. Let us now brieﬂy describ e the relation of this mo del with the Villain mo del. The Villain mo del is an XY mo del on Λ n with a modiﬁed Hamiltonian. It is closely related to the Coulom b gas, as it ma y b e view ed as the pro duct measure of the Coulomb gas and a Gaussian free ﬁeld (GFF) on Λ n . This relation was describ ed in [ GS23a ]. Direct consequences are: • F or an y ﬁxed β , we ha v e Z +1 Villain , Λ n ,β = Z free Coulomb , Λ n ,β · Z 0 GFF , Λ n ,β , (1) see [ GS23a , P age 672, Remark 10], • F or any ﬁxed β , one can tak e a sample from µ free Coulomb , Λ n ,β via the following procedure (detailed in [ GS23a , Section 4]): (1) Sample σ ∼ µ +1 Villain , Λ n ,β , (2) Sample additional i.i.d. random v ariables ζ on the edges of Λ n , (3) The charge q f at any face f ∈ F (Λ n ) is now a function of the spins σ u on the four corners u of f and ζ e on the four edges e of f . In particular, the Coulom b gas can b e sampled lo cally in terms of the Villain model. W e stress that the relation b etw een the Coulom b gas and the Villain mo del is not a F ourier t yp e transform: there is no temp erature inv ersion. Rather, the Coulomb gas is deﬁned on the dual graph simply b ecause it enco des top ological features along the faces of the primal graph on whic h the Villain mo del is deﬁned. Lik e the XY mo del, the Villain mo del in 2D has a sp ecial inv erse temp erature β BKT suc h that correlations deca y exponentially fast for β < β BKT ( Villain ) and p olynomially fast for β ≥ β BKT ( Villain ) . As discussed in [ Ber72 ; FS81 ; GS23a ], this phase transition is driven b y qualitative c hanges in the Coulomb gas measures µ free Coulomb , Λ n ,β . This must necessarily b e the case as the Gaussian free ﬁeld do es not undergo an y phase transition; changing β 6 LUCAS D’ALIMONTE AND PIET LAMMERS simply scales the ﬁeld by a constan t factor. As Theorem 2.2 extends directly to the Villain mo del, we obtain the following results for the Coulomb gas. Theorem 2.4. Fix d = 2 . Then the fr e e ener gy of the Coulomb gas f Coulomb : R ≥ 0 → R , β 7→ lim n →∞ 1 | Λ n | log Z free Coulomb , Λ n ,β , is analytic on [0 , β BKT (Villain)) . The same holds true if we r eplac e fr e e b oundary c onditions by Dirichlet b oundary c ondi- tions, se e [ GS23a ] for details. This theorem follows directly from Equation ( 1 ) , analyticity of the free energy of the Villain mo del on [0 , β BKT ( Villain )) , and analyticity of the free energy of the Gaussian free ﬁeld on [0 , ∞ ) . Changing the b oundary conditions (Dirichlet vs. free) mo diﬁes the partition function of log Z Coulomb , Λ n ,β b y O ( | ∂ Λ n | ) = o ( | Λ n | ) , as can b e observ ed from the relation ( 1 ), and therefore the theorem extends to Dirichlet b oundary conditions. 2.3. F ree energy of the square w ell mo del (SWM). Deﬁnition 2.5 (Square well mo del (SWM)) . Fix d ∈ Z ≥ 1 and β ∈ R ≥ 0 . F or any domain Λ and ζ : Z d → [ − 1 , 1] , the squar e wel l mo del (SWM) on Λ with b oundary condition ζ at in v erse temperature β is the probability measure on α ∈ [ − 1 , 1] Λ giv en by d µ ζ SWM , Λ ,β ( α ) = 1 Z ζ SWM , Λ ,β e − β H ζ SWM , Λ ( α ) d α, where Z ζ Λ ,β is the normalising constant, d α the Leb esgue measure, and H ζ SWM , Λ ( α ) := X { x,y }⊂ Λ x ∼ y ( α x − α y ) 2 + X x ∈ Λ , y ∈ Λ c x ∼ y ( α x − ζ y ) 2 . The lab el SWM is omitted when no confusion is likel y to arise. McBry an and Sp encer prov ed in [ MS77 ] that this mo del exhibits exp onen tial decay of correlations for an y ﬁxed d and β . Consequen tly there exists a unique inﬁnite-volume Gibbs measure, referred to as µ SWM ,β W e deriv e here the following result. Theorem 2.6 (Analyticit y of the free energy of the SWM) . Fix d ∈ Z ≥ 1 . Then f SWM : R ≥ 0 → R , β 7→ lim n →∞ 1 | Λ n | log Z 0 SWM , Λ n ,β , the fr e e ener gy of the mo del, is analytic. 2.4. The three mo dels are exp onential factors of i.i.d. (EFI IDs). The deﬁnition of an exp onential factor of i.i.d. is already present in the literature in spirit, but not y et deﬁned as a distinct mathematical ob ject. Belo w, w e giv e a precise deﬁnition. W e start with a standard deﬁnition of (classical) factors of i.i.d. Deﬁnition 2.7 (F actor of i.i.d.) . Consider a lo cally ﬁnite transitiv e graph G = ( V , E ) (with resp ect to some group Θ ⊂ Aut ( G ) ). Let S denote a measurable space, and let µ denote the distribution of a Θ -stationary random ﬁeld on S V . Let ( T , λ ) denote a probabilit y space, and consider P := λ V . A factor of i.i.d. (FI ID) is a map φ : T V → S V with the follo wing prop erties: • φ is measurable with resp ect to the pro duct σ -algebras on T V and S V , • φ is equiv ariant with resp ect to Θ , • If X ∼ P , then φ ( X ) ∼ µ . FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 7 FI IDs are useful for enco ding correlations in random ﬁelds. F or example, if G = Z d and if φ ( X ) u is measurable with resp ect to X | u +Λ n , then µ is 2 n -dep enden t (w.r.t. ℓ ∞ ). Our mo dels do not fall into this category , as they exhibit long-range correlations (alb eit v ery w eak ones). T o enco de those correlations, we ﬁrst in tro duce the notion of lo c al sets . Deﬁnition 2.8 (Local set) . Let V denote a v ertex set and T a measurable space. Consider the measurable space T V endo w ed with the pro duct σ -algebra. A t ypical elemen t is denoted X ∈ T V . A lo c al set is a random subset L = L ( X ) ⊂ V suc h that for an y L ⊂ V , the even t {L = L } is measurable with resp ect to X | L . Deﬁnition 2.9 (Exp onen tial factor of i.i.d. (EFI ID)) . Consider the setting of Deﬁnition 2.7 . A FI ID is called an exp onential factor of i.i.d. (EFI ID) with tails ( c, C ) for constants c, C ∈ (0 , ∞ ) if for any u ∈ V , there exists a connected local set L u = L u ( X ) ∋ u suc h that: • The v alue of φ ( X ) u is measurable with resp ect to X | L u , • W e ha v e P [ {|L u | ≥ n } ] ≤ C e − cn for any n ∈ Z ≥ 0 . Most work in this article is devoted to proving the follo wing result. Theorem 2.10 (The mo dels are EFI IDs) . Al l thr e e mo dels al lude d to ab ove ar e EFIIDs. Mor e pr e cisely, we obtain the fol lowing r esults. XY Consider the XY mo del in dimension d ∈ Z ≥ 2 at inverse temp er atur e β ∈ [0 , β c ( d )) . Then the inﬁnite-volume Gibbs me asur e µ +1 XY ,β is an EFIID. SWM Consider the SWM in dimension d ∈ Z ≥ 2 at inverse temp er atur e β ∈ R ≥ 0 . Then the inﬁnite-volume Gibbs me asur e µ 0 SWM ,β is an EFIID. Coulom b Consider the two-dimensional Coulomb gas at β ∈ [0 , β BKT ( Villain )) . Then the se- quenc e ( µ free Coulomb , Λ n ,β ) n tends to some inﬁnite-volume me asur e µ free Coulomb ,β on Z F ( Z 2 ) as n → ∞ in the lo c al c onver genc e top olo gy, and this limit is an EFIID. In the c ase of the Coulomb gas, we ar e c ar eful not to c al l this me asur e a Gibbs me asur e, as the Hamiltonian of the mo del is non-lo c al and ther efor e it is not so cle ar what it me ans exactly to b e a Gibbs me asur e. In p articular, these r esults imply that for any of the thr e e mo dels, c orr elations b etwe en b ounde d observables de c ay exp onential ly fast in the distanc e at which they ar e me asur e d. Remark 2.11 (Deb y e screening in the Coulom b gas) . The Hamiltonian in the deﬁnition of the Coulom b gas is long-range. Ho wev er, it w as predicted in the ph ysics literature that c harges of opp osing signs should gather in suc h a w a y that these long-range eﬀects cancel out o v er long distances, for suﬃciently small β . This eﬀect is called Debye scr e ening . Deb ye screening at p erturbativ ely small β w ere previously obtained in [ Bry78 ; Y an87 ], and [ Sep22 ] describ es ho w exp onen tial decay in the dual height function mo del implies Deb y e screening in the Coulom b gas. The current work, in com bination with [ Lam23a ; Lam23b ] sharp ens the picture: the Deb y e phase in the Coulomb gas coincides exactly with the exponential deca y phase in the Villain mo del and the lo calised (exp onential deca y) phase in the dual height function mo del. Moreov er, the EFIID structure implies screening in a strong sense, namely that any ﬁnite n umber of lo cal observ ables of any t yp e, mix exp onentially fast in the distance at whic h they are measured. EFI IDs are useful for studying analyticit y [ Ott20b ; Ott20a ]. In fact, Theorem 2.10 essen tially implies Theorems 2.2 , 2.4 , and 2.6 (see the next section for details; the situation is slightly more complicated in the case of Theorem 2.4 ). The main contribution of this article is therefore to show that the three mo dels are indeed EFI IDs. 8 LUCAS D’ALIMONTE AND PIET LAMMERS 3. Pr oof o ver view and ar ticle or ganisa tion Subsection 3.1 describ es how the EFI IDs (Theorem 2.10 ) are constructed in the remaining parts of this article. The existence of suc h EFI IDs forms the core result. Subsection 3.2 describ es ho w this result implies analyticity . This implication w as already in [ Ott20b ; Ott20a ], and w e merely include a pro of sk etch. 3.1. Organisation of the pro of of Theorem 2.10 . W e execute our construction sep- arately for the SWM and for the XY mo del. The former case is simpler, but already illustrates some of the k ey ideas. It is contained in Part I . The latter case is a bit more delicate (additional complications arise), and the construction is contained in Part I I . Both parts are organised in tandem. • A new Glauber dynamic (Sections 5 and 10 ). At the ﬁrst step, we observ e that the mo dels enjoy certain monotonicit y prop erties, and w e construct a Glaub er dynamic that is compatible with these monotonicity prop erties. The ultimate goal is to p erform a coupling-from-the-past pro cedure. This is more complicated for our mo dels than for models with discrete spins (suc h as the Ising mo del), since the natural monotone Glauber dynamic do es not couple in ﬁnite time. W e introduce an alternative Glaub er dynamic that do es couple in ﬁnite time, and has the correct monotonicit y prop erty . The construction of this alternative Glaub er dynamic is one of the k ey ideas that mak es the construction work. • Spatial mixing (Sections 8 and 13 ). Then, we prov e for each mo del that the ﬁnite-volume Gibbs measures mix exponentially fast. The proofs are model- dep enden t and do not inv olv e the Glaub er dynamic. • Space-time mixing (Sections 6 and 11 ). Then, we pro v e space-time mixing for this Glaub er dynamic. This relies on the ideas in the w ork of Harel and Spink a [ HS22 ], but the pro of is more complicated b ecause the spin space ab ov e each v ertex is contin uous rather than ﬁnite. • Coarse-graining to w ards a sub critical p ercolation (Sections 7 and 12 ). The space-time mixing prop erty indicates that for ﬁxed β , we ma y ﬁx, once and for all, a space-time scale on whic h the Glaub er dynamic couples extremely fast. More precisely , the coarse-grained dep endency clusters in space-time are dominated b y a sub critical Bernoulli p ercolation. In particular, those clusters exhibit exp onential deca y in their volume. This prop ert y remains true under pro jection (of the space- time clusters onto the spatial dimension), which yields the desired EFI ID structure. This prov es Theorem 2.10 for the XY mo del and the SWM, but not for the Coulomb gas. That requires t wo more steps. (1) The Villain mo del has an EFI ID exactly like the XY mo del (Theorem 14.2 ). (2) The Coulom b gas can be lo cally sampled in terms of the Villain model; see our discussion following Deﬁnition 2.3 . This implies immediately that the EFI ID of the Villain mo del also yields an EFI ID for the Coulomb gas. 3.2. Pro of of Theorems 2.2 , 2.4 , and 2.6 conditional on Theorem 2.10 . W e no w explain ho w the existence of an EFI ID (Theorem 2.10 ) implies analyticity of the free energy for the XY mo del and the SWM (Theorems 2.2 and 2.6 ). Similarly , the EFIID for the Villain mo del (Theorem 14.2 ) leads to analyticity of the free energy of the Villain model (Theorem 14.3 ), and th us to analyticit y of the free energy of the Coulomb gas (Theorem 2.4 ). W e record the follo wing general result. Theorem 3.1 (EFI ID implies analyticit y [ Ott20b ; Ott20a ]) . L et µ b e a stationary r andom ﬁeld on S Z d . W rite σ ∼ µ for the r andom element. Consider a ﬁnite set A ⊂ Z d and a b ounde d observable F A that is σ | A -me asur able. If µ is an EFIID, then ther e exists a c onstant FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 9 δ = δ ( µ, F A ) > 0 such that the function ˜ F A : ( − δ, δ ) → R , ε 7→ lim n →∞ 1 | Λ n | log µ h exp  ε X x ∈ Λ n F A ( τ x σ ) i is analytic. Her e τ x ( σ ) me ans that we tr anslate the c onﬁgur ation