On the absence of time-translation symmetry breaking in some non-reversible interacting particle systems

ON THE ABSENCE OF TIME-TRANSLA TION SYMMETR Y BREAKING IN SOME NON-REVERSIBLE INTERA CTING P AR TICLE SYSTEMS JONAS K ¨ OPPL Abstract. The conditions under whic h sto c hastic systems of infinitely many in ter- acting particles can main tain sufficien t spatial order to mo v e coheren tly along a time- p eriodic orbit, thereby breaking the time-translation inv ariance of the underlying dy- namical equation, hav e b een an elusive issue. Via a free energy technique, we prov e that if a non-rev ersible in teracting particle system on Z d , d = 1 , 2, with strictly positive rates admits a pro duct measure as a stationary measure, then it cannot exhibit time- p eriodic b ehaviour. This provides a first step to wards a general conjecture that time- p eriodic b ehaviour cannot o ccur in one- and t wo-dimensional systems with short-range in teractions and constitutes the first result for non-reversible dynamics in dimension t wo. 1. Introduction W e consider in teracting particle systems on the d -dimensional in teger lattice Z d , whic h are Mark o v pro cesses on the state space Ω = { 1 , . . . , q } Z d sp ecified in terms of generators of the form L f ( η ) = X x ∈ Z d X ξ x c x ( η , ξ x ) [ f ( ξ x η x c ) − f ( η )] , η ∈ Ω , for lo cal functions f : Ω → R . The transition rates c x ( η , ξ x ) can be interpreted as the infinitesimal rate at whic h the particle at site x switc hes from the state η x to ξ x , given that the rest of the system is curren tly in state η x c . W e will denote the asso ciated semigroup by ( P t ) t ≥ 0 and the set of probabilit y measures on Ω by M 1 (Ω). The question w e are primarily in terested in is which symmetries of the transition rates are inherited b y the stationary b ehaviour of the dynamics. As an example, consider the Glaub er dynamics for the Ising mo del. Then it is w ell kno wn that the transition rates are b oth inv arian t under a global spin flip and lat- tice translations. Ho w ev er, in d ≥ 2 at sufficiently lo w temp eratures there exist time- stationary measures whic h are not in v ariant under global spin flips and in d ≥ 3 there exist non-translation-in v arian t time-stationary measures. So these t w o symmetries can b e spontaneously brok en in infinite volume systems. Another ob vious and hence often o v erlo ok ed symmetry is that the transition rates do not depend on time, i.e., the gen- erator is autonomous, and thus in v ariant under time shifts. Of course, trivially any time-stationary measure is also inv arian t under time shifts and one needs to b e a bit more careful when defining the notion of time-translation symmetry breaking. W e will sa y that (sp ontane ous) time-tr anslation symmetry br e aking o ccurs if Date : F ebruary 26, 2026. 2020 Mathematics Subje ct Classific ation. Primary 82C22; Secondary 60K35. Key wor ds and phr ases. Interacting particle systems, free energy , relative entrop y , attractor, time- translation symmetry breaking. 1 TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 2 (TTSB) ∃ µ 0 ∈ M 1 (Ω) ∃ τ > 0 suc h that µP τ = µ 0 and for all s ∈ (0 , τ ) w e hav e µP s  = µ 0 . In other words, there exists a non-trivial time-p erio dic orbit ( µP s ) s ∈ [0 ,τ ] in the space M 1 (Ω) of probability measures on Ω. This means that the usual contin uous shift sym- metry by any t > 0 is reduced to a discrete symmetry by t ∈ τ N . First note, that if an in teracting particle system with strictly p ositive transition rates can break the time-translation symmetry then, as a consequence of the ergo dic theorem for con tin uous-time Mark o v c hains on finite state spaces, this can only happen in infinite v olumes, so it is necessarily a many-bo dy phenomenon that arises due to interactions. The conditions under which suc h noisy systems can main tain sufficien t spatial order to mo v e coheren tly along a time-p erio dic orbit, thereb y spontaneously breaking the time- translation inv ariance of the underlying dynamical equation, ha v e been a con ten tious issue in the ph ysics literature [ GMSB93 , BGH + 90 , CM92 ], and a mathematical treatmen t of suc h brok en-symmetry , macroscopically time-dep endent states has b een mostly limited to mean-field like approximations [ CFT16 ]. As it turns out, the p ossibility or non-p ossibility of time-translation symmetry break- ing app ears to hea vily dep end on the dimension d . Non-rigorous arguments [ GMSB93 ] and extensiv e numerical studies of specific mo dels [ AFM + 25b , AFM + 25a , GB24 ] seem to suggest that interacting particle systems with short-range interactions can exhibit ( TTSB ) in dimensions d ≥ 3, but cannot produce stable time-p erio dic b ehaviour in dimensions d = 1 , 2. Mathematically , the literature regarding this issue is muc h scarcer. While a classical result b y Mountford and Ramirez–V aradhan, see [ Mou95 ] and [ R V96 ], sho ws that no suc h system with finite-range transition rates can exist in one spatial dimension, re- cen t constructions show that systems with long-range interactions can indeed exhibit ( TTSB ), even in dimension one and tw o, see [ JK25b ] for d = 1 , 2 and [ JK14 ] for d ≥ 3. Recen tly , in [ JK25a ], it was additionally shown that in one and t w o dimensions, an y r eversible short-range system cannot exhibit ( TTSB ). Since the examples