σ by x . The pro of is en tirely contained in [ Ott20b ; Ott20a ], although the theorem is stated in a less general form. W e now describ e how Theorem 3.1 implies Theorems 2.2 , 2.4 , and 2.6 . A pro of sketc h of Theorem 3.1 is given at the end of this section. Pr o of of analyticity of the fr e e ener gy. W e sho w that the free energy may b e seen as the analytic function app earing in Theorem 3.1 . W e do this for the SWM model; the other cases are similar. Fix d and β . Let A := { 0 , e 1 , . . . , e d } , and F A ( α ) := d X i =1 ( α 0 − α e i ) 2 . Then asymptotically in n → ∞ , w e get sup α    H 0 SWM , Λ n ,β ( α ) − X x ∈ Λ n F A ( τ x α )    = o ( | Λ n | ) , and so, for | ε | < δ , ˜ F A ( ε ) = lim n →∞ 1 | Λ n | log µ 0 SWM ,β [ e − εH 0 SWM , Λ n ] = f SWM ( β + ε ) − f SWM ( β ) . The equalit y on the righ t is standard, for mo dels where the p otential on eac h edges remains uniformly b ounded (which is indeed the case for all mo dels considered in this article). This pro v es analyticit y of f SWM on ( β − δ, β + δ ) . The same pro of holds for the XY mo del, using the measure µ +1 XY ,β when β < β c ( d ) . □ Finally , w e brieﬂy discuss the pro of of Theorem 3.1 . Sketch of pr o of of The or em 3.1 . The existence of an EFI ID for the measure, together with the coarse-graining pro cedure in [ Ott20a ], imply the existence of a dep endency enc o ding me asur e in the sense of [ Ott20a , Theorem 4.1]. W e point out that actually the condition of b eing EFIID is more restrictive than the existence of a dependency enco ding measure à la [ Ott20a ]. The remainder of the pro of is routine, as a con v ergent cluster expansion of the quan tit y G n ( ε ) := µ  exp  ε X x ∈ Λ n F A ( τ x σ )  ma y b e pro duced with the metho d of [ Ott20a ] when ε is suﬃciently small. This uses the fact that F A is b ounded. Once the conv ergent cluster expansion is pro duced, the rest of the pro of is routine — the cluster expansion pro duces an analytic extension of log G n on a small disk which is uniformly b ounded in n , and Vitali’s theorem is used to get the con v ergence and analyticit y of its limit when n tends to inﬁnity . F or using Vitali’s con v ergence theorem, one needs to c hec k the conv ergence of log G n on a subinterv al of the real line con taining 0. Here our setting is slightly more general than [ Ott20a ], but the conv ergence is ensured b y the fact that µ is an EFI ID, and classical sub-aditivity arguments. W e refer to [ Ott20b ; Ott20a ] for details. □ 10 LUCAS D’ALIMONTE AND PIET LAMMERS P art I. The square w ell mo del 4. SWM: Pr oof over view In many diﬀeren t situations, factors of i.i.d. are constructed out of Glaub er dynamics using a coupling-from-the-past (CFTP) pro cedure [ BS99 ; HS00 ; LS16 ; Spi20 ]. Monotonicity of the Glaub er dynamic is often a crucial ingredient to mak e the pro cedure work. The lo cal set L u in the EFI ID can b e constructed explicitly , b y analysing how far back in the past w e must lo ok in space-time to determine the v alue of φ ( X ) u . In our case, we run into a new issue: since the SWM tak es real v alues, the natural candidate for the Glaub er dynamic (using the canonical grand coupling of all probability distributions on R ) will never exactly couple in ﬁnite time. Thus, one m ust lo ok bac k inﬁnitely far in space-time, and the lo cal sets are therefore to o big to yield an EFI ID. W e circum v ent this issue in three steps (explained later in further detail). • First, we prop ose a new Glaub er dynamic for the SWM, which satisﬁes the follo wing prop erties: (1) The monotonicity prop ert y mentioned ab ov e, (2) When the vert ex u is resampled, with very large probability , the digits in the decimal expansion of α u after the ﬁrst k digits, are sampled uniformly at random indep endently of ev erything else, (3) The Marko v prop ert y . F ormal deﬁnitions may b e found in Subsection 5.1 . Prop ert y 2 is the key mo diﬁcation, as it allo ws us to couple the dynamics up to the ﬁrst k digits in ﬁnite time, and then use the randomness in the remaining digits to ac hiev e an exact coupling. • Second, we show that the original Gibbs measure satisﬁes some v ery strong spatial mixing prop erties. • Third, w e prov e that the mixing together with Prop erties 1 and 2 of the new dynamic, imply that the Glaub er dynamic has go o d space-time mixing. This pro of closely follo ws the w ork of Harel and Spink a [ HS22 ] but we need to mak e some sligh t mo diﬁcations to deal with the real-v alued nature of the mo del. • Finally , w e com bine space-time mixing of the Glaub er dynamic with Prop ert y 3 to construct an EFI ID represen tation of the inﬁnite-volume SWM. The remaining sections are organised as follows. Spatial mixing of the Gibbs measure is stated here, but prov ed at the end, in Section 8 . In Section 5 , we introduce the new Glaub er dynamic, and recall some general statement concerning coupling from the past. Section 6 adapts the Harel–Spink a argumen t to our setup. Finally , in Section 7 , we use the space-time mixing of the Glaub er dynamic and the Mark o v prop ert y to construct the EFI ID representation of the inﬁnite-v olume SWM, and conclude the pro of of Theorem 2.10 for the square-w ell mo del. The main ingredient for proving the ab ov e-men tioned fast space-time mixing of the Glaub er dynamics is the follo wing mixing result, whose pro of is postp oned to Section 8 . In what follows, when µ and ν are tw o probabilit y measures on a measurable space (Ω , F ) , we use the notation ∥ ν − µ ∥ TV to denote the usual total variation distanc e b etw een µ and ν . Lemma 4.1 (Spatial mixing of the Gibbs measure of the SWM) . Fix the dimension d ∈ Z ≥ 1 and the inverse temp er atur e β ∈ R ≥ 0 . Then ther e exist c onstants C < ∞ and c > 0 such that    µ +1 Λ 2 n ,β | Λ n − µ − 1 Λ 2 n ,β | Λ n    TV ≤ C e − cn for al l n ∈ Z ≥ 1 . FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 11 5. SWM: Gla uber dynamic This section is organised as follo ws: • In Subsection 5.1 we deﬁne Glaub er dynamics and their abstract prop erties, • In Subsection 5.2 we construct such a Glaub er dynamic for the SWM, • In Subsection 5.3 we recall some relev ant information on c oupling fr om the p ast . W e shall alw a ys w ork in the following setting: G = ( V , E ) is a transitive lo cally ﬁnite graph, and S a measurable space. In the case of the SWM, G will b e Z d and S will b e the interv al [ − 1 , 1] . 5.1. Abstract description of the Glaub er dynamic. The idea of the Glaub er dynamic is to construct a Marko v chain con verging to the Gibbs measure, b y p erforming lo cal up dates at each v ertex on the conﬁguration. The lo cal updates are performed using an indep enden t source of randomness. In the abstract setting, this sample space for this indep enden t source of randomness is denoted b y Σ , a generic element is denoted ι ∈ Σ , and the probability measure is denoted d ι . A technical requiremen t for the measures µ to b e sampled via Glaub er dynamics is that µ should hav e the ﬁnite ener gy pr op erty , that means that for any v ∈ V , σ ∈ S V , the conditional densit y of µ ( σ v = · | σ u  = v ) is strictly positive almost everywhere on S . This is not a problem for us, as this prop erty is clearly true in all the mo dels considered. Deﬁnition 5.1 (Glaub er dynamic) . Let µ denote a stationary measure on S V with ﬁnite energy . A Glaub er dynamic for µ is a function R : S V × V × Σ → S V with the follo wing prop erties. • Lo cal up date. F or any α ∈ S V , u ∈ V and ι ∈ Σ , we hav e R ( α, u, ι ) v = α v for an y v  = u . • Consistency . F or any α ∈ S V and u ∈ V , the distribution of R ( α, u, ι ) u in d ι is µ [ · | ( α v ) v  = u ] . • Equiv ariance. F or an y α ∈ S V , u ∈ V , ι ∈ Σ and θ ∈ Θ , we hav e R ( α, u, ι ) ◦ θ = R ( α ◦ θ , u ◦ θ , ι ) . W e shall also write R u,ι : S V → S V , α 7→ R ( α , u, ι ) for any u ∈ V and ι ∈ Σ . In addition to the ab ov e prop erties, w e shall quic kly list some other prop erties that may or ma y not hold true for a given Glaub er dynamic. Of course, all these prop erties shall ev en tually apply to the Glaub er dynamic that we construct in the next subsection. Deﬁnition 5.2 (Prop erty 1 : Monotonicity) . Supp ose that S is a partially ordered set. Then w e call a Glaub er dynamic R monotone if for an y u ∈ V and ι ∈ Σ , the map R u,ι : S V → S V preserv es the partial order at the spin u . This means that R u,ι ( α ) u ⪯ R u,ι ( α ′ ) u for any α, α ′ ∈ S V suc h that α v ⪯ α ′ v for all v ∈ V . F or an y x ∈ R , write ⌊ x ⌋ k := 10 − k ⌊ 10 k x ⌋ and { x } k := x − ⌊ x ⌋ k . Deﬁnition 5.3 (Prop ert y 2 : Matc hing digits) . Suppose that S ⊂ R . Consider ﬁxed constan ts ε ∈ [0 , 1] and k ∈ Z ≥ 0 . W e sa y that a Glaub er dynamic R matches digits up to ( ε, k ) if we ma y ﬁnd an ev en t M ⊂ Σ of d ι -probability at least 1 − ε suc h that: • F or an y u ∈ V and ι ∈ M , w e hav e  { R u,ι ( α ) u } k = { R u,ι ( α ′ ) u } k for all α, α ′ ∈ S V  • F or an y u ∈ V and α ∈ S V , the even t M and the random v ariable ⌊ R u,ι ( α ) u ⌋ k are d ι -indep enden t. The even t M is called the matching event . 12 LUCAS D’ALIMONTE AND PIET LAMMERS Deﬁnition 5.4 (Prop ert y 3 : Marko v prop ert y) . W e say that a Glaub er dynamic R has the Markov pr op erty if for any α ∈ S V , u ∈ V and ι ∈ Σ , the v alue of R ( α, u, ι ) u is measurable with resp ect to α | N u and ι , where N u denotes the set of neighbours of u in G . 5.2. Construction of the Glaub er dynamic. Lemma 5.5 (Glaub er dynamic for the SWM with matching digits) . Fix any lo c al ly ﬁnite tr ansitive gr aph G = ( V , E ) and any β ∈ R ≥ 0 . Then for any ε > 0 , ther e exists a c onstant k ∈ Z ≥ 0 and a Glaub er dynamic R for the SWM on G at inverse temp er atur e β , which has Pr op erties 1 – 3 , matching digits up to ( ε, k ) . The pro of of the lemma requires t w o simple intermediate results. T o state them, we ﬁrst need a notion of sto chastic domination . Deﬁnition 5.6. Let µ and ν denote tw o probabilit y measures on R . Then we sa y that µ is sto chastic al ly dominate d by ν if their cum ulativ e distribution functions F µ and F ν satisfy F µ ≥ F ν . Lemma 5.7 (Grand coupling) . Consider the set M of pr ob ability me asur es on R . Then we may c onstruct a c oupling of al l distributions in M such that if µ is sto chastic al ly dominate d by ν , then the µ -distribute d r andom variable is almost sur ely less than or e qual to the ν -distribute d r andom variable. Mor e pr e cisely, we r e quir e that we have a family of r andom variables ( X µ ) µ ∈M such that X µ ∼ µ for e ach µ ∈ M and such that X µ ≤ X ν whenever µ is sto chastic al ly dominate d by ν . Pr o of. Sample a uniform random v ariable U on [0 , 1] , and set X µ := F − 1 µ ( U ) for each µ ∈ M . If F − 1 µ ( U ) is m ulti-v alued then we break ties b y taking the inﬁm um of this set. □ Lemma 5.8 (Grand coupling for almost uniform measures) . L et ε > 0 and c onsider the set M of pr ob ability me asur es µ on [0 , 1] such that µ ≥ (1 − ε ) · d x (the L eb esgue me asur e on [0 , 1] ) as me asur es. Then al l distributions in M c an b e c ouple d to gether in such a way that: • If µ is sto chastic al ly dominate d by ν , then X µ ≤ X ν , • With pr ob ability 1 − ε , al l the r andom variables ( X µ ) µ ∈M ar e e qual. Pr o of. Consider the cum ulativ e distribution functions F µ : [0 , 1] → [0 , 1] of the measures µ ∈ M . By assumption, the functions ˜ F µ : [0 , 1] → [0 , 1] , x 7→ F µ ( x ) − (1 − ε ) x ε are also cumulativ e distribution functions. Moreov er, the op eration F µ 7→ ˜ F µ preserv es the partial order of sto chastic domination (this is easy to see from the explicit expression). The desired coupling is now constructed as follows. First, sample a uniform random v ariable U on [0 , 1] and another Bernoulli random v ariable B with parameter ε , indep enden tly of each other. • If B = 0 , then w e set X µ := U for each µ ∈ M . • If B = 1 , then w e set X µ := inf ˜ F − 1 µ ( U ) for each µ ∈ M . This coupling satisﬁes the desired properties. □ Pr o of of L emma 5.5 . Let D denote the degree of G . Fix u ∈ V . F or an y conﬁguration α ∈ [ − 1 , 1] V \{ u } , let ν α denote the la w of α u in µ α { u } ,β . By deﬁnition, it is immediate that ν α is given by the normal distribution N  1 D X v ∼ u α v , 1 2 β D  , (2) conditioned to b e in the in terv al [ − 1 , 1] . The map α 7→ ν α has the follo wing properties: • The la w ν α only dep ends on α | N u , FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 13 • The la w ν α is sto chastically increasing in α . F or the construction of the