in [ JK25b ] heavily rely on the long-range nature of the in terac- tions, the current theoretical evidence leads to the same conjecture as in the physics literature: ( TTSB ) cannot occur in one- and t w o-dimensional systems with short-range in teractions, ev en in non-reversible systems. In this article, we wan t to take a first step to w ards verifying this conjecture and in v estigate the p ossible long-time b ehaviour of in teracting particle systems in one and t w o spatial dimensions without the additional ass umption of reversibilit y , at least for a sp ecific class of systems. F or this, w e com bine the pro of strategy of [ JK25a ] together with key ideas from [ JK23 ] and [ Ram02 ] that allow us to go b eyond rev ersible systems. Organisation of the man uscript. In Section 2 , we introduce the precise framew ork in whic h we are w orking and setup the required notation b efore we state our main results. W e then comment on p ossible extensions to more general non-reversible systems in dimensions d = 1 , 2 in Section 3 . After this we finally start with the main work and pro vide an outline of the pro of strategy in Section 4 b efore diving into the details in Section 5 . 2. Setting and main resul ts Let q ∈ N and consider the configuration space Ω := Ω Z d 0 = { 1 , . . . , q } Z d , which w e will equip with the usual product top ology and the corresp onding Borel sigma-algebra F . F or Λ ⊂ Z d let F Λ b e the sub-sigma-algebra of F that is generated by the op en sets TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 3 in Ω Λ := { 1 , . . . , q } Λ . W e will use the shorthand notation Λ ⋐ Z d to signify that Λ is a finite subset of Z d . In the follo wing w e will often denote for a given configuration ω ∈ Ω b y ω Λ its pro jection to the volume Λ ⊂ Z d and write ω Λ ω ∆ for the configuration on in Λ ∪ ∆ comp osed of ω Λ and ω ∆ for disjoin t Λ , ∆ ⊂ Z d . F or the sp ecial case Λ = { x } we will also write x c = Z d \ { x } and ω x ω x c . The set of probability measures on Ω will b e denoted by M 1 (Ω) and the space of contin uous functions by C (Ω). F or a configuration η ∈ Ω w e will denote by η x,i the configuration that is equal to η everywhere except at the site x where it is equal to i . Moreov er, for Λ ⊂ Z d w e will denote the corresp onding cylinder sets by [ η Λ ] = { ω : ω Λ ≡ η Λ } . Whenev er we are taking the probabilit y of such a cylinder even t with resp ect to some measure ν ∈ M 1 (Ω), we will omit the square brac k ets and simply write ν ( η Λ ). 2.1. In teracting particle systems. W e will consider time-contin uous Mark o vian dy- namics on Ω, namely in teracting particle systems characterized by time-homogeneous generators L with domain dom( L ) and its asso ciated Mark o v semigroup ( P t ) t ≥ 0 . F or in teracting particle systems we adopt the notation and exp osition of the classical text- b o ok [ Lig05 , Chapter 1]. In our setting, the generator L is given via a collection of single-site transition rates c x ( η , ξ x ), which are con tin uous in the starting configuration η ∈ Ω. These rates can be in terpreted as the infinitesimal rate at whic h the particle at site x switches from the state η x to ξ x , giv en that the rest of the system is currently in state η x c . The full dynamics of the interacting particle system is then given as the sup erp osition of these lo cal dynamics, i.e., L f ( η ) = X x ∈ Z d X ξ x c x ( η , ξ x )[ f ( ξ x η x c ) − f ( η )] . In [ Lig05 , Chapter 1] it is sho wn that the follo wing tw o conditions are sufficien t to guaran tee the well-definedness. (L1) The rate at which the particle at a particular site c hanges its spin is uniformly b ounded, i.e., sup x ∈ Z d X ξ x ∥ c x ( · , ξ x ) ∥ ∞ < ∞ (L2) and the total influence of all other particles on a single particle is uniformly b ounded, i.e., sup x ∈ Z d X y  = x X ξ x δ y ( c x ( · , ξ x )) < ∞ , where δ y ( f ) := sup η ,ξ : η y c = ξ y c | f ( η ) − f ( ξ ) | is the oscillation of a function f : Ω → R at the site y ∈ Z d . Under these conditions, one can then sho w that the op erator L , defined as ab ov e, is the generator of a w ell-defined Mark o v pro cess and that a core of L is given by the space of functions with finite total oscillation, i.e. D (Ω) := n f ∈ C (Ω) : X x ∈ Z d δ x ( f ) < ∞ o . Let us emphasise briefly that w e will not assume translation-inv ariance of the rates. TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 4 2.2. Relativ e entrop y loss. F or µ, ν ∈ M 1 (Ω) and a finite volume Λ ⋐ Z d define the relativ e en trop y of ν with resp ect to µ in Λ via h Λ ( ν | µ ) := ( P ω Λ ∈ Ω Λ ν ( ω Λ ) log ν ( ω Λ ) µ ( ω Λ ) , if ν Λ ≪ µ Λ , ∞ , else, where we use the con v en tion that 0 log 0 = 0. No w recall that ( P t ) t ≥ 0 denotes the Mark o v semigroup corresp onding to the Marko v generator L . W e write ν t := ν P t for the time-evolv ed measure ν ∈ M 1 (Ω). The finite-volume relativ e entrop y loss in Λ ⋐ Z d is defined by g L Λ ( ν | µ ) := d dt    t =0 h Λ ( ν t | µ ) . Usually , one then works with the density limits of the relative en trop y and the relativ e en trop y loss and sho ws that the latter can still be used as a Ly apuno v function for the dynamics. How ev er, the sub-additivity arguments that are used to sho w that the relativ e en trop y loss density actually exists as a limit are only a v ailable for translation-inv arian t measures. W e do not wan t to make any suc h assumptions and therefore we will hav e to instead work with the family of finite-volume relative en trop y losses. Note that calling the finite-volume deriv ativ es loss is not entirely correct, since they are not necessarily non-p ositiv e. Ho w ev er one can sho w that the positive con tributions are of b oundary order and v anish in the densit y limit, see [ JK23 , Lemma 21 and Lemma 23]. 