Glaub er dynamic, it now suﬃces to construct a grand coupling of all measures ( ν α ) α preserving the monotonicit y and matching the digits. W e do so in tw o steps. • First, we consider the grand coupling of ( ν α ) α giv en by Lemma 5.7 : w e ﬁrst sample a uniform random v ariable U ′ on [0 , 1] , and then set X ′ α := inf F − 1 α ( U ′ ) . • Next, w e would like to r esample the digits after the ﬁrst k ones. More precisely , we w an t to couple the distributions ν ′ α := ν α [ · |{⌊ 10 k X α ⌋ = ⌊ 10 k X ′ α ⌋} ] . Again, the map α 7→ ν ′ α is sto c hastically increasing in α . Moreov er, ν ′ α is just given b y the distribution in Equation ( 2 ) conditioned to b e in the interv al h 10 − k ⌊ 10 k X ′ α ⌋ , 10 − k ( ⌊ 10 k X ′ α ⌋ + 1)  . If k is large enough, then each measure ν ′ α is bigger (in the sense of measures) than (1 − ε ) times the uniform distribution on the conditioning interv al. W e ma y then apply Lemma 5.8 to construct a grand coupling of the measures ν ′ α preserving the monotonicit y and matching the digits up to ( ε, k ) . Since this Glaub er dynamic never insp ects the v alues of α outside of N u , it also has the Mark o v property . □ 5.3. Coupling from the past. Ab ov e, we discussed how to up date a single spin α u . In practice, w e w an t to up date all spins man y times in order to mix to the Gibbs measure. W e w an t to enco de the information ab out this contin uous Marko v c hain in terms of a set Π ⊂ V × Σ × R ≤ 0 , where the ﬁrst co ordinate is the vertex to b e up dated, the second co ordinate is the randomness used for the up date, and the third co ordinate is the time of the up date. In this context, w e call V × R ≤ 0 the sp ac e-time . W e sa y that Π has c ol lisions if t w o distinct p oin ts in Π ha v e the same time co ordinate, and we say that Π is lo c al ly ﬁnite if for any u ∈ V and t ∈ R ≤ 0 , only ﬁnitely man y p oin ts of Π hav e the form ( u, ι, s ) with s ∈ [ t, 0] . W rite Ω :=  Π ⊂ V × Σ × R ≤ 0 : Π is lo cally ﬁnite and has no collisions  . In practice, Π is sampled according to a P oisson p oin t pro cess on V × Σ × R ≤ 0 , in whic h case Π almost surely b elongs to Ω . F or an y Π ∈ Ω and any b ounded space-time subset B ⊂ V × R ≤ 0 , we write Π( B ) := { ( u, ι, t ) ∈ Π : ( u, t ) ∈ B } . Moreo v er, w e think of Π( B ) as an ordered set, b y ordering the p oints in Π( B ) according to their time co ordinate, with the lo west (most negativ e) time coming ﬁrst (this is well-deﬁned since Π has no collisions). Finally , we write R Π( B ) := ( R u n ,ι n ◦ · · · ◦ R u 1 ,ι 1 ) : S V → S V , α 7→ R u n ,ι n ◦ · · · ◦ R u 1 ,ι 1 ( α ) , where (( u i , ι i , t i )) i is the ordered enumerate of the triples in Π( B ) . W e now record a num b er of useful prop erties of this construction, that go es back to [ PW96 ]. Theorem 5.9 (Coupling from the past) . Consider a Glaub er dynamic R for a me asur e µ on S V . Supp ose that S is p artial ly or der e d and that R is monotone. Supp ose mor e over S has a lar gest element α + and a smal lest element α − . W e also write α ± for the c orr esp onding elements in S V . Then for any ﬁxe d Π ∈ Ω : • F or any b ounde d B ⊂ V × R ≤ 0 , the map R Π( B ) : S V → S V is monotone, • The element R Π(Λ × [ − t, 0]) ( α + ) is de cr e asing in Λ and t , • The element R Π(Λ × [ − t, 0]) ( α − ) is incr e asing in Λ and t . 14 LUCAS D’ALIMONTE AND PIET LAMMERS In p articular, if R Π(Λ × [ − t, 0]) ( α + ) u = R Π(Λ × [ − t, 0]) ( α − ) u then this value wil l not change if we r eplac e α + by any other α ∈ S V , Λ by any lar ger set, and t by any lar ger time. If Π is a random set, then w e shall also write: • R Λ , − t := R Π(Λ × [ − t, 0]) , • R Λ , − t, − s := R Π(Λ × [ − t, − s ]) . In what follows, we shall use this construction (Glaub er dynamics coupled through the coupling-from-the-past). In particular, α + (resp. α − ) will refer to the conﬁguration constan t equal to 1 (resp. − 1 ) on Z d . 6. SWM: Sp a ce-time mixing The goal of this section is to pro ve that the mixing property stated in Lemma 4.1 implies that the Glaub er dynamics deﬁned ab ov e mixes b oth in space and in time. Our main goal is to pro ve the following space-time mixing estimate, stating that with high probability , the maximal and minimal dynamics for α in the coupling previously constructed remain equal for a p ositive prop ortion of the av ailable space-time when the b oundary conditions are suﬃciently far aw a y in space-time. Prop osition 6.1. Ther e exists δ > 0 such that: P [ \ − r ∈ [ − δ n, 0] { R Λ 2 n , − n, − r ( α + ) | Λ n = R Λ 2 n , − n, − r ( α − ) | Λ n } ] − → n →∞ 1 . W e divide the pro of of this statement in tw o subsections. The ﬁrst one is dedicated to reducing the pro of to a k ey statemen t, namely Proposition 6.3 . The second subsection adapts an argument due to Harel and Spink a [ HS22 ], which itself is generalisation of an original argument of [ MO94 ], to pro vide a pro of of this k ey statement. 6.1. Reduction to Prop osition 6.3 . W e start b y deﬁning some “coupling even ts” that w e shall app eal to reap eteadly in what follows. Deﬁnition 6.2. F or an y n ≥ 0 and − t < − s < 0 , deﬁne the following coupling even ts. C ( n, t, s ) := { R Λ n , − t, − s ( α + ) 0 = R Λ n , − t, − s ( α − ) 0 } and C ( n, t, ≥ s ) := \ − r ∈ [ − s, 0] C ( n, t, r ) , and ⌊ C ( n, t, s ) ⌋ k := {⌊ R Λ n ( v ) , − t, − s ( α + ) 0 ⌋ k = ⌊ R Λ n ( v ) , − t, − s ( α − ) 0 ⌋ k } and ⌊ C ( n, t, ≥ s ) ⌋ k := \ − r ∈ [ − s, 0] ⌊ C ( n, t, r ) ⌋ k . Finally , when s = 0 , w e remov e it from the notation e.g. w e write: C ( n, t, 0) := C ( n, t ) . Observ e that those even ts are the ones app earing in Theorem 5.9 , which explains their name of “coupling even ts”. W e will also name their complementary even ts b y replacing C b y NC (“non-coupling”) for the four even ts just deﬁned. W e ﬁrst state the key statement that will allo w us to pro v e Prop osition 6.1 . FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 15 Prop osition 6.3. Ther e exists δ > 0 such that for any n ≥ 0 , and v ∈ Z d and any exp onent β > 0 , it is the c ase that n β P [ NC ( n, n (1 − δ ))] − → n →∞ 0 . The goal of the remainder of this subsection is to derive Prop osition 6.1 from Propo- sition 6.3 , that is to prov e that a small ﬁxed time probability of non-coupling implies a small probability of non-coupling during a time interv al. As Prop osition 6.3 implies a subp olynomial decay of the ﬁxed time probability of non-coupling, w e can b e quite rough and p erform a straigh tforw ard union b ound, as explained in the follo wing lemma. Lemma 6.4. Ther e exist c > 0 and δ > 0 such that for any n ≥ 0 , P [ NC ( n, n, ≥ δ n )] ≤ 100 δ n P [ NC ( n, n (1 − δ ))] + exp( − cn ) . The same r esult holds when r eplacing NC ( n, n, ≥ δ n ) (r esp. P [ NC ( n, n, δ n )] ) by ⌊ NC ( n, n, δ n ) ⌋ k (r esp. ⌊ NC ( n, n, ≥ δ n ) ⌋ k ). Pr o of. In the time interv al [ − δ n, 0] , the spin at 0 is resampled P times, where P is a Poisson v ariable of exp ectation δ n . Eac h time that the spin at 0 is resampled, the probabilit y of the non-coupling ev en t is b ounded by P [ NC ( n, n, δ n )] = P [ NC ( n, (1 − δ ) n )] due to the monotonicit y of the dynamics in time. The result follows b y a union b ound together with a classical Chernoﬀ-type b ound for the tails of Poisson v ariables. The pro of also applies in the case of the coupling of the k -th ﬁrst digits of α . □ W e no w explain ho w the proof of Prop osition 6.1 follows from Prop osition 6.3 . This is nothing but a quite crude union b ound. Pr o of of Pr op osition 6.1 . Let δ be given b y Prop osition 6.3 . W rite P [  \ − r ∈ [ − δ n, 0] { R Λ 2 n , − n, − r ( α + ) | Λ n = R Λ 2 n , − 2 n, − r ( α − ) | Λ n }  c ] ≤ n d sup v ∈ Λ n P [ NC v (2 n, n, ≥ δ n )] ≤ n d P [ NC ( n, n, ≥ δ n )]] ≤ n d (100 δ n P [ NC ( n, n (1 − δ ))] + exp( − cδ n )) . W e used a union b ound for the ﬁrst inequality , monotonicit y in space for the second inequalit y , and Lemma 6.4 at the third line. By Prop osition 6.3 , this quantit y tends to 0, whic h concludes the pro of. □ 6.2. The Harel-Spink a t yp e argumen t. W e turn to the proof of Proposition 6.3 . Similarly to [ HS22 ], our strategy is to pro vide a renormalization inequality for the coupling probabilit y of the maximal and minimal dynamics. How ev er, the task is complicated b y the inﬁniteness of the spin space, and the pro of crucially relies on the “ ( ε, k ) -matc hing of the digits” prop erty satisﬁed b y the dynamic. W e will need three preparatory lemmas. The ﬁrst tw o ones are a standard fact about disagreemen t probabilities of ordered random v ariables, and a basic calculus estimate that w e shall use to establish the sup erp olynomial decay of P [ NC ( n, n (1 − δ ))] . Lemma 6.5 (Lemma 15, [ HS22 ]) . L et X and Y b e two r andom variables taking values in a total ly or der e d and ﬁnite spin sp ac e S , such that X ≤ Y almost sur ely. Then, P [ X  = Y ] ≤ ( | S | − 1) ∥ X − Y ∥ TV . Lemma 6.6. L et (Ψ( n )) n ≥ 0 b e a p ositive se quenc e that we assume monotone de cr e asing to zer o. Assume that ther e exists κ > 0 , C, c > 0 , and a p olynomial P such that for any 1 ≤ m ≤ n , Ψ(2 n ) ≤ P ( m )Ψ((1 − κ ) n ) 2 + C e − cm . (3) 16 LUCAS D’ALIMONTE AND PIET LAMMERS Then, Ψ( n ) tends sup erp olynomial ly fast to 0, that is, for any A > 0 , n A Ψ( n ) − → n →∞ 0 . Pr o of. W e basically rep eat the argumen t of [ HS22 , Lemma 17]. Call α = 2 1 − κ . W e ﬁrst rewrite the h yp othesis as Ψ( αn ) ≤ P ( m )Ψ( n ) 2 + C e − cm , v alid for any (1 − κ ) m ≤ n . By naming a n := − log Ψ( n ) , w e observe that a αn ≥ 2 cm − log C for an y m ≤ n suc h that cm + log P ( m ) ≤ 2 a n . (4) No w observe that either a n ≥ c (1 − κ ) − 1 n/ 2 for inﬁnitely many v alues of n , either an y solution of cm + log P ( m ) satisﬁes (1 − κ ) m ≤ n . Observ e that the former case implies exp onen tial decay for Ψ ; we fo cus on the latter. F or any ε > 0 , observe that cm + log P ( m ) ≤ 2 x for all m ≤ (2 /c − ε ) x , and large x . As a n tends to + ∞ b y assumption, ev aluate the latter expression in x = a n and use ( 4 ) to obtain a αn ≥ c (2 /c − ε ) a n − log C ≥ (2 − cε − ε ) a n . W e conclude that lim a α n 2 − δ n = ∞ for any δ ∈ (0 , 1) . It is clear that in that case, Ψ( n ) tends to 0 faster than any p olynomial. □ The third preparatory lemma states that, up to reducing linearly the time-space windo w, in the Glaub er dynamics the probabilit y of coupling only the ﬁrst k digits is comparable to the probability of coupling all the digits. This lemma would b e false if w e were to consider the “classical” Glaub er dynamics for α : we crucially use the prop erty that our dynamics has the “matc hing of the digits” prop ert y . Lemma 6.7. F or ε > 0 smal l enough, for any δ ∈ (0 , 1) , ther e exist c ε , C ε > 0 and k ≥ 0 such that • The dynamics has the ( ε, k ) -matching of the digits pr op erty. • F or any n ≥ 0 , P [ NC ( n, n )] ≤ exp( − c ε n ) + C ε P [ ⌊ NC ((1 − δ ) n, (1 − δ ) n ) ⌋ k ] . Pr o of. F or a space-time p oint ( − t, v ) ∈ R ≤ 0 × Z d , we construct its “matching tree” T M ( − t, v ) with the following recursiv e pro cedure. Start from the space-time p oint ( − t, v ) , and denote b y − t 1 the smallest negative time b elow − t at whic h the spin α v w as resampled. W e examine the o ccurence of the matching ev en t M ( − t 1 ,v ) in the deﬁnition of Prop erty 2 , and note that it is indep endent of the k -th ﬁrst digits of the up dated v alue of the spin at ( − t 1 , v ) b y construction. No w, • If M ( − t 1 ,v ) o ccurs, we add ( − t 1 , v ) to the tree. In that case, we shall sa y that ( t 1 , v ) is a le af of the tree. • If M c ( − t 1 ,v ) o ccurs, we add ( − t 1 , v ) and T M ( − t 1 , u ) to the tree, for eac h u ∼ v . This provides a w ell-deﬁned notion of matching tree. W e are in terested in the matching tree of the point (0 , 0) that b e call T M in short. Also call ∂ T M the set of lea v es of the matc hing tree of (0 , 0) . W e claim the follo wing tw o prop erties: • On the ev ent {T M ∩ ∂ (( − n, 0] × Λ n ) = ∅} , the random v ariable R Λ n , − n ( α ± ) 0 is measurable with resp ect to – The data of R Λ n , − n, − t ′ ( α ± ) u ′ , for all ( − t ′ , u ′ ) ∈ ∂ T M . – The data of Π(Λ n × ( − n, 0]) . This is due to the Marko v property of the Glaub er dynamics: at each resampling time, the dynamics insp ect the v alues of the neighbours of the spin to b e resampled, and uses the P oisson randomness to resample. A v alue on the b oundary of the b o xe is never explored by assumption. FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 17 • On the even t that