2.3. Time-stationary measures, orbits, and the attractor. If one is in terested in the long-term behaviour of an in teracting particle system, a natural ob ject to study is the so-called attr actor of the measure-v alued dynamics which is defined as A = n ν ∈ M 1 (Ω) : ∃ ν 0 ∈ M 1 (Ω) and t n ↑ ∞ such that lim n →∞ ν t n = ν o . This is the set of all accum ulation p oints of the measure-v alued dynamics induced b y L . In the language of dynamical systems this is the ω -limit set. This enco des (most of ) the dynamically relev ant information about the long-time behaviour of the system. In this article, we are particularly in terested in tw o subsets of the attractor, namely the time-stationary me asur es giv en by S := { ν ∈ M 1 (Ω) : ∀ s ≥ 0 : ν P s = ν } , and the measures which lie on a stationary orbit O := { ν ∈ M 1 (Ω) : ∃ T > 0 : ν P T = ν } . The relation b etw een these sets can b e summarised as follows S ⊂ O ⊂ A . In general, the first inclusion is strict as can b e seen b y considering the non-trivial examples constructed in [ JK14 ] and [ JK25b ] or the (from a probabilistic p oint of view) trivial example given in [ Lig05 , p.12]. Historically , most atten tion has b een paid to in v estigating the set of time-stationary measures and their prop erties, but not muc h w as kno wn ab out the b ehaviour of in teracting particle systems outside of this set. TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 5 2.4. Main results. Before w e can state our main results, let us introduce some stronger conditions on the transition rates ( c x ( · , ξ x )) x ∈ Z d ,ξ x ∈ Ω 0 that will turn out to be crucial for our results. Conditions for the rates. (R1) The rate at which the particle at a particular site changes its spin is uniformly b ounded, i.e., sup x ∈ Z d X ξ x ∥ c x ( · , ξ x ) ∥ ∞ < ∞ . (R2) F or ev ery x ∈ Z d and ξ x ∈ Ω 0 the function Ω ∋ η 7→ c x ( η , ξ x ) ∈ [0 , ∞ ) is contin uous. (R3) The transition rates are b ounded aw a y from zero, i.e., inf x ∈ Z d , η ∈ Ω , ξ x ∈ Ω 0 c x ( η , ξ x ) =: c > 0 . (R4) W e ha v e X y ∈ Z d | y | sup x ∈ Z d δ x + y c x ( · ) < ∞ , where c x ( η ) = P i  = η x c x ( η , i ) is the total rate at which the particle at site x c hanges its state when the system is in configuration η . The condition ( R4 ) essentially ensures that the specification and the transition rates are short-r ange and is satisfied if the dep endence of c x on the spin at site y ∈ Z d deca ys faster than | x − y | − 2 d . Let us emphasise again that we do not assume that the rates translation-in v ariant. Our main result is the follo wing no-go result that essentially states that the presence of a pro duct measure as a fixed p oint for the measure-v alued dynamics makes it imp ossible to also p ossess perio dic orbits. Note that admitting a product measure as a time- stationary measure does not imply that the dynamics are in some sense trivial or non- in teracting. The example of kinetically constrained mo dels, see e.g. [ HT25 ], shows that ev en interacting particle system s with v ery strong in teractions can admit a product measure as a time-stationary measure. Theorem 2.1. L et d ∈ { 1 , 2 } and assume that L is the gener ator of an inter acting p article system that satisfies assumptions ( R1 ) − ( R4 ) and that it admits a pr o duct me asur e µ as time-stationary me asur e.Then we have that O = S = { µ } . While the first identit y can b e seen as the main result of this pap er, the second iden tity also extends the results of [ Ram02 ] from binary to general finite lo cal state spaces and to unbounded range interactions up until p ow er-law deca y with exp onent α > 2 d , see assumption ( R4 ). This structure theorem in particular implies the follo wing no-go result that essen tially states that the presence of a pro duct measure as fixed p oint for the measure-v alued dynamics mak es it imp ossible to also p ossess p erio dic orbits. The precise formulation is as follows. TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 6 Corollary 2.2. L et d ∈ { 1 , 2 } and assume that L satisfies the ab ove assumptions and that ( P t ) t ≥ 0 is the Markov semigr oup gener ate d by L with some pr o duct me asur e µ as time-stationary me asur e. Then, the me asur e-value d dynamics given by [0 , ∞ ) × M 1 (Ω) ∋ ( t, ν ) 7→ ν P t ∈ M 1 (Ω) do es not c ontain non-trivial time-p erio dic orbits, i.e., ther e is no pr ob ability me asur e ν ∈ M 1 (Ω) such that ( ν P t ) t ≥ 0 is non-c onstant and such that ther e exists a T > 0 with ν t = ν t + T . The proof strategy for Theorem 2.1 and commen ts on p ossible extensions to more general time-stationary measures µ is discussed in Section 4 . 3. Outlook Before diving into pro of of the main result, let us briefly commen t on some op en problems related to time-p erio dic b eha viour in interacting particle systems and p ossible strategies to establish (parts of ) the conjecture mentioned in the introduction. 