the t w o truncated dynamics coincide on the leav es set ∂ T M , i.e. on the formal even t {T M ∩ ∂ (( − n, 0] × Λ n ) = ∅} ∩ { \ ( − t ′ ,u ) ∈ ∂ T M ⌊ C u ( n, n, − t ′ ) ⌋ k } , then it is the case that R Λ n , − n ( α + ) 0 = R Λ n , − n ( α − ) 0 . Indeed, at the lea v es of the tree, α + and α − coincide: their k ﬁrst digits are the same by assumption, and b y deﬁnition of the lea ves of the tree, their further digits are matched. The claim is then a consequence of the ﬁrst item, with the fact that the dynamics nev er explores v alues on the b oundary of the b o x b y assumption. No w recall that 0 < δ < 1 , and assume that ε > 0 is suﬃcien tly small, and k = k ( ε ) is c hosen so that: • The dynamics is ( ε, k ) -matching the digits. • There exists c ε > 0 such that P [ T M ∩ (Λ δ n × ( − δ n, 0]) c  = ∅ ] ≤ exp( − c ε δ n ) . Suc h a c hoice can alwa ys be made, by comparing the law of the oﬀspring of the matching tree with that of a sub critical branc hing process in con tin uous time and using classical results ab out those pro cesses, see [ AN04 ] for instance. It is no w easy to conclude. Indeed, the ev en t NC ( n, n, 0) is partitioned as w ether the ev en t T M ∩ (( − δ n, 0] × Λ δ n ) c  = ∅ o ccurs or not. W e obtain P [ NC ( n, n )] ≤ exp( − c ε δ n ) + P [ {T M ⊂ (( − δ n, 0] × Λ δ n ) } , [ ( − t ′ ,u ) ∈ ∂ int T M ⌊ NC u ( n, n, − t ′ ) ⌋ k ] . Call C ε the exp ected size of the leav es set of T M (whic h is ﬁnite by the fact that we chose ε > 0 so small that this tree is dominated by a sub critical branc hing pro cess), and observe that by monotonicity in space and in time, P [ ⌊ NC u ( n, n, − t ′ ) ⌋ k ] ≤ P [ ⌊ NC ((1 − δ ) n, n, − δ n ) ⌋ k ] when t ′ ≤ δ n and u ∈ Λ δ n . Using the indep endence b etw een the o ccurence of the matching ev en t and the ﬁrst k digits of the dynamics, w e obtain P [ NC ( n, n )] ≤ exp( − c ε δ n ) + C ε P [ ⌊ NC ((1 − δ ) n, n, − δ n ) ⌋ k ] = exp( − c ε δ n ) + C ε P [ ⌊ NC ((1 − δ ) n, (1 − δ ) n ) ⌋ k ] . whic h concludes the pro of. □ W e are ready to prov e Prop osition 6.3 . Pr o of of Pr op osition 6.3 . Fix δ ∈ (0 , 1) . In ligh t of Lemma 6.7 , one has P [ NC ( n, n, δ n )] ≤ P [ NC ((1 − δ ) n, (1 − δ ) n )] ≤ exp( − c ε δ n ) + P [ ⌊ NC ((1 − δ ) 2 n, (1 − δ ) 2 n ) ⌋ k ] . It is th us suﬃcient to argue that the quantit y Ψ( n ) := P [ ⌊ NC ( n, n ) ⌋ k ] decays sup er- p olynomially fast. W e are now in the context of a ﬁnite spin space, and will mimic the Harel–Spink a argument. How ev er the absence of “temp oral Marko v prop erty” for the truncated ﬁeld requires yet another argumen t in the pro of, that w e shall mak e explicit. Our goal is to use Lemma 6.6 and thus to argue that Ψ satisﬁes a renormalization inequalit y . Fix κ > 0 , the v alue of whic h will b e set later on. Also deﬁne φ ( n, s ) := P [ ⌊ NC ( n, s ) ⌋ k ] . 18 LUCAS D’ALIMONTE AND PIET LAMMERS In tro duce tw o v ariables ξ + , ξ − , such that ξ + ∼ µ + Λ m , ξ − ∼ µ − Λ m and ξ − ≤ ξ + almost surely . W e ﬁrst write, using Lemma 6.5 : φ ( n, s ) ≤ φ + ( n, s ) + φ − ( n, s ) + 10 k ∥ µ + Λ n ( ⌊ α 0 ⌋ k ∈ · ) − µ − Λ n ( ⌊ α 0 ⌋ k ∈ · ) ∥ TV , (5) where φ ± ( n, s ) = P [ R Λ n ,s ( α ± ) 0  = R Λ n ,s ( ξ ± ) 0 ] . By the mixing prop ert y of Lemma 4.1 , the total v ariation term is upp er b ounded by e − cn . W e will now pro v e that for any m, n ∈ N an y t, s ∈ R > 0 suc h that t ≥ n , the follo wing holds: φ ± ( m + n, t + s ) ≤ 10 k exp( − cm ) + m d exp( − c ε κn ) + C ε m d φ ( n, s ) φ ((1 − κ ) n, (1 − κ ) n ) . (6) The exact v alues in this inequalit y are not so important. Indeed, to conclude, it will b e suﬃcien t to observe that for s = t = n and m ≤ n , ( 5 ) and ( 6 ) give φ ( m + n, 2 n ) ≤ 2 C ε m d Ψ( n )Ψ((1 − κ ) n ) + m d e − cn + C e − cm , and as m ≤ n , the term m d e − cn can b e absorb ed b y C e − cm , altering the v alues of c, C > 0 . Monotonicit y in space-time allows to b ound Ψ( n ) ≤ Ψ((1 − κ ) n ) , so that Ψ satisﬁes the remormalisation equation ( 3 ) . As is is clear that Ψ is monotone decreasing to 0, the pro of is complete. W e th us fo cus on proving ( 6 ) for φ + , and ﬁx n, m, s, t and κ accordingly . Introduce an indep enden t cop y of the P oisson p oin t pro cess on the volume Λ n + m in the time in terv al [ − t, 0] , and denote b y ˜ R Λ n + m ,t the asso ciated Glauber map. Standard indep endence prop erties of Poisson p oin t pro cesses imply that: φ + ( n + m, t + s ) = P [ ⌊ R Λ n + m ,s ( ˜ R Λ n + m ,t ( α + )) 0 ⌋ k  = ⌊ R Λ n + m ,s ( ˜ R Λ n + m ,t ( ξ + )) 0 ⌋ k ] . W e decomp ose ov er the ev en t: E = \ v ∈ Λ m { ˜ R Λ n + m ,t ( α + ) v = ˜ R Λ n + m ,t ( ξ + ) v } . Indeed, observe that: • If E o ccurs, then R Λ n + m ,t ( α + ) reac hed the inv arian t measure on Λ m . Consequently , conditioned on that ev en t, the worst situation for coupling is if R Λ n + m ,t ( α + ) (resp. R Λ n + m ,t ( ξ + ) ) is maximal (resp. minimal) outside of the b ox Λ m . Call ζ + , + (resp. ζ + , − ) the conﬁguration equal to R Λ n + m ,t ( ξ + ) in Λ m and maximal (resp. minimal) outside of it. This discussion yields (we still w ork conditionally on E , and write φ E ( · , · ) for the conditional probability) φ E ( n + m, t + s ) ≤ P [ ⌊ R Λ m ,s ( ζ + , + ) 0 ⌋ k  = ⌊ R Λ m ,s ( ζ + , − ) 0 ⌋ k ] . But observ e that ζ + , − ≥ ζ − , − , where ζ − , − is constructed by the same pro cedure with R Λ n + m ,t ( ξ − ) instead of R Λ n + m ,t ( ξ + ) . This is due to the monotonicity in the coupled dynamics. Th us, φ E ( n + m, r + s ) ≤ P [ ⌊ R Λ m ,s ( ζ + , + ) 0 ⌋ k  = ⌊ R Λ m ,s ( ζ − , − ) 0 ⌋ k ] By the fact that ζ + , + (resp. ζ − , − ) is in v ariant for the dynamics with + (resp. − ) b oundary conditions on Λ m and using Lemmas 6.5 and 4.1 , we obtain φ E ( n + m, r + s ) ≤ 10 k ∥ µ + Λ m ( α 0 ∈ · ) − µ − Λ m ( α 0 ∈ · ) ∥ TV ≤ 10 k exp( − cm ) . (7) FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 19 • No w, if E do es not o ccur, in whic h case we write φ E c ( · , · ) for the conditional proba- bilit y , then the worst situation for coupling is if R Λ n + m ,t ( α + ) (resp. R Λ n + m ,t ( ξ + ) ) is maximal (resp. minimal) in Λ n + m . In that case, w e may b ound the conditional probabilit y of failing to couple at time 0 by φ ( n + m, s ) . Thus φ E c ( m + n, t + s ) ≤ φ ( n + m, s ) . (8) T o conclude it remains to pro ve an upp er bound on P [ E c ] . Here a new argumen t is required. Fix κ > 0 to b e the size of a “buﬀer zone” that we are going to use for applying Lemma 6.7 . Indeed, we observe that, due to the c hoice t ≥ n and by monotonicity in time and space, P [ E c ] ≤ m d sup v ∈ Λ m P [ NC v ( n + m, n )] ≤ m d P [ NC ( n, n )] . No w, Lemma 6.7 implies that P [ NC v ( n, n )] ≤ exp( − c ε κn ) + C ε φ ((1 − κ ) n, (1 − κ ) n ) (9) Putting ( 7 ), ( 8 ), and ( 9 ) together yields: φ ( n + m, t + s ) ≤ 10 k exp( − cm ) + m d φ ( n + m, s )(exp( − c ε κn ) + C ε φ ((1 − κ ) n, (1 − κ ) n )) ≤ 10 k exp( − cm ) + m d exp( − c ε κn ) + C ε m d φ ( n, s ) φ ((1 − κ ) n, (1 − κ ) n ) . This is exactly ( 6 ), which concludes the pro of. □ 7. SWM: Constr uction of the EFI ID 7.1. Strong sup ercriticalit y of the pro cess of mixed b o xes. Let δ > 0 b e the quan tit y giv en b y Prop osition 6.1 . In tro duce L ∈ N . This num ber will b e though t ab out as the coarse-graining scale of the space-time set R − × Z d . By conv enience, assume that Lδ − 1 := n L is alwa ys an in teger. W e ﬁrst introduce a notion of mixe d p oint . Deﬁnition 7.1. F or any ( t, x ) ∈ L · ( Z ≤ 0 × Z d ) , w e sa y that ( t, x ) is ( L, δ ) -mixed if the ev en t \ − r ∈ [ − L ( t +1) , − Lt ] { R Λ 2 n L ( v ) , − ( t +1) n L , − r ( α + ) | Λ n L ( v ) = R Λ 2 n L ( v ) , − ( t +1) n L , − r ( α − ) | Λ n L ( v ) } o ccurs. In other words, w e ask that the maximal and minimal dynamics remain coupled in a b o x of size n L during a time L , when the b oundary conditions are at space time distance 2 n L from ( t, v ) . The scales are exactly c hosen so as to apply Prop osition 6.1 . W e start b y observing that the en vironment of mixed b oxes can b e made extremely sub critical when L is large. Introduce the site p ercolation pro cess on L · ( Z ≤ 0 × Z d ) deﬁned b y Θ δ,L ( t,x ) = 1 { ( t,x ) is ( L,δ ) − mixed } . Lemma 7.2. Ther e exists a se quenc e ε L tending to 0 when L → ∞ such that the law of Θ δ,L sto chastic al ly dominates P L 1 − ε L , wher e P L 1 − ε L is the law of a Bernoul li b ond p er c olation of p ar ameter 1 − ε L on the lattic e L · ( Z ≤ 0 × Z d ) . Pr o of. F or any ( t, x ) ∈ L · ( Z ≤ 0 × Z d ) , observ e that the ev en t { Θ L,δ ( t,x ) = 1 } is measurable with resp ect to Π(Λ 2 n L ( v ) , [ − ( t + 1) n L , tL ]) . This implies that the pro cess Θ is a 2 δ − 1 + 1 dep endent pro cess on L · ( Z ≤ 0 × Z d ) , endo wed with nearest-neighbour connectivit y . 20 LUCAS D’ALIMONTE AND PIET LAMMERS F urthermore, by the choice of n L , Prop osition 6.1 implies that for any ( t, v ) ∈ L · ( Z ≤ 0 × Z d ) , P [Θ L,δ ( t,x ) = 1] − → L →∞ 1 . A classical result of [ LSS97 ] then implies the result. □ W e also gather the prop erties of the set of mixed b oxes that w e shall use to construct the EFI ID representation of the mo del. Lemma 7.3. L et ( t, x ) ∈ L · ( Z ≤ 0 × Z d ) such that ( t, x ) is ( L, δ ) -mixe d. Then, the fol lowing statements ar e true: • F or any r ∈ [ − L ( t + 1) , − Lt ] , the distributions of R Λ 2 n L ( v ) , − ( t +1) n L , − r ( α ± ) | Λ n L ( x ) ar e given by the me asur e µ SWM Λ 2 n L ( ·| Λ n L ( x ) ) . • F or any r ∈ [ − L ( t + 1) , − Lt ] , the r andom variables R Λ 2 n L ( x ) , − ( t +1) n L , − r ( α ± ) | Λ n L ( x ) ar e indep endent of the Poisson pr o c ess outside of the set Λ 2 n L ( x ) × [ − ( t + 1) n L , tL ] . Pr o of. Both of these statements come from the prop erties of the coupling from the past stated in Theorem 5.9 . □ 7.2. Construction of the lo cal sets for Θ . F or any L ≥ 0 , w e endo w the space-time set L · ( Z ≤ 0 × Z d ) with the usual ∗ -connectivit y: t w o v ertices are considered neighbours when their ℓ ∞ norm is equal to L in the underlying graph Z ≤ 0 × Z d . In a { 0 , 1 } -v alued site p ercolation pro cess on L · ( Z ≤ 0 × Z d ) , we denote b y C the 1-cluster of the vertex 0 for the ∗ -connectivit y , and by C ∗ the 0-cluster of the v ertex 0 for the ∗ -connectivit y . W e start by ﬁxing the v alue of L that w e shall use to construct the lo cal sets. Lemma 7.4. Ther e exists a value of L ≥ 0 lar ge enough and two c onstants c, C > 0 such that for any n ≥ 1 , P L 1 − ε L [ |C ∗ | > n ] ≤ C exp( − cn ) . Pr o of. This follows from a standard Peierls arguments for Bernoulli b ond p ercolation of small parameter ε L in the graph L · ( Z ≤ 0 × Z d ) . □ The v alue of L is no w ﬁxed and given b y Lemma 7.4 , as w ell as the v alues of the constan ts c, C > 0. W e will also drop the dep endency in L and in δ in the notation Θ δ,L . F or an in teger N ≥ 0 , w e also in tro duce the follo wing usual notion of N -external c omplement . F or a connected set C ⊂ L · ( Z ≤ 0 × Z d ) con taining 0, we deﬁne its N -external complemen t by: ∂ N ext C := { y ∈ L · ( Z ≤ 0 × Z d ) \ C , inf x ∈ C ∥ x − y ∥ ∞ > LN and y L · ( Z ≤ 0 × Z d ) \ C ← → ∞ . } where we recall that the inﬁnite norm is tak en with resp ect to the underlying graph Z ≤ 0 × Z d . The notation y L · ( Z ≤ 0 × Z d ) \ C ← → ∞ means that there exists an inﬁnite self-a voiding nearest-neigh b our path in L · ( Z ≤ 0 × Z d ) \ C emanating from y . W e no w call C ∗ the 0-cluster of the vertex (0,0) in the pro cess Θ . The k ey input for constructing the lo cal sets for α 0 is the follo wing observ ation. Lemma 7.5. The r andom variable R ∞ , −∞ , 0 ( α + ) 0 is indep endent of the Poisson p oint pr o c ess Π r estricte d to the set ∂ 2 n L ext C ∗ × R < 0 , wher e we r e c al l that n L = ⌈ δ − 1 L ⌉ . Pr o of. The lemma follo ws b y the observ ation that the external b oundary of C ∗ consists in a space-time surface of mixed b oxes shielding 0 from ∂ n L ext C ∗ . When up dating the state of a p oin t on the in terior of that surface, the Glauber rule inspects the v alue of the ﬁeld: • Either on a v ertex con tained on the inside of the surface, in which case it do es not dep end on the P oisson randomness outside of it, FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 21 • Or on a vertex contained in the surface, in which case by Lemma 7.3 , it is also indep enden t of the P oisson pro cess on ∂ 2 n L ext C ∗ × R < 0 . As (0 , 0) is contained in the space-time surface, the pro of is complete. □ W e conclude by explaining how this prop erty implies Theorem 2.10 in the case of the SWM. Pr o of of The or