3.1. T o w ards a no-go theorem in d = 1 , 2 . At least on an in tuitiv e lev el one could be led to b elieve that the stationary measures of in teracting particle system with sufficiently nice rates, say finite-range and strictly p ositive, should also b e somewhat regular. F rom this persp ective, it is not muc h of a stretch to conjecture that generically there should b e at least one stationary measure µ which is quasilocal, i.e., it admits a v ersion of its lo cal conditional distributions µ Λ [ ·|· ] whic h is c ontinuous with r esp e ct to the b oundary c ondition . In the present setting of finite local state spaces this in particular implies that µ is a Gibbs measure for some absolutely summable Hamiltonian H . The set of translation-in v ariant Gibbs measures for a Hamiltonian H will b e denoted by G θ ( H ). Assuming that this quasilo calit y is generically given, if one now restricts attention to the translation-in v ariant case, then the following prop osition is a consequence of a more general result in [ JK23 ]. Prop osition 3.1. If a tr anslation-invariant inter acting p article system with gener ator L that satisfies ( L1 ) − ( L2 ) and ( R3 ) admits a time-stationary Gibbs me asur e µ ∈ G θ ( H ) , wher e H is absolutely summable, then any time-p erio dic orbit that c onsists of tr anslation-invariant me asur es is ne c essarily c ontaine d in G θ ( H ) . By using an argument from [ K ¨ u84 ] based on prop erties of extremal measures we obtain the follo wing connection b et w een time-p erio dic b eha viour and a prop ert y of the set of extremal Gibbs measures. Prop osition 3.2. If a tr anslation-invariant inter acting p article system satisfying the c onditions ( L1 ) − ( L2 ) and ( R3 ) admits a time-stationary me asur e µ ∈ G θ ( H ) , wher e H is absolutely summable, and if ex G θ ( H ) is finite, then every ν ∈ G θ ( H ) is time-stationary for the dynamics. In p articular, ther e c an b e no non-trivial time-p erio dic orbits. Pr o of. A translation-inv ariant Gibbs measure ν is an extremal p oint if and only if it is ergo dic with resp ect to spatial shifts. Moreov er, by [ Lig05 , Theorem I.4.15] if ν 0 is ergo dic, then so is ν t for every t ≥ 0. So if we choose an extremal Gibbs measure ν as initial condition, then by monotonicit y of the free energy densit y , see [ JK23 , Theorem 6], we will remain in the set of extremal Gibbs measures for all times t ≥ 0. But since ( ν t ) t ≥ 0 is contin uous as a function of time and ex G θ ( H ) is finite and in particular totally disconnected, this can only b e the case if ν t = ν for all times t ≥ 0. By using the extremal decomp osition for Gibbs measures, see e.g. [ FV17 , Section 6.8.4], the time-stationarit y of all extremal translation-in v arian t Gibbs measures implies that any TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 7 translation-in v ariant Gibbs measure is time-stationary for the pro cess. In particular, there can b e no non-trivial time-p erio dic orbits. □ There is a general conjecture, see e.g. [ VE14 ], that in tw o dimensions finite-range mo dels should alwa ys p ossess a finite num b er of extremal Gibbs states, all of which are translation-inv arian t. In particular, this would imply that all Gibbs states are translation-in v ariant. This would then also imply that there can b e no non-trivial time-p erio dic b eha viour in tw o-dimensional interacting particle systems that admit a time-stationary Gibbs measure. In [ DS85 ], the authors make use of Pirogo v–Sinai the- ory to show that at least for v ery low temp eratures, this general conjecture holds. But the intermediate temp erature regime is still op en. F or the P otts mo del, the main re- sult of [ CDCIV14 ] in particular implies that for an y temp erature β ≥ 0, there are only finitely man y extremal Gibbs measures, but the technique do es not easily extend to more general situations. In higher dimensions, this do es not need to b e the case as e.g. the example con- structed in [ Sla74 , Section 4] shows. Indeed, despite only having a finite lo cal state space and finite-range interactions in d ≥ 3, there is not just an infinite but even an unc ountable n um b er of translation-inv arian t extremal Gibbs me asures at sufficiently lo w temp eratures. T o sum up the preceding discussion, at least in the translation-inv ariant case, one p ossible strategy to rule out stable time-perio dic orbits in one and t w o dimensions w ould b e to show that (a) an y sufficien tly nice interacting particle system admits a time- stationary Gibbs measure (with a sufficiently regular Hamiltonian) and (b) for sufficiently nice Hamiltonians, the set of translation-in v ariant extremal Gibbs measures is alwa ys finite in tw o dimensions. 3.2. T o w ards a minimal example. While the examples constructed in [ JK14 ] and [ JK25b ] sho w that ( TTSB ) is indeed p ossible in finite dimensions, even in non-degenerate in teracting particle systems, their definition is quite implicit and not particularly well suited for further analysis. Therefore, it is desirable to find a sufficiently simple to y mo del, that (a) exhibits time-p erio dic b ehaviour, (b) is amenable to mathematical anal- ysis, and (c) is non-trivial. Numerical simulations suggest that p erio dic b ehavior is not a rare phenomenon. Sev- eral interacting particle systems with explicit dynamics are kno wn to exhibit p erio dic b eha vior in simulations, at least in three and higher dimensions. Three notable examples are the nonr e cipr o c al Ising mo del , see e.g.