em 2.10 for the SWM. The EFI ID is constructed as follo ws. The ﬁeld ( X v ) v ∈ Z d is giv en by the data of the Poisson p oint pro cess Π( { v } , R ≤ 0 ) v ∈ Z d , and the map φ is given b y φ ( X ) v := R ∞ , ∞ , 0 ( α + ) v , for an y v ∈ Z d . Moreo v er, for any v ∈ Z d , deﬁne the set L v := Z d \ π ( ∂ n L ext C ∗ v ) , where π : Z ≤ 0 × Z d → Z d , ( t, x ) 7→ x is the spatial pro jection. • By construction, φ is clearly measurable, equiv ariant with resp ect to the automor- phism group of Z d , and φ ( X ) is distributed as µ SWM . • That { Θ L,δ ( t,x ) = 1 } is measurable with resp ect to Π(Λ 2 n L ( v ) , [ − ( t + 1) n L , tL ]) . implies that L v is a lo cal set for Π . • Lemma 7.5 implies that the v alue of φ ( X ) v is measurable with resp ect to X | L v . • Finally , b y Lemma 7.4 together with the sto c hastic domination giv en by Lemma 7.2 , P [ |L v | > n | ≤ C exp( − cn ) . Th us the measure µ SWM is a EFIDD. □ 8. SWM: Sp a tial mixing of the Gibbs measure This section contains a pro of of Lemma 4.1 . The quadratic in teraction in the Hamiltonian of µ +1 Λ 2 n ,β mak es that w e can the asso ciated Br ownian interp olation . Let C Λ ⊂ R d denote the c able gr aph of Λ ⊂ Z d , and write ˜ α for the Brownian interpolation of α on C Λ . The corresp onding Gibbs measure is denoted ˜ µ +1 Λ 2 n ,β . Refer to [ Lup16 ] for details. By ﬂip-symmetry of Bro wnian motion, we get    µ +1 Λ 2 n ,β | Λ n − µ − 1 Λ 2 n ,β | Λ n    TV ≤ ˜ µ +1 Λ 2 n ,β [ { Λ n { ˜ α  =0 } ← − − → ∂ Λ 2 n } ] ≤ X x ∈ ∂ Λ n ˜ µ +1 Λ 2 n ,β [ { x { ˜ α  =0 } ← − − → ∂ Λ 2 n } ] ≍ X x ∈ ∂ Λ n µ +1 Λ 2 n ,β [ α x ] . F or Lemma 4.1 , it suﬃces to pro v e that this last expression decays exponentially in n . This is prov ed by comparison with a massive Gaussian free ﬁeld. More precisely , we deﬁne the massiv e Gaussian free ﬁeld ν ζ Λ ,β ,m on Λ ⊂ Z d with mass m ∈ (0 , ∞ ) and in v erse temperature β ∈ [0 , ∞ ) as the measure on R Λ with density 1 Z ζ Λ ,β ,m e − β H ζ Λ ( α ) − m P u ∈ Λ α 2 u d α. Since all interactions are quadratic, this measure is Gaussian, whic h means that it is quite easy to make explicit calculations. The desired result follows from the following t w o lemmas. Lemma 8.1. Fix a dimension d ≥ 1 , an inverse temp er atur e β ∈ [0 , ∞ ) and a mass m ∈ (0 , ∞ ) . Then ther e exist a c onstant c ∈ (0 , ∞ ) such that for any ﬁnite Λ ⊂ Z d and for any u ∈ Λ , we have ν +1 Λ ,β ,m [ α u ] ≤ e − c Distance( u,∂ Λ) . 22 LUCAS D’ALIMONTE AND PIET LAMMERS Lemma 8.2. Fix a dimension d ≥ 1 and an inverse temp er atur e β ∈ [0 , ∞ ) . Then ther e exists a mass m ∈ (0 , ∞ ) such that for any ﬁnite Λ ⊂ Z d and for any u ∈ Λ , we have µ +1 Λ ,β [ α u ] ≤ ν +1 Λ ,β ,m [ α u ] . Join tly the tw o lemmas clearly imply the desired exp onential deca y . W e provide a pro of for b oth lemmas; a version of the second lemma already app eared in the work of McBry an–Sp encer [ MS77 ]. W e provide an alternative pro of using the FKG inequality that ma y b e of indep enden t interest. This pro of mak es it slightly easier to handle boundary conditions. Pr o of of L emma 8.1 . Let X denote a simple random w alk in Z d started from u , which measure is written P . Let T denote the ﬁrst time that X hits ∂ Λ . Then the Gaussian structure implies that ν +1 Λ ,β ,m [ α u ] = E [(1 + m 2 dβ ) − T ] ≤ (1 + m 2 dβ ) − Distance( u,∂ Λ) . This is the desired result. □ Pr o of of L emma 8.2 . W e present here a v ariation of the McBryan–Spencer argumen t. W e presen t the argumen t in the most general context possible. Let G denote a ﬁnite connected graph, and let g ∈ V ( G ) denote a ﬁxed vertex (w e think of g as the b oundary of G , it is t ypically a vertex with a v ery large degree). No w consider the following setup. F or any u ∈ V ( G ) , λ u denotes an arbitrary probabilit y measure on R that is symmetric under a sign ﬂip. W e write λ := ⊗ u λ u . Then, let φ λ denote the probabilit y measure on R V ( G ) with density 1 Z ( λ ) e − β P uv ∈ E ( G ) ( α ( u ) − α ( v )) 2 d λ ( α ) . No w set λ := λ A where A ⊂ V ( G ) , and λ A u :=      ( δ +1 + δ − 1 ) / 2 if u = g , U ([ − 1 , 1]) if u  = g and u ∈ A , N (0 , 1 / 2 m ) if u  = g and u ∈ A . No w let G denote the ﬁnite graph with vertex set V ( G ) = Λ ∪ { g } , and whose edge set is the one inherited from Λ . T o ﬁnish the pro of, it suﬃces to pro v e that µ +1 Λ ,β [ α v ] = φ λ ∅ [ α g α v ] ≤ φ λ V ( G ) [ α g α v ] = ν +1 Λ ,β ,m [ α v ] , where the equalities are obvious. T o ﬁnish the pro of, it suﬃces the demonstrate that φ λ A [ α g α v ] ≤ φ λ A ∪{ u } [ α g α v ] (10) for any A ⊂ V ( G ) and for any u ∈ Λ for a suitably c hosen m = m ( d, β ) . The rest of the pro of is dedicated to deriving Equation ( 10 ) . By standard results on absolute-v alue-FKG, it is easy to see that Equation ( 10 ) holds true if the la w of | α u | under φ λ A is sto chastically dominated by its law under φ λ A ∪{ u } . F or this purp ose, we compare its law in thr e e measures: the measures φ := φ λ A , φ ′ := φ λ A ∪{ u }  ·   {| α u | ≤ 1 }  , and φ ′′ := φ λ A ∪{ u } . It is easy to see how they are related: φ ′ is a conditioned v ersion of φ ′′ , and φ ′ ma y b e written as the measure φ with a Radon–Nik o dym deriv ativ e prop ortional to e − mα 2 u . Supp ose now that the conditioning even t (in the deﬁnition of φ ′ ) has a φ ′′ -probabilit y of at most 1 − ε for some ε = ε ( d, β ) > 0 . If m = m ( ε ) is suﬃciently small, then the Radon–Nik o dym deriv ative d φ ′ / d φ satisﬁes d φ ′ / d φ ≤ (1 − ε ) − 1 . FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 23 It is then easy to see that the la w of | α u | in φ is sto c hastically dominated by its law under φ ′′ . It suﬃces to ﬁnd an appropriate ε . Notice that by FKG, the c onditional probabilit y of {| α u | ≤ 1 } is the lar gest when α t = 0 for all neigh b ours t ∼ u , In that case, the la w of α u is N (0 , 1 / 2(2 dβ + m )) , where 2 d is the degree of u in the graph Z d . In particular, the means that (when m ≤ 2 dβ ) φ ′′ [ {| α u | ≤ 1 } ] ≤ P [ {| X | ≤ 1 } ] where X ∼ N (0 , 1 / 8 dβ ) in P . This is the desired uniform bound in terms of d and β . □ P art I I. The XY mo del 9. XY: Pr oof over view and monotone represent a tion The spins of the XY mo del take v alues in the circle S 1 ⊂ C . This is not naturally a (partially) ordered set with a minimal and maximal elemen t. In this part, we ﬁrst in tro duce a representation of the XY mo del that is appropriately partially ordered. Then, w e introduce a Glaub er dynamic that works w ell with this partial ordering. Once the Glaub er dynamic has b een established, we follow the pro of steps of Part I . The main diﬀerence lies in the fact that our Glaub er dynamic do es not hav e the Mark o v prop erty (Deﬁnition 5.4 ), and w e m ust perform some steps to replace the Marko v prop erty by something sligh tly weak er (to b e more precise: we sho w that in equilibrium, the Glaub er dynamic is Marko v with resp ect to connected lo cal sets that ha v e exp onen tial tails). 9.1. Monotone represen tation of the XY mo del. The represen tation for the XY mo del is most easily introduced on ﬁnite graphs. Let G = ( V , E ) denote a ﬁnite graph and ﬁx β ∈ [0 , ∞ ) . The XY mo del on G with inv erse temp erature β is the probability measure µ G,β on σ ∈ ( S 1 ) V with a densit y d µ G,β ( σ ) ∝ e − P uv ∈ E β 2 ∥ σ u − σ v ∥ 2 2 · d σ with resp ect to the pro duct Leb esgue measure. Here, we consider the c o or dinate r epr esentation of the XY mo del, whic h w as, to the b est kno wledge of the authors, ﬁrst introduced by Cha y es [ Cha98 ] and later studied in [ DF22 ]. Eac h spin σ u ∈ S 1 ma y b e written in terms of its t w o co ordinates (or in terms of its real and imaginary part) via σ u = ξ u cos α u + i · ζ u sin α u where ξ u , ζ u ∈ {− 1 , +1 } , α u ∈ [0 , π / 2] . The decomp osition is unique unless α u ∈ { 0 , π / 2 } whic h clearly happ ens almost nev er in µ G,β as the la w of α u has a densit y with resp ect to the Leb esgue measure. By writing out the Hamiltonian in terms of the triple ( α, ξ , ζ ) , it is easy to see that, conditional on α , the spins ξ and ζ b eha ve like tw o indep endent Ising mo dels, whose couplings dep end on α . W e may consider simultaneously the indep enden t FK–Ising couplings of the tw o indep endent Ising mo dels with the asso ciated F ortuin–Kasteleyn (FK) p ercolation mo dels. This leads to the coupling of α with t w o p ercolations, denoted ω and η b elow. Another wa y to obtain the same coupling ( α, ω , η ) is by attaching attac hing a Brownian in terp olation (in C ) to eac h edge uv of the graph from σ u to σ v , and recording the edges of G where this Brownian in terp olation do es not hit the imaginary axis (this is ω ⊂ E ) and the edges where the interpolation do es not hit the real axis (this is η ⊂ E ). Both approac hes (which are in fact equiv alent) lead to the follo wing lemma. Lemma 9.1 (Co ordinate representation) . L et G b e a ﬁnite gr aph and β ∈ [0 , ∞ ) . Deﬁne µ ′ G,β as the pr ob ability me asur e on τ := ( α, ω , η ) ∈ [0 , π 2 ] V × { 0 , 1 } E × { 0 , 1 } E := Π G 24 LUCAS D’ALIMONTE AND PIET LAMMERS with density d µ ′ G,β ( τ ) ∝  Q u ∈ V d α u  · 2 k ( ω )  Q uv ∈ ω (1 − e − 2 β (cos α u )(cos α v ) )  Q uv ∈ ω e − 2 β (cos α u )(cos α v )  · 2 k ( η )  Q uv ∈ η (1 − e − 2 β (sin α u )(sin α v ) )  Q uv ∈ η e − 2 β (sin α u )(sin α v )  . Her e k ( ρ ) denotes the numb er of c onne cte d c omp onents of ( V , ρ ) . Now obtain a sample σ ∈ ( S 1 ) V as fol lows: • Sample τ ∼ µ ′ G,β , • Sample ξ ∈ {− 1 , +1 } V by ﬂipping a fair c oin for e ach c onne cte d c omp onent of ω , • Sample ζ ∈ {− 1 , +1 } V by ﬂipping a fair c oin for e ach c onne cte d c omp onent of η , • Set σ u := ξ u cos α u + i · ζ u sin α u for e ach u ∈ V . Then σ ∼ µ G,β . Pr o of. This straigh tforw ard calculation is left to the reader. □ T o construct an EFI ID for the XY mo del, it now suﬃces to: • Construct an EFIID for the measure the co ordinate representation, • Show that all the ω -clusters and η -clusters hav e exp onen tial tails. T o mimic the strategy of Part I , it is essential that w e can construct a Glaub er dynamic that has some monotonicity prop erties. This is discussed now. Deﬁnition 9.2 (Ordering of triples) . Consider tw o triples τ = ( α, ω , η ) and τ ′ = ( α ′ , ω ′ , η ′ ) . W e say that the ﬁrst is smaller than the second, and write τ ⪯ τ ′ , whenever all of the follo wing conditions hold: α ≤ α ′ , − ω ≤ − ω ′ , η ≤ η ′ , where the inequalities are understo o d comp onen twise. This deﬁnes a partial order on Π G . Lemma 9.3 (Monotonicit y of angles) . Fix a ﬁnite gr aph G , some β ∈ [0 , ∞ ) . L et u denote some vertex, and let E u denote the set of e dges incident to u . Pick τ ′ = ( α ′ , ω ′ , η ′ ) ∈ [0 , π / 2] V \{ u } × { 0 , 1 } E \ E u × { 0 , 1 } E \ E u , L et µ τ ′ denote the me asur e µ G,β c onditione d on { τ | ( V \{ u } ) × ( E \ E u ) × ( E \ E u ) = τ ′ } . Then the density of α u on [0 , π / 2] under µ τ ′ is pr op ortional to  Q Γ ∈ C ( ω ′ ) 2 cosh( β cos α u P v ∈ Γ cos α ′ v )   Q Γ ∈ C ( η ′ ) 2 cosh( β sin α u P v ∈ Γ sin α ′ v )  , wher e C ( ρ ) is the p artition of { v ∈ V : v ∼ u } into ρ -c onne cte d c omp onents. In p articular, if µ α u τ ′ denotes the law of α u under µ τ ′ , then τ ′ ⪯ τ ′′ = ⇒ µ α u τ ′ ≤ stoch µ α u τ ′′ . Pr o of. The formula for the densit y follows from Lemma 9.1 via a calculation. F or the sto c hastic domination, one simply chec ks the Holley criterion. □ Lemma 9.4 (Monotonicit y of the edges) . Fix a ﬁnite gr aph G and some β ∈ [0 , ∞ ) . L et E ′ ⊂ E and uv ∈ E \ E ′ . Deﬁne G ′ := ( V , E ′ ) . F or any τ ′ = ( α ′ , ω ′ , η ′ ) ∈ Π G ′ , write µ ω uv τ ′ and µ η uv τ ′ for the laws of ω uv and η uv in the c onditional me asur e µ G,β [ · |{ τ | G ′ = τ ′ } ] . Then τ ′ ⪯ τ ′′ = ⇒ ( µ ω uv τ ′′ ≤ stoch µ ω uv τ ′ and µ η uv τ ′ ≤ stoch µ η uv τ ′′ ) Notic e also that ω and η ar e indep endent in this c onditional me asur e. Mor e over, µ ω uv τ ′ only dep ends on the values of α u , α v , and the p artition of { x ∈ V : xy ∈ E \ E ′ } into ω ′ -c onne cte d c omp onents. The same holds true for µ η uv τ ′ (mutatis mutandis). FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 25 Pr o of. Conditional on α , the percolations ω and η are just indep enden t FK p ercolation mo dels with cluster w eight q = 2 and with edge w eigh ts dep ending on α . Moreo v er, the edge w eights for ω are increasing in α , while those for η are decreasing in α . The sto chastic domination then simply follows from w ell-kno wn results for FK p ercolation. □ 9.2. Statemen t of the spatial mixing estimate. Finally , w e state our external spatial mixing input that is pro v ed at the end, in Section 13 . The b oundary conditions +1 , + i ∈ S 1 ⊂ C are chosen to represent the minimal and maximal b oundary conditions resp ectiv ely in the monotone representation introduced abov e. Lemma 9.5 (Spatial mixing of the Gibbs measure of the XY mo del) . Fix the dimension d ∈ Z ≥ 1 and the inverse temp er atur e β ∈ [0 , β c ( d )) . Then ther e exists c onstants C < ∞ and c > 0 such that (1) F or any n ∈ Z ≥ 1 ,    µ +1 Λ 2 n ,β | Λ n − µ + i Λ 2 n ,β | Λ n    TV ≤ C 5 e − cn . (2) F or any n ∈ Z ≥ 1 , sup ζ ∈{ +1 , + i } , ν ∈{ ω ,η } µ ζ Λ 2 n ,β [ { Λ n ν ← → ∂ Λ 2 n } ] ≤ C 5 e − cn . 