[ CFT16 , AFM + 25b , GB24 ], the cycle c onform mo del , see [ Swa26 , Section 1.9], and (self-)driven clo ck mo dels , see [ MS11 , WY26 ]. One feature all of these examples share is that they are all non-reversible. While the nu- merical evidence for the existence of time-p erio dic b ehaviour in all these mo dels is quite substan tial, at the moment only their mean-field counterparts can b e rigorously sho wn to admit time-p erio dic solutions. 4. Proof stra tegy The pro of strategy for Theorem 2.1 is inspired b y [ JK25a ] and essen tially consists of t w o main steps that w e will no w explain briefly b efore w e start with the actual mathematics. Let us already note that apart from the very last step in the proof of Prop osition 4.1 all the tec hnical results derived in the forthcoming sections hold in an y dimension d ∈ N and can therefore b e used for future inv estigations in arbitrary dimensions. Moreo v er, only one building blo ck, namely Lemma 5.6 , seems to require the pro duct structure of µ . TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 8 4.1. Finite-v olume relative entrop y loss and stationary orbits. The first tech- nical result is the follo wing time-aver age d version of the results in [ Ram02 ] which also extends the classical result beyond finite-range interactions and from binary to general finite lo cal state spaces. The main difference to [ JK25a , Prop osition 4.1] is that we replace the assumption of rev ersibilit y , whic h provides a lo cal relation b etw een the rates and the rev ersible measure, b y the assumption that the pro cess lea v es some pro duct measure inv ariant. Prop osition 4.1 (Time-av eraged entrop y loss principle) . Assume that d ∈ { 1 , 2 } , that L satisfies ( R1 ) − ( R4 ) and admits a stationary me asur e µ which is a pr o duct me asur e. If ν ∈ M 1 (Ω) is such that (M1) for al l η ∈ Ω , Λ ⋐ Z d , and s ≥ 0 it holds that ν P s ( η Λ ) > 0 , and (M2) for al l Λ ⋐ Z d we have R T 0 g L Λ ( ν P s | µ ) ds = 0 , then we have ν = µ . The pro of of this can b e found in Section 5 and morally follo ws a similar strategy as in [ JK25a ] but the details are quite different, since w e do not ha v e reversibilit y at our hands. Therefore, we make use of a differen t rewriting of the finite-v olume relativ e en trop y loss which first app eared in [ Ram02 ]. Since condition ( M2 ) is directly implied by p erio dicity and the fundamental theorem of calculus, our main result Theorem 2.1 will follo w b y sho wing that ( M1 ) is satis- fied along time-p erio dic orbits. But this was already established in [ JK25a ]. Let us nev ertheless recall the main ideas. 4.2. The p ositiv e-mass prop erty. In the simple setting of an irreducible contin uous- time Marko v chain on finite state space X , it is easy to show that there exist constants ρ, τ > 0 such that for an y initial distribution ν and all states x ∈ X and times t ≥ τ > 0, the probability of b eing in state x at time t is at least ρ , i.e, we ha v e ν t ( x ) ≥ ρ > 0. In the setting of infinite-v olume in teracting particle systems that are irreducible in a suitable wa y , an analogous prop ert y should b e true, but here it is not as straightforw ard to see. The key idea to make this in tuition precise for infinite volume systems is to compare the dynamics to an interacting particle system in which all of the sites inside of a finite v olume Λ b eha v e indep enden tly and flip with the minimal transition rate. A Girsanov- t yp e formula then allows one to compare this finite-v olume p erturbation with the original system and yields the following result. Prop osition 4.2 (Positiv e-mass prop erty) . Assume that the r ates of an inter acting p article systems satisfy assumptions ( R1 ) − ( R4 ) . Then, for al l τ > 0 and Λ ⋐ Z d , ther e exists a c onstant C ( τ , Λ) > 0 such that, for any starting me asur e ν and any time t ∈ [ τ , ∞ ) , we have ∀ η ∈ Ω ν t ( η Λ ) ≥ C ( τ , Λ) . In p articular, for al l subse quential limits ν ∗ = lim n →∞ ν t n with t n ↑ ∞ , we have ∀ η ∈ Ω ∀ Λ ⋐ Z d ν ∗ ( η Λ ) ≥ C ( τ , Λ) > 0 . The proof of this can be found in [ JK25a ]. Note that this in particular implies that the condition ( M1 ) holds along time-perio dic orbits and that the positive-mass prop erty can b e in terpreted as a somewhat quantitativ e version of the noisy nature of the dynamics. Ev en if w e start our process with a p oint mass δ ω at some configuration ω ∈ Ω as initial condition, the distribution of the pro cess at any p ositiv e time t > 0 will already put p ositiv e mass on any cylinder set [ η Λ ]. TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 9 5. Proof of the time-a veraged rela tive entr opy loss principle Unless stated otherwise, we will from now on alw a ys assume that the transition rates of our dynamics satisfy ( R1 ) − ( R4 ). 5.1. Rewriting the finite-volume relativ e en trop y loss. W e start our pro of b y obtaining a more con v enient representation of the relative en trop y loss in finite v olumes Λ ⋐ Z d . Recall that the relative entrop y loss in the finite v olume Λ is defined b y g L Λ ( ν | µ ) = d dt | t =0 h Λ ( ν | µ ) , ν ∈ M 1 (Ω) . The first step is to rewrite this in a suitable w a y that will allow us to separate the (negativ e) contribution from the bulk from the (p ossibly p ositive) finite v olume error terms. The expression w e deriv e here is differen t from the one in [ JK25a , Lemma 5.3] and more similar to the one in [ JK23 , Lemma 23]. Ho w ev er, in the latter reference, w e w ere just interested in the rough asymptotics of the error term and prov ed that it is of b oundary order. This is not sufficient for our goal in this article and we therefore hav e to keep track of the error term and will later mak e use of its explicit form. Lemma 5.1. F or Λ ⋐ Z d and ν ∈ M 1 (Ω) with ν ( η Λ ) > 0 for al l η Λ ∈ Ω Λ we have g L Λ ( ν | µ ) = − X η Λ X x ∈ Λ X j  = η x F ν ( η Λ ) ν ( η x,j Λ ) µ ( η x,j Λ ) µ ( η Λ ) ! µ ( η Λ ) µ ( η x,j Λ ) ν ( η x,j Λ ) ν ( η Λ ) Z η Λ c x ( ω , j ) ν ( dω ) − X η Λ X x ∈ Λ X j  = η x " Z η Λ c x ( ω , j ) ν ( dω ) − µ ( η Λ ) µ ( η x,j Λ ) ν ( η x,j Λ ) ν ( η Λ ) Z η Λ c x ( ω , j ) ν ( dω ) # wher e we use the notation F ( x ) = x log x − x + 1 for x ≥ 0 with the c onvention that 0 log 0 = 0 . Pr o of. A direct calculation using the definition of the generator yields g L Λ ( ν | µ ) = X η Λ ν ( L 1 η Λ ) log  ν ( η Λ ) µ ( η Λ )  = X η Λ X x ∈ Λ X j Z c x ( ω , j )  1 η Λ ( ω x,j ) − 1 η Λ ( ω )  ν ( dω ) log  ν ( η Λ ) µ ( η Λ )  (5.1) = X η Λ X x ∈ Λ X i  = η x " Z [ η x,i Λ ] c x ( ω , η x ) ν ( dω ) − Z [ η Λ ] c x ( ω , i ) ν ( dω ) # log  ν ( η Λ ) µ ( η Λ )  , where we used that if j  = η x , then Z c x ( ω , j )  1 η Λ ( ω x,j ) − 1 η Λ ( ω )  ν ( dω ) = − Z [ η Λ ] c x ( ω , j ) ν ( dω ) , and if j = η x , then Z c x ( ω , j )  1 η Λ ( ω x,j ) − 1 η Λ ( ω )  ν ( dω ) = X i  = η x Z [ η x,i Λ ] c x ( ω , η x ) ν ( dω ) . By rearranging the summation slightly , the identit y ( 5.1 ) yields g L Λ ( ν | µ ) = X η Λ X x ∈ Λ X i  = η x log µ ( η Λ ) ν ( η x,i Λ ) ν ( η Λ ) µ ( η x,i Λ ) ! Z [ η Λ ] c x ( ω , i ) ν ( dω ) . By adding and subtracting suitable terms this now yields the claimed expression. □ TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 10 5.2. The time-a v eraged zero-loss estimate. The previously deriv ed representation of the relativ e en trop y loss in finite volumes Λ ⋐ Z d leads to the following inequality in case the time-av eraged relative entrop y loss v anishes. Lemma 5.2. L et Λ ⋐ Z d and ν ∈ M 1 (Ω) b e such that ther e exists T > 0 with i. ν P s ( η Λ ) > 0 for al l η ∈ Ω Λ and s ∈ [0 , T ] , ii. and R T 0 g L Λ ( ν P s | µ ) ds = 0 . Then for the c onstant c > 0 fr om ( R3 ) , that do es in p articular not dep end on Λ and ν , we have c 2 Z T 0 X η Λ X x ∈ Λ X j  = η x   v u u t ν s ( η x,j Λ ) µ ( η x,j Λ ) − s ν s ( η Λ ) µ ( η Λ )   2 µ ( η Λ ) ds ≤ 2 Z T 0 X η Λ X x ∈ Λ X j  = η x γ Λ ( x )      ν s ( η x,j Λ ) µ ( η x,j Λ ) − ν s ( η Λ ) µ ( η Λ )      µ ( η Λ ) ds, wher e we use the notation γ Λ ( x ) := X y / ∈ Λ q X j =1 δ y ( c x ( · )) . In the c ase wher e Λ = Λ n = [ − n, n ] d ∩ Z d , we wil l use the shorthand γ n inste ad of γ Λ n . F or the pro of of Lemma 5.2 we make use of the following technical help er which is reminiscen t of Leb esgue’s differen tiation theorem and already app eared in [ JK25a ]. It will also come in handy later, so w e record it for future reference and give a quic k pro of. Lemma 5.3 (Quantitativ e differentiation lemma) . L et ν b e a pr ob ability me asur e such that ν ( η Λ ) > 0 for al l η ∈ Ω and Λ ⋐ Z d . Then, for any function f : Ω → R with the pr op erty X x ∈ Z d δ x f < ∞ , the fol lowing uniform err or estimate holds for al l η ∈ Ω      1 ν ( η Λ ) Z [ η Λ ] f ( ω ) ν ( dω ) − f ( η )      ≤ X x / ∈ Λ δ x f . Pr o of. Fix η ∈ Ω and Λ ⋐ Z d . Then we can fix an enumeration of the v ertices in Λ c and write [ n ] = Λ ∪ { x 1 , . . . , x n } . By a telescop e sum we see f ( η ) − f ( ω ) = ∞ X n =1  f ( η [ n ] ω [ n ] c ) − f ( η [ n − 1] ω [ n − 1] c )  ≤ X x / ∈ Λ δ x f . The claim now follows via in tegration. □ TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 11 Pr o of of L emma 5.2 . F rom the represen tation deriv ed in Lemma 5.1 and the assumption that R T 0 g n L ( ν P s | µ ) ds = 0, we directly obtain the iden tit y Z T 0 X η Λ X x ∈ Λ X j  = η x F ν s ( η Λ ) ν s ( η x,j Λ ) µ ( η x,j Λ ) µ ( η Λ ) ! µ ( η Λ ) µ ( η x,j Λ ) ν s ( η x,j Λ ) ν s ( η Λ ) Z η Λ c x ( ω , j ) ν s ( dω ) ds (5.2) = − Z T 0 X η Λ X x ∈ Λ X j  = η x " Z η Λ c x ( ω , j ) ν s ( dω ) − µ ( η Λ ) µ ( η x,j Λ ) ν s ( η x,j Λ ) ν s ( η Λ ) Z η Λ c x ( ω , j ) ν s ( dω ) # ds. Since F ( x ) ≥ 1 2 (1 − √ x ) 2 for all x > 0, the left-hand side of this iden tit y can b e b ounded from b elow by Z T 0 X η Λ X x ∈ Λ X j  = η x F ν s ( η Λ ) ν s ( η x,j Λ ) µ ( η x,j Λ ) µ ( η Λ ) ! µ ( η Λ ) µ ( η x,j Λ ) ν s ( η x,j Λ ) ν ( η Λ ) Z η Λ c x ( ω , j ) ν s ( dω ) ds ≥ 1 2 Z T 0 X η Λ X x ∈ Λ X j  = η x   1 − v u u t ν s ( η Λ ) ν s ( η x,j Λ ) µ ( η x,j Λ ) µ ( η Λ )   2 µ ( η Λ ) µ ( η x,j Λ ) ν s ( η x,j Λ ) ν s ( η Λ ) Z η Λ c x ( ω , j ) ν s ( dω ) ds ≥ c 2 Z T 0 X η Λ X x ∈ Λ X j  = η x   v u u t ν s ( η x,j Λ ) µ ( η x,j Λ ) − s ν s ( η Λ ) µ ( η Λ )   µ ( η Λ ) ds T o deal with the righ t-hand side of the iden tit y ( 5.2 ) we first note that by stationarit y and non-nullness of µ we ha v e for any ρ ∈ M 1 (Ω) 0 = Z Ω L  dρ dµ    F Λ  ( ω ) µ ( dω ) = X η Λ X x ∈ Λ X j  = η x Z [ η Λ ] c x ( ω , j ) ρ ( η x,j Λ ) µ ( η x,j Λ ) − ρ ( η Λ ) µ ( η Λ ) ! µ ( dω ) (5.3) = − X η Λ X x ∈ Λ X j  = η x µ ( η Λ ) c Λ ,µ x ( η Λ , j ) " ρ ( η x,j Λ ) ν ( η x,j Λ ) − µ ( η