10. XY: Gla uber dynamic 10.1. Abstract description of the Glaub er dynamic. The following deﬁnition of a Glaub er dynamic is essentially equiv alent to Deﬁnition 5.1 . W e must mak e a small technical mo diﬁcation: in Deﬁnition 5.1 we only enco ded a spin α u at each vertex, while now we also enco de some p ercolations around each v ertex. In the current section, we imp ose that a Glaub er up date resamples the v alue of α u at eac h vertex u , as well as all p ercolation edges inciden t to u . Deﬁnition 10.1 (Glaub er dynamic) . Consider the square lattice graph Z d (also denoted G = ( V , E ) ). Let µ denote a stationary measure on Π G . F or any u ∈ V , let G u denote the subgraph of G induced by the vertices V \ { u } . A Glaub er dynamic for µ is a function R : Π G × V × Σ → Π G , with the follo wing prop erties. • Lo cal update. F or an y τ ∈ Π G , u ∈ V and ι ∈ Σ , we hav e R ( τ , u, ι ) | G u = τ | G u . • Consistency . F or an y τ ∈ Π G and u ∈ V , the distribution of R ( τ , u, ι ) in d ι is µ [ · | τ | G u ] . • Equiv ariance. F or any τ ∈ Π G , u ∈ V , ι ∈ Σ and θ ∈ Θ , we ha ve R ( τ , u, ι ) ◦ θ = R ( τ ◦ θ , u ◦ θ , ι ) . W e shall also write R u,ι : Π G → Π G , τ 7→ R ( τ , u, ι ) for an y u ∈ V and ι ∈ Σ . W e stress again that this deﬁnition is essen tially equiv alen t to Deﬁnition 5.1 . Next, we extend the prop erties of the Glaub er dynamic from Deﬁnition 5.1 to the curren t setting. Deﬁnition 10.2 (Prop erty 1 : Monotonicity) . A Glaub er dynamic R is monotone if for any u ∈ V and ι ∈ Σ , the map R u,ι : Π G → Π G preserv es the partial order ⪯ of Deﬁnition 9.2 . Deﬁnition 10.3 (Property 2 : Matc hing digits) . Consider ﬁxed constants ε ∈ [0 , 1] and k ∈ Z ≥ 0 . W e sa y that a Glaub er dynamic R matches digits up to ( ε, k ) if w e may ﬁnd an ev en t M ⊂ Σ of d ι -probabilit y at least 1 − ε suc h that: • F or an y u ∈ V and ι ∈ M , w e hav e  { R u,ι ( τ ) 1 ,u } k = { R u,ι ( τ ′ ) 1 ,u } k for all τ , τ ′ ∈ Π G  26 LUCAS D’ALIMONTE AND PIET LAMMERS • F or any u ∈ V and τ ∈ Π G , the even t M and the random v ariable ⌊ R u,ι ( τ ) 1 ,u ⌋ k are d ι -indep enden t. The even t M is called the matching event . F or deﬁning the mo diﬁcation of the Mark o v prop ert y enjo yed by the dynamics, it is con v enien t to in tro duce the following notation. F or τ ∈ Π G and v ∈ V ( G ) , denote by C ω v ( τ ) (resp. C η v ( τ ) ) the ω (resp η )-cluster of v . Also call C ω N v ( τ ) (resp. C ω N v ( τ ) ) the union of the ω (resp. η )-clusters of the neighbours of v . Deﬁnition 10.4 (Prop erty 3 b: almost Mark o v prop ert y) . W e sa y that a Glaub er dynamic R has the almost Markov pr op erty if for any τ ∈ Π G , u ∈ V and ι ∈ Σ , the v alue of R ( τ , u, ι ) at u and on the edges incident to u , is measurable with respect to ι , together with: • The data of C ω N u ( τ | G u ) , • The data of C η N u ( τ | G u ) • The data of α | N ( u ) in τ . Compared to Part I and the case of the SWM, the relaxation of a “true” Marko v prop erty 3 in to the “almost” Marko v prop ert y of Deﬁnition 10.4 is an imp ortant change. Indeed, the argumen ts relying on the Marko v prop erty need to be adapted. As an example for the reader, the “shielding” argument used to construct the lo cal sets in Section 7 is not v alid an ymore: ev en though a vertex is shielded by a surface of mixed b o xes, the Glaub er dynamics could lo ok for information further a w ay , through the existence of a very long connection in ω or η . Ho w ev er, those t w o p ercolations are pro v ed to ha v e exp onen tially small clusters, and the arguments can b e adapted. 10.2. Construction of the Glaub er dynamic. Lemma 10.5 (Glaub er dynamic for the XY model with matc hing digits) . Fix d ∈ Z ≥ 1 and β ∈ R ≥ 0 . Then for any ε > 0 , ther e exists a c onstant k ∈ Z ≥ 0 and a Glaub er dynamic R for the monotone r epr esentation of the XY mo del on Z d at inverse temp er atur e β , which has Pr op erties 1 , 2 , and 3 b, matching digits up to ( ε, k ) . Pr o of. The Glaub er dynamic is constructed as follo ws. First, one erases the v alue of α u , as w ell as ω and η on all edges incident to u . Then, one ﬁrst resamples α u according to the conditional distribution of α u giv en b y Lemma 9.3 . This can b e done in a monotone fashion (Lemma 9.3 ) and with matching digits (b y follo wing the same strategy as in Subsection 5.2 ). Then, one simply needs to resample ω and η on all edges incident to u in a monotone fashion; this can b e done thanks to Lemma 9.4 . By the ab ov e construction, the Glaub er dynamic clearly satisﬁes Prop erties 1 and 2 . Since the conditional la w of the new v alues only dep ends on α at the neigh b ours of u , as well as the connectivity of the neighbours of u in ω and η , the almost Marko v prop erty (Prop ert y 3 b) is also automatically satisﬁed. □ W e can imp ort the sim ultaneous coupling of the Glaub er dynamics in any volume, time and b oundary conditions giv en by the coupling-from-the-past of Theorem 5.9 . In what follo ws, the notation R Λ , − t, − s ( τ ± ) will b e used to refer to this construction. It will also b e con v enient to distinguish b etw een the law of the p ercolation pro cesses ω and η and of the ﬁeld α under this dynamics. This is done as follo ws. Deﬁnition 10.6. Fix a subgraph G of Z d , a conﬁguration τ ∈ Π G , and −∞ ≤ − t < − s ≤ 0 . • The ω (resp. η )-marginal of R Λ , − t, − s ( τ ) will b e denoted b y R Λ , − t, − s ω ( τ ) (resp. R Λ , − t, − s η ( τ ) ). • The ( ω , η ) -marginal of R Λ , − t, − s ( τ ) will b e denoted by R Λ , − t, − s ( ω ,τ ) ( τ ) . • The α -marginal of R Λ , − t, − s ( τ ) will b e denoted by R Λ , − t, − s α ( τ ) . FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 27 Our goal is no w to imp ort the structure used in the pro of of Theorem 2.10 in Part I . The ma jorit y of the arguments is preserved, and w e shall reprov e and adapt the ones that use the exact Mark ov property . 11. XY: Sp a ce-time mixing Our goal is to prov e the analog of Prop osition 6.1 with our new Glaub er map. In what follo ws, with distinguish an edge e 0 adjacen t to the vertex 0. It will b e con v enient to write 0 = (0 , e 0 ) . W e also imp ort all the notation regarding the coupling even ts C and NC in tro duced in Section 6 , with the correct adaptation. In particular, for any p oin t v ∈ Λ n , any edge e ∈ Λ n , • C ( n, n ) (resp. NC ( n, n ) ) will refer to C ( n, n ) 0 (resp. NC ( n, n ) 0 ). • C ω ,η ,e ( n, n ) will refer to the coupling ev ent at the edge e , only for the dynamics on ( ω , η ) giv en b y R Λ n , − n, 0 ( ω ,η ) ( τ ± ) . • C α,v ( n, n ) (resp. ⌊ C α,v ( n, n ) ⌋ k ) will refer to the coupling even t at v , only for the dynamics on α (resp. ⌊ α ⌋ k ) given by R Λ n , − n, 0 α ( τ ± ) . • Finally , when considering the “global” truncated coupling even t ⌊ C ( n, n ) ⌋ k and its complement ⌊ NC ( n, n ) ⌋ k , the truncation only concerns the ﬁeld α , as ω and η already hav e a ﬁnite spin space. F or con v enience, we rewrite the target prop osition. Prop osition 11.1. Ther e exists δ > 0 such that P  \ − r ∈ [ − δ n, 0] { R 2 n, − n, − r ( τ + ) 0  = { R 2 n, − n, − r ( τ + ) 0 }  − → n →∞ 1 . The pro of of this statement requires an adequate adaptation. W e ﬁrst observe that when forgetting ab out the ﬁeld α , the argumen t of Harel–Spink a directly imply the mixing of the dynamics for the marginal on ( ω , η ) , as this random v ariable en ters in to their framew ork. 1 As, such the following prop osition is a direct consequence of [ HS22 ]. Prop osition 11.2. Ther e exist c, C > 0 and δ > 0 such that for any n ≥ 0 , P [ NC ( ω ,η ) ( n, − n, ≥ − δ n )] ≤ C e − cn . W e stress on the fact that this statement do es not a priori say an ything on α . Pr o of of Pr op osition 11.2 . The statement for δ = 0 directly from [ HS22 ] as the law of ( ω , η ) en ters their framew ork of up wards-do wnw ards speciﬁcations, is monotone, and inherits the weak mixing property of the whole triple ( α, ω , η ) . F or extending it to the space-time b o x Λ n × ( − δ n, 0] , w e use Lemma 6.4 that do es not rely on the Mark ov prop erty of the mo del. □ W e now explain ho w to adapt the argumen t of Section 6 . The careful reader might chec k that the only result of that section relying on the Marko v property of the dynamics is Lemma 6.7 . Indeed, the spin space of the truncated dynamics still is ﬁnite, and the Harel– Spink a argument do es not rely on the Marko v prop ert y . Thus, the pro of of Prop osition 11.1 b oils down to the following lemma, that w e pro v e right after. Lemma 11.3. F or ε > 0 , for any δ ∈ (0 , 1) , ther e exist c ′ ε , C ′ ε > 0 such that • The dynamics has the ( ε, k ) matching of the digits pr op erty • F or any n ≥ 0 , P [ NC ( n, n )] ≤ exp( − c ′ ε n ) + C ′ ε P [ ⌊ NC ((1 − δ ) n, (1 − δ ) n ) ⌋ k ] . 1 A ctually ,the authors of [ HS22 ] work in the muc h more general context of upwards-b ackwar ds sp e ciﬁc ations , and in particular do not mak e any use of a form of Marko v property . 28 LUCAS D’ALIMONTE AND PIET LAMMERS T o prov e this statemen t, we adapt the construction of the matching tree as to also incorp orate all the p oints b elonging to the ω and η clusters of the neighbours of a p oint to b e resampled at each step. This will restore the actual Mark ov prop ert y , and w e will chec k that this tree still has an exp onen tial probability to die out in linear time and space. More formally , for a space-time p oint ( − t, v ) ∈ R ≤ 0 × Z d , construct its “Marko v-matc hing tree”, still noted T M ( − t, v ) , with the follo wing recursive pro cedure. Start with the space-time p oint ( − t, v ) and add it to the tree. Next, denote by − t 1 the ﬁrst negative time b elow − t at which v w as resampled. Again we examine the o ccurence of the matching even t M ( − t 1 , v ) . • If M ( − t 1 , v ) o ccurs, we add ( − t 1 , v ) to the tree. As previously we sa y that this space-time p oint is a le af of the tree. • If M ( − t 1 , v ) do es not o ccur, then the follo wing are added to the tree: – The v ertex ( − t 1 , v ) . – The v ertices of C ω N u ( τ | G u ) , – The v ertices of C η N u ( τ | G u ) , – All the trees T M ( − t 1 , u ) , for all u ∼ v . The set of v ertices that are either lea ves of the tree or that hav e a subtree attac hed to them will b e called the trunk of the tree. As previously , the matc hing tree of 0 is simply denoted by T M , and its leaf set that is denoted by ∂ T M . This ob ject in hand, we turn to the pro of of the lemma. Pr o of of L emma 11.3 . W e claim the following t w o prop erties. • On the ev en t {T M ⊂ ( − n, 0] × Λ n } , the random v ariables R Λ n , − n, 0 ( τ ± ) 0 are measurable with resp ect to: – The data of R Λ n , − n, − t α ( τ ± ) u for all the ( − t, u ) b elonging to the lea ves set of the tree. – The data of the ω and η -op en clusters of u in R Λ n , − n, − t ( ω ,η ) ( τ ± ) for all the ( − t, u ) b elonging to the trunk of the tree. – The data of Π(Λ n × ( − n, 0]) . • On the even t that T M ⊂ ( − n, 0] × Λ n , that the tw o truncated dynamics for α + and α − coincide on the leav es set and the even t that the ω and η op en clusters of all the vertices of the trunk of the Marko v tree coincide, then the tw o dynamics for α coincide at the vertex 0 at time 0. The ﬁrst statemen t is a reformulation of the almost-Marko v prop erty , and the second consists in the observ ation that on the leaf set on the tree the dynamics are matc hed by construction, together with the ﬁrst property ab ov e mentioned. No w let δ ∈ (0 , 1) and assume that ε > 0 is suﬃciently small, and k = k ( ε ) is chosen small enough so that: • The dynamics is ( ε, k ) -matching the digits. • There exists c ε > 0 such that P [ {T M ⊂ ( − δ n, 0] × Λ δ n ] } ∩ \ − r ∈ ( − δ n, 0] { R − Λ n , − n, − r ( ω ,η ) ( τ + ) | Λ δn = R − Λ n , − n, − r ( ω ,η ) ( τ − ) | Λ δn } ≥ 1 − exp( − c ε δ n ) . W e brieﬂy explain ho w such a choice can be done. First, the exponential probabilit y of coincidence of the dynamics for ( ω , η ) in the space-time windo w ( − δ n, 0] × Λ n is giv en b y Proposition 11.2 . Conditionally on this ev en t, the p ercolations ( ω , η ) reached their in v ariant measure on Λ δ n , so that they exhibit exp onential deca y of their clusters size by Lemma 9.5 . As the trunk of the tree can b e compared with a v ery sub critical branc hing pro cess b y chosing ε > 0 small enough, the probability that the whole tree is not con tained FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 29 in ( − δ n, 0] × Λ δ n is exp onentially small by a basic union b ound on all the vertices of the tree. The proof is no w concluded exactly as in Lemma 11.3 . □ The reader might no w chec k that once this adaptation is done, the rest of the pro of follo ws as in Section 6 in tro ducing φ ( n, t ) := P [ ⌊ NC ( n, n ) ⌋ k ] This provides a pro of of Prop osition 11.1 . 