Λ ) µ ( η x,j Λ ) # (5.4) where we use the notation c Λ ,ρ x ( η Λ , j ) := 1 ρ ( η Λ ) Z [ η Λ ] c x ( ω , j ) ρ ( dω ) , η Λ ∈ Ω Λ , for a probability measure ρ ∈ M 1 (Ω), Λ ⋐ Z d , and j ∈ Ω 0 . By using the same notation, the right-hand side of ( 5.2 ) can b e rewritten as Z T 0 X η Λ X x ∈ Λ X j  = η x µ ( η Λ ) c Λ ,ν s x ( η Λ , j ) ν s ( η Λ ) µ ( η Λ ) − ν s ( η x,j Λ ) µ ( η x,j Λ ) ! . So we can use ( 5.3 ) for every ν s , where s ∈ [0 , T ], and com bine it with ( 5.2 ) to obtain Z T 0 X η Λ X x ∈ Λ X j  = η x F ν s ( η Λ ) ν s ( η x,j Λ ) µ ( η x,j Λ ) µ ( η Λ ) ! µ ( η Λ ) µ ( η x,j Λ ) ν s ( η x,j Λ ) ν s ( η Λ ) Z η Λ c x ( ω , j ) ν s ( dω ) ds (5.5) = Z T 0 X η Λ X x ∈ Λ X j  = η x µ ( η Λ )  c Λ ,µ x ( η Λ , j ) − c Λ ,ν s x ( η Λ , j )  ν s ( η x,j Λ ) µ ( η x,j Λ ) − ν s ( η Λ ) µ ( η Λ ) ! . (5.6) TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 12 By Lemma 5.3 , we get the following uniform estimate for any ρ ∈ M 1 (Ω) and η Λ ∈ Ω Λ   c Λ ,µ x ( η Λ , j ) − c Λ ,ρ x ( η Λ , j )   ≤ 2 X y / ∈ Λ δ y ( c x ( · , j )) , x ∈ Λ , j ∈ Ω 0 . No w com bining this with a simple application of the triangle inequality to ( 5.5 ) yields the claimed estimate. □ Motiv ated by the inequality deriv ed in Lemma 5.2 , let us introduce the follo wing notation for ρ ∈ M 1 (Ω), Λ ⋐ Z d , and x ∈ Λ: α Λ ( x, ρ ) := X η Λ X j  = η x   v u u t ρ ( η x,j Λ ) µ ( η x,j Λ ) − s ρ ( η Λ ) µ ( η Λ )   2 µ ( η Λ ) , β Λ ( x, ρ ) := X η Λ X j  = η x      ρ ( η x,j Λ ) µ ( η x,j Λ ) − ρ ( η Λ ) µ ( η Λ )      µ ( η Λ ) . In the case where Λ = Λ n , w e will use the shorthand α n and β n instead of α Λ n and β Λ n . With this notation at hand, w e can express the result from Lemma 5.2 as c 2 X x ∈ Λ Z T 0 α Λ ( x, ν s ) ds ≤ 2 X x ∈ Λ γ Λ ( x ) Z T 0 β Λ ( x, ν s ) ds. (5.7) T o get some intuition on the ab ov e inequalit y , note that in the case of nearest-neighbour in teractions, the co efficients γ Λ ( x ) v anish if x / ∈ ∂ Λ, so the sum on the righ t-hand side of ( 5.7 ) can b e seen as a b oundary con tribution, whereas the sum on the left is the bulk con tribution. Moreov er, the α Λ ( x, · ) are clearly non-negative, contin uous with resp ect to the weak top ology on M 1 (Ω) and one has ρ = µ ⇐ ⇒ ∀ Λ ⋐ Z d ∀ x ∈ Λ : α Λ ( x, ρ ) = 0 . It will therefore b e our goal to show that these co efficients actually do v anish along time-p erio dic orbits. 5.3. Prop erties of the γ -co efficients. In the general case, where the in teraction range is not b ounded, the intuition as stated ab ov e can b e made precise as follows. Lemma 5.4. Under assumption ( R4 ) it holds that C 1 := sup x ∈ Z d ∞ X n =1 γ n ( x ) < ∞ , and C 2 := sup n ∈ N 1 n d − 1 X x ∈ Λ n γ n ( x ) < ∞ . Pr o of. A d C 1 : F or fixed x ∈ Z d w e ha v e ∞ X n =1 γ n ( x ) = ∞ X n =1 X y / ∈ Λ n δ y c x ( · ) = X y ∈ Z d δ y c x ( · ) |{ n ∈ N : x ∈ Λ n , y / ∈ Λ n }| ≤ X y ∈ Z d δ y c x ( · ) | x − y | . No w assumption ( R4 ) yields a uniform in x upp er b ound on this quantit y . TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 13 A d C 2 : Here w e ha v e for fixed n ∈ N X x ∈ Λ n X y / ∈ Λ n δ y c x ( · ) ≤ X v ∈ Z d X x ∈ Λ n : x + v / ∈ Λ n δ x + v c x ( · ) ≤ d (2 n + 1) d − 1 X v ∈ Z d | v | sup x ∈ Z d δ x + v c x ( · )) . This can b e b ounded from ab o v e, indep enden t of n , by assumption ( R4 ). □ 5.4. P oin t wise b ound of the β -co efficients in terms of the α -co efficients. Lemma 5.5. Ther e exists a c onstant C > 0 such that for al l ρ ∈ M 1 (Ω) , Λ ⋐ Z d and x ∈ Λ it holds that β Λ ( x, ρ ) 2 ≤ C · α Λ ( x, ρ ) . Pr o of. By definition of β Λ ( x, ρ ) and the Cauch y–Sch w arz inequality we hav e β Λ ( x, ρ ) 2 =   X η Λ X j  = η x µ ( η Λ )   v u u t ρ ( η x,j Λ ) µ ( η x,j Λ ) − s ρ ( η Λ ) µ ( η Λ )     v u u t ρ ( η x,j Λ ) µ ( η x,j Λ ) + s ρ ( η Λ ) µ ( η Λ )     2 ≤ X η Λ X j  = η x µ ( η Λ )   v u u t ρ ( η x,j Λ ) µ ( η x,j Λ ) − s ρ ( η Λ ) µ ( η Λ )   2 + X η Λ X j  = η x µ ( η Λ )   v u u t ρ ( η x,j Λ ) µ ( η x,j Λ ) + s ρ ( η Λ ) µ ( η Λ )   2 = α Λ ( x, ρ ) X η Λ X j  = η x µ ( η Λ )   v u u t ρ ( η x,j Λ ) µ ( η x,j Λ ) + s ρ ( η Λ ) µ ( η Λ )   2 So it suffices to b ound the second factor on the righ t-hand side b y a constant which do es not dep end on ρ, Λ and x . T o do this, first note that for all a, b ≥ 0 one has ( a + b ) 2 ≤ 2( a 2 + b 2 ), hence X η Λ X j  = η x µ ( η Λ )   v u u t ρ ( η x,j Λ ) µ ( η x,j Λ ) + s ρ ( η Λ ) µ ( η Λ )   2 ≤ 2 X η Λ X j  = η x µ ( η Λ ) ρ ( η x,j Λ ) µ ( η x,j Λ ) + ρ ( η Λ ) µ ( η Λ ) ! ≤ X η Λ X j  = η x  µ ( η x ) µ ( j x ) ρ ( η x,j Λ ) + ρ ( η Λ )  ≤ 2  1 δ + q  , where we used that µ is a pro duct measure with non-degenerate marginals and ρ | F Λ is a probability measure. □ 5.5. P oin t wise monotonicit y of the α -co efficien ts. As a last step, we no w prov e that considering larger v olumes do es not decrease the α -co efficients. This is the only place in which we really need the pro duct structure of µ . TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 14 Lemma 5.6. If µ is a pr o duct me asur e, then for any ρ ∈ M 1 (Ω) and x ∈ ∆ ⊂ Λ ⋐ Z d it holds that 0 ≤ α ∆ ( x, ρ ) ≤ α Λ ( x, ρ ) . Pr o of. The pro duct structure of µ gives us the factorisation µ ( η Λ ) = µ ( η x ) µ ( η Λ \ x ) , so the α co efficients for x ∈ ∆ ⊂ Λ ⋐ Z d can b e written as α ∆ ( x, ρ ) = X η ∆ X j  = η x µ ( η x )   s ρ ( η x,j ∆ ) µ ( j x ) − s ρ ( η ∆ ) µ ( η x )   2 , α Λ ( x, ρ ) = X η ∆ X j  = η x X ξ Λ : ξ ∆ = η ∆ µ ( η x )   s ρ ( ξ x,j Λ ) µ ( j x ) − s ρ ( ξ Λ ) µ ( η x )   2 . F or fixed η ∆ and j  = η x w e can write ρ ( η ∆ ) = X ξ Λ : ξ ∆ = η ∆ ρ ( ξ Λ ) , ρ ( η x,j ∆ ) = X ξ Λ : ξ ∆ = η ∆ ρ ( ξ x,j Λ ) . So in order to show that 0 ≤ α ∆ ( x, ρ ) ≤ α Λ ( x, ρ ) , it suffices to prov e a subadditivity prop ert y for functions of the form Φ( u, v ) =  r u c − r v d  2 , for c, d > 0. F or u 1 , u 2 , v 1 , v 2 > 0 we hav e Φ( u 1 , v 1 ) + Φ( u 2 , v 2 ) − Φ( u 1 + u 2 , v 1 + v 2 ) = 2 r ( u 1 + u 2 )( v 1 + v 2 ) cd − 2 r u 1 v 1 cd − 2 r u 2 v 2 cd . Therefore, to show that Φ is subadditive, it suffices to sho w that p ( u 1 + u 2 )( v 1 + v 2 ) ≥ √ u 1 v 1 + √ u 2 v 2 . First, observe that ( v 1 u 2 − v 2 u 1 ) 2 ≥ 0 , th us in particular ( v 1 u 2 ) 2 + ( v 2 u 1 ) 2 ≥ 2 v 1 u 2 v 2 u 1 . This implies that ( v 1 u 2 + v 2 u 1 ) 2 = ( v 1 u 2 ) 2 + ( v 2 u 1 ) 2 + 2 v 1 u 2 v 2 u 1 ≥ 4 v 1 u 2 v 2 u 1 . Since b oth sides are non-negative, w e can take square ro ots and obtain v 1 u 2 + v 2 u 1 ≥ 2 √ v 1 u 2 v 2 u 1 . This in turn implies that ( v 1 + v 2 )( u 1 + u 2 ) ≥ ( √ v 1 u 1 + √ v 2 u 2 ) 2 . But up to taking square roots on b oth sides, this is precisely what w e w an ted to sho w. □ TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 15 With all of these rather technical estimates in place, we are finally ready to provide the pro of of Prop osition 4.1 . Pr o of of Pr op osition 4.1 . By combining Lemma 5.2 with the p oin t wise estimates in Lemma 5.5 we obtain c 2 X x ∈ Λ Z T 0 α Λ ( x, ν s ) ds ≤  2 q δ  1 / 2 X x ∈ Λ γ Λ ( x ) Z T 0 p α Λ ( x, ν s ) ds. (5.8) By Lemma 5.6 and Lemma 5.4 w e hav e the following p oint wise estimates for all n ∈ N and s ∈ [0 , T ] X x ∈ Λ n α n ( x, ν s ) ≥ C − 1 1 X x ∈ Λ n α n ( x, ν s ) n X k =1 γ k ( x ) ≥ C − 1 1 n X k =1 X x ∈ Λ k α k ( x, ν s ) γ k ( x ) . (5.9) F or k ∈ N we now define δ k := X x ∈ Λ k γ k ( x ) Z T 0 α k ( x, ν s ) ds. Note that by definition of α k ( · , · ) and γ k ( · ) w e ha v e δ k ≥ 0 for all k ∈ N . By combining ( 5.8 ) and ( 5.9 ) with the Cauc h y–Sch warz inequality for sums we obtain " n X k =1 δ k # 2 ≤ 2 C 1 2 q δ X x ∈ Λ n γ n ( x ) ! X x ∈ Λ n γ n ( x )  Z T 0 p α n ( x, ν s ) ds  2 ! Another application of the Cauc h y–Sch warz inequalit y to the in tegrals on the righ t-hand side yields  Z T 0 p α n ( x, ν s ) ds  2 ≤ T Z T 0 α n ( x, ν s ) ds. Com bining this with Lemma 5.4 w e finally obtain " n X k =1 δ k # 2 ≤ C δ n n d − 1 for some constant C > 0. If there were an index n 0 ∈ N suc h that δ n 0 > 0, then for all n > n 0 it would hold that 1 n d − 1 ≤ C " 1 P n − 1 k =1 δ k − 1 P n k =1 δ k # . The series o v er the terms on the right-hand side conv erges because of a standard telescop- ing argumen t and monotonicity . How ever, for d ∈ { 1 , 2 } , this leads to a con tradiction. Therefore, w e m ust hav e δ n = 0 for all n ∈ N and hence by contin uity α n ( x, · ) ≡ 0 for all n ∈ N and x ∈ Λ n . But this implies that ν s = µ for all s ∈ [0 , T ] and the claim follo ws. □ The final argument of the proof can be condensed in to the follo wing elemen tary an- alytical lemma that we first state and prov e b efore commenting on what it en tails for higher dimensions. TIME-PERIODIC BEHA VIOUR WITHOUT REVERSIBILITY 16 Lemma 5.7. F or any C > 0 and d ∈ N , ther e is a se quenc e ( δ n ) n ∈ R that satisfies ∀ n ∈ N : δ n ≥ C n − d +1 n X k =1 δ k ! 2 (5.10) which is not identic al to zer o if and only if d ≥ 3 . Pr o of. F or d = 1 , 2: Let ( δ n ) n ∈ N b e a sequence that satisfies ( 5.10 ). If there is an index m such that δ m > 0, then for all n > m we obtain n − d +1 ≤ C − 1 1 P n − 1 k =1 δ k − 1 P n k =1 δ k ! . No w by a telescoping argument the sum o ver the terms on the righ t hand side conv erges to a finite v alue whereas the sum ov er the left-hand side diverges. In particular, there can b e no such index m where δ m is not equal to zero and w e obtain that δ n ≡ 0. F or d ≥ 3: Since d ≥ 3 we kno w that d − 1 ≥ 2, in particular the series with terms n − ( d − 1) con v erges. So choosing the candidate series ˆ δ n = an − ( d − 1) , we see that the gro wth b ound ( 5.10 ) simplifies to 1 a ≥ C n X k =1 k − d +1 ! 2 . F or fixed C > 0, this is clearly satisfied for sufficien tly small v alues of a > 0. □ In particular, the abov e strategy cannot be extended to dimensions d ≥ 3. How ever, one ma y still ask whether we hav e lost to o m uc h information when we applied the Cauc h y–Sc hw arz inequality for the first time, and finer estimates could still mak e the strategy w ork in d ≥ 3. But this is also not the case. T o see this, let’s restrict to the case of nearest-neighbour interactions, i.e., γ Λ ( x ) ≲ 1 ∂ Λ ( x ), and consider the sequences ( α n ( x )) n ∈ N that are defined by α n ( x ) = ( 0 if x / ∈ Λ n , a · k − 2 d +2 if x ∈ (Λ k \ Λ k − 1 ) ⊂ Λ n , where a > 0 is some constant whic h will b e determined later. Then for fixed x ∈ Z d , the sequence ( α n ( x )) n ∈ N satisfies the required monotonicity in n and additionally we hav e X x ∈ Λ n α n ( x ) ≤ c ( d ) a n X k =1 k − d +1 and X x ∈ ∂ Λ n p α n ( x ) ≤ c ( d ) √ a, where c ( d ) is a dimension dep endent constan t. F or d ≥ 3, this tells us that b y c ho osing a sufficiently small, this non-v anishing sequence satisfies the growth b ound ( 5.8 ). Ac kno wledgemen ts. 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