12. XY: Constr uction of the EFI ID The argument is similar to the pro of in the case of the SWM, but the fact the Glaub er dynamics in not Marko vian an ymore needs to b e tak en into account, as a space-time surface of mixed b o xes around 0 is not suﬃcient to decouple the state of R ∞ , −∞ , 0 ( τ + ) 0 from the P oisson pro cess in the n L -complemen t of the surface. 12.1. Subcriticality of the set of go o d b oxes . Similarly as in Section 7 , w e ﬁx a coarse-graining scale L ≥ 0 and assume for conv enience that δ − 1 L is an integer. W e say that a p ercolation conﬁguration is cr ossing a given b ox if t w o opp osite faces are linked by an op en path. The notion of mixed p oin t needs to b e replaced by the following deﬁnition. Deﬁnition 12.1. A p oint ( t, x ) ∈ L · ( Z ≤ 0 × Z d ) is said to b e ( L, δ ) -go o d is the follo wing three conditions are satisﬁed: • F or all r ∈ [ − L ( t + 1) , − Lt ] , the tw o conﬁgurations R Λ 2 n L ( v ) , − ( t +1) n L , − r ( τ ± ) coincide on the space b o x Λ n L ( v ) . • F or all − r ∈ [ − L ( t +1) , − Lt ] , there exists no ω -op en cluster of R Λ 2 n L ( v ) , − ( t +1) n L , − r ( τ + ) crossing the space b o x Λ L ( v ) . • F or all − r ∈ [ − L ( t +1) , − Lt ] , there exists no η -op en cluster of R Λ 2 n L ( v ) , − ( t +1) n L , − r ( τ + ) crossing the space b o x Λ L ( v ) . If ( t, x ) satisﬁes the ﬁrst condition amongst those three, we still say that it is ( L, δ ) -mixed. As previously , introduce the site p ercolation pro cess on L · ( Z ≤ 0 × Z d ) deﬁned b y Θ δ,L ( t,x ) = 1 { ( t,x ) is ( L,δ ) − go o d } . Lemma 12.2. Ther e exists a se quenc e ε L tending to 0 when L → ∞ such that the law of Θ δ,L sto chastic al ly dominates P L 1 − ε L , wher e P L 1 − ε L is the law of a Bernoul li b ond p er c olation of p ar ameter 1 − ε L on the lattic e L · ( Z ≤ 0 × Z d ) . Pr o of. The proof is the same as the corresp onding lemma in P art I , as the conditions on ω and η are lo cal. Indeed, it is still the case that for an y ( t, x ) ∈ L · ( Z ≤ 0 × Z d ) , the even t { Θ L,δ ( t,x ) = 1 } is measurable with resp ect to Π(Λ 2 n L ( v ) , [ − ( t + 1) n L , tL ]) . This implies that the pro cess Θ is a 2 δ − 1 + 1 dep endent pro cess on L · ( Z ≤ 0 × Z d ) , endo wed with nearest-neighbour connectivit y . F urthermore, b y the choice of n L , Prop osition 11.1 implies that for an y ( t, x ) ∈ L · ( Z ≤ 0 × Z d ) , P [( t, x ) is ( L, δ ) -mixed ] − → L → 0 1 . Th us, the pro of will be concluded b y the result of [ LSS97 ] pro vided that we prov e that conditionally on fact that ( t, x ) is ( L, δ ) -mixed, then the probability of ﬁnding either an ω -op en crossing cluster or an η -crossing cluster in R Λ 2 n L ( v ) , − ( t +1) n L , − r ( τ + ) for some r ∈ [( − t + 1) L, − tL ] tends to 0 as L tends to 0. W e do it for ω and conclude with a union b ound. 30 LUCAS D’ALIMONTE AND PIET LAMMERS Observ e that conditionally on the fact that ( t, x ) is ( L, δ ) -mixed, R Λ 2 n L ( v ) , − ( t +1) n L , − r ( τ + ) follo ws the in v ariant measure µ b y Theorem 5.9 . Call Cross L ( x ) the ev en t that the ω -cluster of the point v ∈ Λ L ( x ) is crossing. F or a giv en p oint v of the box Λ n L ( x ) , observe that an y edge b ordering v is resampled P times in the time interv al [ − L ( t + 1) , − Lt ] , where P is a P oisson v ariable of mean L . Thus, similarly as the pro of of Lemma 6.4 , w e may use Chernoﬀ type upp er b ounds on the tail of a Poisson v ariable to upp er bound the probability that there exists an ω -op en cluster of R Λ 2 n L ( v ) , − ( t +1) n L , − r ( τ + ) crossing the b o x Λ L ( x ) by X v ∈ Λ n L { 100 Lµ [ Cross L ( v )] + exp( − cL ) } − → L →∞ 0 , as the ω -p ercolation in µ has exp onentially small connection probabilities b y Lemma 9.5 . □ The set of ( δ, L ) -go o d b o xes also enjoy the same set of prop erties as in P art I . Lemma 12.3. L et ( t, x ) ∈ L · ( Z ≤ 0 × Z d ) such that ( t, x ) is ( L, δ ) -mixe d. Then, the fol lowing statements ar e true: • F or any r ∈ [ − L ( t + 1) , − Lt ] , the distributions of R Λ 2 n L ( v ) , − ( t +1) n L , − r ( α ± ) | Λ n L ( x ) ar e given by the me asur e µ SWM Λ 2 n L ( ·| Λ n L ( x ) ) . • F or any r ∈ [ − L ( t + 1) , − Lt ] , the r andom variables R Λ 2 n L ( x ) , − ( t +1) n L , − r ( α ± ) | Λ n L ( x ) ar e indep endent of the Poisson pr o c ess on the set (Λ 2 n L ( x ) × [ − ( t + 1) n L , tL ]) c . Pr o of. Both of these statements come from the prop erties of the coupling from the past stated in Theorem 5.9 . □ 12.2. Construction of the lo cal sets for Θ . The end of the pro of is extremely similar, when noticing that the t wo additional prop erties in the deﬁnition of ( δ, L ) -go o d b oxes allow to decouple the inside and the outside of space-time surfaces of go o d b o xes. As previously , for any L ≥ 0 , we endow the space-time set L · ( Z ≤ 0 × Z d ) with the usual ∗ -connectivit y: tw o vertices are considered neigh b ours when their L ∞ norm is equal to L in the underlying graph Z ≤ 0 × Z d . In a { 0 , 1 } -v alued site p ercolation pro cess on L · ( Z ≤ 0 × Z d ) , w e denote by C the 1-cluster of the vertex 0 for the ∗ -connectivit y , and b y C ∗ the 0-cluster of the v ertex 0 for the ∗ -connectivit y . W e start b y ﬁxing the v alue of L that we shall use to construct the lo cal sets. Lemma 12.4. Ther e exists a value of L ≥ 0 lar ge enough and two c onstants c, C > 0 such that for any n ≥ 1 , P L 1 − ε L [ |C ∗ | > n ] ≤ C exp( − cn ) . Pr o of. This follo ws from a standard P eierls argument for Bernoulli b ond percolation of small parameter ε L in the graph L · ( Z ≤ 0 × Z d ) . □ The v alue of L is now ﬁxed and giv en b y Lemma 12.4 , as w ell as the v alues of the constan ts c, C > 0. W e will also drop the dep endency in L and in δ in the notation Θ δ,L . W e no w call C ∗ the 0-cluster of the vertex (0,0) in the pro cess Θ . The k ey input for constructing the lo cal sets for α 0 is the following observ ation (remem b er the notion of N -external complemen t that was in tro duce in Section 7 ). Lemma 12.5. The r andom variable R ∞ , −∞ , 0 ( α + ) 0 is indep endent of the Poisson p oint pr o c ess Π r estricte d to the set ∂ 2 n L ext C ∗ × R < 0 , wher e we r e c al l that n L = ⌈ δ − 1 L ⌉ . Pr o of. The lemma follo ws b y the observ ation that the external b oundary of C ∗ consists in a space-time surface of mixed b oxes shielding 0 from ∂ n L ext C ∗ . When up dating the state of a p oin t on the in terior of that surface, the Glauber rule inspects the v alue of: FREE ENERGY ANAL YTICITY OF XY AND DEBYE SCREENING IN THE COULOMB GAS 31 • α on the neigh b ours of a point on the interior of the surface, and the η and ω -clusters of these neighbours, wic h are a priori non-lo cal. Ho w ev er, the second and third conditions on the deﬁnition of a ( δ, L ) -go o d b o x prev en t those connections to cross the surface. It means that in an y case, the randomness that has b een explored lies: – On the in terior of the surface, in whic h case it is indep enden t of the Poisson p oin t process on ∂ 2 n L ext C ∗ × R < 0 – In the surface, in which case it is indep endent of the Poisson p oint pro cess on ∂ 2 n L ext C ∗ × R < 0 b y construction. W e conclude that this up date is indep enden t of the P oisson p oint • Or ( α, ω , η ) inside of the surface, whic h by assumption is indep endent of the Poisson p oin t process on ∂ 2 n L ext C ∗ × R < 0 b y construction. As (0 , 0) is contained in the space-time surface, the pro of is complete. □ W e conclude by explaining how this prop erty implies Theorem 2.10 in the case of the XY mo del. Pr o of of The or em 2.10 for the XY mo del. The pro of is exactly the same as in P art I , by setting • ( X v ) v ∈ Z d = Π( { v } , R ≤ 0 ) v ∈ Z d , • φ ( X ) v := R ∞ , ∞ , 0 ( σ +1 ) v , • L v := Z d \ π ( ∂ n L ext C ∗ v ) , where π : Z ≤ 0 × Z d → Z d , ( t, x ) 7→ x is the spatial pro jection. The reader might chec k that the argumen ts follow mutatis mutandis . Thus the measure µ +1 XY is a EFIDD. □ 13. XY: Sp a tial mixing of the Gibbs measure Pr o of of L emma 9.5 . By rotating the system b y an angle of − π / 4 , it suﬃces to pro v e that    µ e iπ/ 4 Λ 2 n ,β | Λ n − µ e − iπ/ 4 Λ 2 n ,β | Λ n    TV ≤ C e − cn . But in the co ordinate representation, these b oundary conditions diﬀer b y ﬂipping the sign of ζ . Th us, the ab ov e total v ariation estimate is b ounded by ( µ e iπ/ 4 Λ 2 n ,β ) ′ [ { Λ n η ← → ∂ Λ 2 n } ] . By the monotonicit y properties, this is upp er b ounded by ( µ i Λ 2 n ,β ) ′ [ { Λ n η ← → ∂ Λ 2 n } ] . Just like at the b eginning of Section 8 , we now get    µ e iπ/ 4 Λ 2 n ,β | Λ n − µ e − iπ/ 4 Λ 2 n ,β | Λ n    TV ≤ X x ∈ ∂ Λ n ( µ + i Λ 2 n ,β ) ′ [ { x η ← → ∂ Λ 2 n } ] ≍ X x ∈ ∂ Λ n µ +1 Λ 2 n ,β [ σ x ] . But this deca ys to zero exp onentially fast in n , by the deﬁnition of β c = β c ( d ) > β . This also implies the desired exp onential decay of the connection probabilities in the statement of the lemma. □ 32 LUCAS D’ALIMONTE AND PIET LAMMERS 14. The Villain model and the 2D Coulomb gas Deﬁnition 14.1 (Villain mo del) . Fix d ∈ Z ≥ 1 and β ∈ R ≥ 0 . F or any domain Λ and ζ : Z d → S 1 , the Vil lain mo del on Λ with b oundary condition ζ at inv erse temp erature β is the probability measure on σ ∈ ( S 1 ) Λ giv en by d µ ζ Villain , Λ ,β ( σ ) = 1 Z ζ Villain , Λ ,β e − H ζ Villain , Λ ,β ( σ ) d σ, where Z ζ Villain , Λ ,β is the normalising constant, d σ the Haar measure, and e − H ζ Villain , Λ ,β ( σ ) =  Y { x,y }⊂ Λ x ∼ y p β ( σ x , σ y )  Y x ∈ Λ , y ∈ Λ c x ∼ y p β ( σ x , ζ y )  ; p β ( z , z ′ ) := X m ∈ Z e − β 2  arg( z ) − arg ( z ′ )+2 π m  2 . Notice that the Hamiltonian is deﬁned somewhat atypically and incorp orates the dep endency on β ; more on this later. The critic al inverse temp er atur e is deﬁned as the largest β Villain c = β Villain c ( d ) ∈ (0 , ∞ ) such that µ +1 Villain , Λ n ,β [ σ 0 ] decays exp onentially fast in n for any ﬁxed β ∈ [0 , β Villain c ) . Classical argumen ts based on correlations inequalities imply the existence of an inﬁnite-v olume Gibbs measure with +1 b oundary conditions at in v erse temp erature β ≥ 0 , that we refer to as µ +1 Villain ,β . The lab el Villain is omitted from notations when it is clear from the context. Theorem 14.2 (The Villain mo del is an EFI ID) . Consider the Vil lain mo del in dimension d ∈ Z ≥ 2 at inverse temp er atur e β ∈ [0 , β Villain c ( d )) . Then the inﬁnite-volume Gibbs me asur e µ +1 Villain ,β is an EFIID. Pr o of. W e claim that the pro of for the XY model applies to the Villain mo del with minor mo diﬁcations. This holds true essentially b ecause the Villain model ma y b e seen as an XY mo del on the cable graph, or alternatively as a limit of XY mo dels on graphs where the edges are replaced by m ultiple edges connected in series (this was already observ ed by Berezinskii). The only problem with the Villain model, is that if we consider the same monotone represen tation (on v ertices and edges) as for the XY mo del, then ω and η are no longer indep enden t conditional on α . More generally , the precise form ulas for the densities mu st b e mo diﬁed, but this p oses no real problem. More precisely: • The formulas for the densities in Lemmas 9.1 and 9.3 must b e mo diﬁed, but the qualitativ e monotonicity prop erties still hold true, • The independence of ω and η in Lemmas 9.1 and 9.4 is lost, but the edges can still b e resampled in a monotone fashion, by sampling b oth ω and η from the conditional distribution, on one edge of the underlying graph at a time. The Glaub er dynamic with matc hing digits can still be constructed as for the XY mo del. The rest of the pro of remains identical. □ Theorem 14.3 (Analyticit y of the free energy of the Villain mo del) . Fix d ∈ Z ≥ 1 . Then f Villain : R ≥ 0 → R , β 7→ lim n →∞ 1 | Λ n | log Z +1 Villain , Λ n ,β , the fr e e ener gy of the mo del, is analytic on [0 , β Villain c ( d )) . Pr o of. W e may not directly apply Theorem 3.1 b ecause the Hamilontian of the Villain mo del does not scale linaerly with β . This is not really a problem for the applicabilit y REFERENCES 33 of Theorem 3.1 : the situation is exactly the same for the random-cluster mo del handled in [ Ott20a ], and we refer to that work for details. Indeed, the only prop erty on the Hamiltonian that w e require, is the prop erty that H 0 Villain , Λ n ,β + ε − H 0 Villain , Λ n ,β tends to zero uniformly in σ when ε → 0 , and that the the dep endence on ε is itself analytic. □ A c kno wledgemen ts. The authors thank Diederik v an Engelenburg, Nathan de Mon t- golﬁer, Sébastien Ott, and A velio Sepúlveda for useful discussions and commen ts on the man uscript, as w ell as Christophe Garban for suggesting to incorp orate the square well mo del in the analysis. This research was supp orted by the F renc h National Researc h Agency (ANR), pro ject num ber ANR-23-CPJ1-0150-01 . This researc h w as supp orted in part b y the In ternational Centre for Theoretical Sciences (ICTS) for participating in the programme Ge ometric metho ds in Per c olation and Spin Mo dels 2026 (co de: ICTS/GPSM2026/03 ). References [Abb+18] Emman uel Abb e, Lauren t Massoulié, Andrea Mon tanari, Allan Sly, and Nikhil Sriv astav a. “ Group Sync hronization on Grids”. In: Mathematic al Statistics and L e arning 1.3 (2018), pp. 227–256. [AD21] Mic hael Aizenman and Hugo Duminil-Copin. “ Marginal T rivialit y of the Scaling Limits of Critical 4D Ising and λφ 4 4 Mo dels”. In: Annals of Mathematics 194.1 (2021), pp. 163–235. [Aiz+22] Mic hael Aizenman, Matan Harel, Ron P eled, and Jacob Shapiro. Depinning in Inte ger-R estricte d Gaussian Fields and BKT Phases of Two-Comp onent Spin Mo dels . 2022. arXiv: 2110.09498 [math] . Pre-published. [Aiz82] Mic hael Aizenman. “ Geometric Analysis of φ 4 Fields and Ising Mo dels. P arts I and I I”. In: Communic ations in Mathematic al Physics 86.1 (1982), pp. 1–48. [AN04] K.B. Athrey a and P .E. Ney. Br anching Pr o c esses . Dov er Bo oks on Mathematics. Do v er Publications, 2004. [BEF86] J. Bricmont, A. El Mellouki, and J. F röhlich. “ Random surfaces in statistical mec hanics: roughening, rounding, wetting, . . . ”. In: J . Statist. Phys. 42.5-6 (1986), pp. 743–798. [Ber71] V. L. Berezinskii. “ Destruction of Long-Range Order in One-Dimensional and T wo-Dimensional Systems Having a Con tin uous Symmetry Group I. Classical Systems”. In: Soviet Physics JETP 32 (1971), pp. 493–500. [Ber72] V. L. Berezinskii. “ Destruction of Long-Range Order in One-Dimensional and T wo-Dimensional Systems Possessing a Contin uous Symmetry Group. I I. Quan tum Systems”. In: Soviet Physics JETP 34.3 (1972), pp. 610–616. [BPR24a] Roland Bauersc hmidt, Jiw o on P ark, and Pierre-F rançois Rodriguez. “ The Discrete Gaussian Mo del, I. Renormalisation Group Flow at High T emp erature”. In: The A nnals of Pr ob ability 52.4 (2024), pp. 1253–1359. [BPR24b] Roland Bauerschmidt, Jiw o on Park, and Pierre-F rançois Ro driguez. “ The Dis- crete Gaussian Mo del, I I. Inﬁnite-v olume Scaling Limit at High T emp erature”. In: The A nnals of Pr ob ability 52.4 (2024), pp. 1360–1398. [Bry78] Da vid C. Brydges. “ A Rigorous Approac h to Deby e Screening in Dilute Classical Coulom b Systems”. In: Communic ations in Mathematic al Physics 58.3 (1978), pp. 313–350. [BS24] Jeanne Boursier and Sylvia Serfaty. Dip ole F ormation in the Two-Comp onent Plasma . 2024. Pre-published. [BS25] Jeanne Boursier and Sylvia Serfat y . Multip ole and Ber ezinskii-Kosterlitz-Thouless T r ansitions in the Two-c omp onent Plasma . 2025. Pre-published. 34 REFERENCES [BS99] J. v an den Berg and J. E. Steif. “ On the existence and nonexistence of ﬁnitary co dings for a class of random ﬁelds”. In: Ann. Pr ob ab. 27.3 (1999), pp. 1501– 1522. [Cha98] L. Cha y es. “ Discontin uit y of the spin-w a v e stiﬀness in the t wo-dimensional X Y mo del”. In: Comm. Math. Phys. 197.3 (1998), pp. 623–640. [DF22] Julien Dub édat and Hugo F alconet. R andom Clusters in the Vil lain and XY Mo dels . 2022. arXiv: 2210.03620 [math] . Pre-published. [DR T19] Hugo Duminil-Copin, Aran Raouﬁ, and Vincent T assion. “ Sharp Phase T ransi- tion for the Random-Cluster and P otts Mo dels via Decision T rees”. In: Annals of Mathematics 189.1 (2019), pp. 75–99. JSTOR: 10.4007/annals.2019.189. 1.2 . [DST17] Hugo Duminil-Copin, Vladas Sidoravicius, and Vincen t T assion. “ Contin uit y of the phase transition for planar random-cluster and Potts mo dels with 1 ≤ q ≤ 4 ”. In: Comm. Math. Phys. 349.1 (2017), pp. 47–107. [Dum+21] Hugo Duminil-Copin, Maxime Gagnebin, Matan Harel, Ioan Manolescu, and Vincen t T assion. “ Discontin uit y of the phase transition for the planar random- cluster and Potts mo dels with q > 4 ”. In: Ann. Sci. Éc. Norm. Sup ér. (4) 54.6 (2021), pp. 1363–1413. [Dum+22] Hugo Duminil-Copin, Karol Ka jetan Kozlo wski, Dmitry Krach un, Ioan Manolescu, and T atiana Tikhonovsk aia. “ On the Six-V ertex Mo del’s F ree Energy”. In: Com- munic ations in Mathematic al Physics 395.3 (2022), pp. 1383–1430. [Ehr33] P Ehrenfest. “ Phasenum w andlungen im ueblic hen und erw eiterten Sinn, classi- ﬁziert nach den en tsprec henden Singularitaeten des thermo dynamisc hen P oten- tiales”. In: Pr o c. R oyal A c ad. Amster dam 36 (153 1933). [EL23] Diederik v an Engelen burg and Marcin Lis. “ An Elemen tary Pro of of Phase T ransition in the Planar XY Mo del”. In: Communic ations in Mathematic al Physics 399.1 (2023), pp. 85–104. [EL25] Diederik v an Engelenburg and Marcin Lis. “ On the Duality b etw een Height F unctions and Con tinuous Spin Mo dels”. In: Pr ob ability and Mathematic al Physics 6.4 (2025), pp. 1291–1325. [FK G71] C. M. F ortuin, P . W. Kasteleyn, and J. Ginibre. “ Correlation Inequalities on Some Partially Ordered Sets”. In: Communic ations in Mathematic al Physics 22.2 (1971), pp. 89–103. [F rö82] Jürg F röhlic h. “ On the T rivialit y of λφ 4 d Theories and the Approac h to the Critical Poin t in d ≥ 4 Dimensions”. In: Nucle ar Physics B 200.2 (1982), pp. 281–296. [FS81] Jürg F röhlich and Thomas Sp encer. “ The Kosterlitz-Thouless T ransition in T w o- Dimensional Ab elian Spin Systems and the Coulom b Gas”. In: Communic ations in Mathematic al Physics 81.4 (1981), pp. 527–602. [FSS76] J. F röhlic h, B. Simon, and Thomas Sp encer. “ Infrared Bounds, Phase T ransi- tions and Contin uous Symmetry Breaking”. In: Communic ations in Mathemat- ic al Physics 50.1 (1976), pp. 79–95. [Gin88] P . Ginsparg. “ Curiosities at c = 1”. In: Nucle ar Physics B 295.2 (1988), pp. 153– 170. [GP23] Agelos Georgak opoulos and Christoforos Panagiotis. “ Analyticit y results in Bernoulli p ercolation”. In: Mem. Amer. Math. So c. 288.1431 (2023), pp. v+102. [Gri69] Rob ert B. Griﬃths. “ Nonanalytic Beha vior Ab ov e the Critical Poin t in a Random Ising F erromagnet”. In: Phys. R ev. L ett. 23 (1 1969), pp. 17–19. [GS22] Christophe Garban and Thomas Spencer. “ Contin uous Symmetry Breaking along the Nishimori Line”. In: Journal of Mathematic al Physics 63.9 (2022), p. 093302. REFERENCES 35 [GS23a] Christophe Garban and A velio Sepúlveda. “ Quan titativ e Bounds on V ortex Fluctuations in 2 d Coulom b Gas and Maximum of the Integer-V alued Gaussian F ree Field”. In: Pr o c e e dings of the L ondon Mathematic al So ciety 127.3 (2023), pp. 653–708. [GS23b] Christophe Garban and A v elio Sepúlveda. “ Statistical Reconstruction of the GFF and KT T ransition”. In: Journal of the Eur op e an Mathematic al So ciety 26.2 (2023), pp. 639–694. [HS00] Olle Häggström and Jeﬀrey E. Steif. “ Propp-Wilson algorithms and ﬁnitary co dings for high noise Marko v random ﬁelds”. In: Combin. Pr ob ab. Comput. 9.5 (2000), pp. 425–439. [HS22] Matan Harel and Yinon Spink a. “ Finitary co dings for the random-cluster mo del and other inﬁnite-range monotone mo dels”. In: Ele ctr on. J. Pr ob ab. 27 (2022), P ap er No. 51, 32. [KP17] Vital Kharash and Ron P eled. The F r öhlich-Sp enc er Pr o of of the Ber ezinskii- Kosterlitz-Thouless T r ansition . 2017. arXiv: 1711 . 04720 [math-ph] . Pre- published. [KT73] J. M. K osterlitz and D. J. Thouless. “ Ordering, Metastability and Phase T ransitions in T wo-Dimensional Systems”. In: Journal of Physics C: Solid State Physics 6.7 (1973), p. 1181. [Lam22] Piet Lammers. “ Heigh t F unction Delo calisation on Cubic Planar Graphs”. In: Pr ob ability The ory and R elate d Fields 182.1 (2022), pp. 531–550. [Lam23a] Piet Lammers. A Dichotomy The ory for Height F unctions . 2023. arXiv: 2211. 14365 [math] . Pre-published. [Lam23b] Piet Lammers. Bije cting the BKT T r ansition . 2023. arXiv: 2301.06905 [math] . Pre-published. [LO24] Piet Lammers and Sébastien Ott. “ Delocalisation and Absolute-V alue-FK G in the Solid-on-Solid Mo del”. In: Pr ob ability The ory and R elate d Fields 188.1 (2024), pp. 63–87. [LS16] Ey al Lub etzky and Allan Sly. “ Information p ercolation and cutoﬀ for the sto c hastic Ising mo del”. In: J. A mer. Math. So c. 29.3 (2016), pp. 729–774. [LSS97] T. M. Liggett, R. H. Sc honmann, and A. M. Stacey. “ Domination b y pro duct measures”. In: Ann. Pr ob ab. 25.1 (1997), pp. 71–95. [Lup16] Titus Lupu. “ F rom Lo op Clusters and Random In terlacements to the F ree Field”. In: The Annals of Pr ob ability 44.3 (2016), pp. 2117–2146. [L Y52a] T. D. Lee and C. N. Y ang. “ Statistical theory of equations of state and phase transitions. I. Theory of condensation”. In: Phys. R ev. (2) 87 (1952), pp. 404– 409. [L Y52b] T. D. Lee and C. N. Y ang. “ Statistical theory of equations of state and phase transitions. I I. Lattice gas and Ising mo del”. In: Phys. R ev. (2) 87 (1952), pp. 410–419. [MO94] F. Martinelli and E. Olivieri. “ Approach to equilibrium of Glaub er dynamics in the one phase region. I. The attractive case”. In: Comm. Math. Phys. 161.3 (1994), pp. 447–486. [MS77] Oliv er A. McBryan and Thomas Sp encer. “ On the Decay of Correlations in SO(n)-Symmetric F erromagnets”. In: Communic ations in Mathematic al Physics 53.3 (1977), pp. 299–302. [MW66] N. D. Mermin and H. W agner. “ Absence of F erromagnetism or Antiferromag- netism in One- or T wo-Dimensional Isotropic Heisen berg Mo dels”. In: Phys. R ev. L ett. 17 (22 1966), pp. 1133–1136. [OS25] Sébastien Ott and Florian Sc h weiger. Quantitative Delo c alization for Solid-on- Solid Mo dels at High T emp er atur e and Arbitr ary Tilt . 2025. arXiv: 2505.16804 [math] . Pre-published. 36 REFERENCES [Ott20a] Sébastien Ott. W e ak mixing and analyticity in Random-Cluster and low tem- p er atur e Ising mo dels . 2020. arXiv: 2003.05879 [ma th-ph] . [Ott20b] Sébastien Ott. “ W eak mixing and analyticity of the pressure in the Ising mo del”. In: Comm. Math. Phys. 377.1 (2020), pp. 675–696. [Pra83] Chetan Prak ash. “ High-temperature diﬀeren tiabilit y of lattice Gibbs states by Dobrushin uniqueness tec hniques”. In: J. Statist. Phys. 31.1 (1983), pp. 169– 228. [PS22] Christoforos Panagiotis and F ranco Sev ero. “ Analyticit y of Gaussian free ﬁeld p ercolation observ ables”. In: Comm. Math. Phys. 396.1 (2022), pp. 187–223. [PW96] James Gary Propp and Da vid Bruce Wilson. “ Exact sampling with coupled Mark o v c hains and applications to statistical mechanics”. In: Pr o c e e dings of the Seventh International Confer enc e on Random Structur es and Algorithms (Atlanta, GA, 1995) . V ol. 9. 1-2. 1996, pp. 223–252. [Sep22] A velio Sepúlv eda. “ Scaling Limit of the Discrete Coulomb Gas, Based on Join t W ork with Christophe Garban”. Random Geometry and Statistical Ph ysics (Online Seminar). 2022. [She05] Scott Sheﬃeld. “ Random Surfaces”. In: Astérisque 304 (2005). [SK66] H. E. Stanley and T. A. Kaplan. “ P ossibility of a Phase T ransition for the T wo-Dimensional Heisenberg Mo del”. In: Physic al R eview L etters 17.17 (1966), pp. 913–915. [Spi20] Yinon Spink a. “ Finitary co dings for spatial mixing Marko v random ﬁelds”. In: A nn. Pr ob ab. 48.3 (2020), pp. 1557–1591. [Y an87] W ei-Shih Y ang. “ Deb y e Screening for T w o-Dimensional Coulomb Systems at High T emp eratures”. In: Journal of Statistic al Physics 49.1 (1987), pp. 1–32. CNRS, Sorbonne Université, LPSM Email addr ess : dalimonte@lpsm.paris CNRS, Sorbonne Université, LPSM Email addr ess : piet.lammers@cnrs.fr

Free energy analyticity of the disordered XY model and Debye screening in the 2D Coulomb gas

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment