Restriction and mixing properties of interacting particle systems with unbounded range

RESTRICTION AND MIXING PR OPER TIES OF INTERA CTING P AR TICLE SYSTEMS WITH UNBOUNDED RANGE BENEDIKT JAHNEL AND JONAS K ¨ OPPL Abstract. W e consider in teracting particle systems with unbounded interaction range on general countably infinite graphs S and prov e explicit non-asymptotic error b ounds for approximations of the infinite-volume dynamics by systems of finitely man y inter- acting particles. Moreo ver, w e also provide non-asymptotic quantitativ e bounds on the spatial decay of correlations at times t > 0 and then apply these results to sho w that in teracting particle systems on Z whose in teraction strengths deca ys exp onen tially fast cannot sp on taneously break the time-translation symmetry , neither in the strong, nor in the w eak sense. 1. Introduction W e consider in teracting particle systems, which are Mark ov processes on the state space Ω = { 0 , . . . , q − 1 } S , where the set of sites is some coun tably infinite set S equipp ed with a metric d , specified in terms of generators of the form L f ( η ) = X ∆ ⋐ S X ξ ∆ ∈ Ω ∆ c ∆ ( η , ξ ∆ ) [ f ( ξ ∆ η ∆ c ) − f ( η )] , η ∈ Ω , for lo cal functions f : Ω → R . W e write ∆ ⋐ S to signify that ∆ is a finite subset of S and Ω ∆ := { 0 , . . . , q − 1 } ∆ . The transition rates c ∆ ( η , ξ ∆ ) can be in terpreted as the infinitesimal rate at whic h the particles inside of the finite v olume ∆ switc h from the lo cal state η ∆ to ξ ∆ , giv en that the rest of the system is currently in the state η ∆ c . W e will denote the asso ciated semigroup b y ( S ( t )) t ≥ 0 and the set of probability measures on Ω by M 1 (Ω). A prototypical example of suc h a system is the Glaub er dynamics for the Ising mo del, i.e., single-site spin-flip dynamics on the configuration space Ω = {± 1 } S with rates giv en b y c x ( η , − η x ) = [1 + exp ( H ( − η x η x c ) − H ( η ))] − 1 , (1.1) where H ( ω ) = − X x,y ∈ S J x,y ω x ω y , ω ∈ Ω . In man y ph ysical applications, the coupling constan ts J x,y are not strictly finite range but instead exhibit a spatial decay , e.g., p ow er-law or exp onen tial. While finite-range systems allow for intuitiv e graphical represen tations via Harris’ con- struction, see, e.g., [ Swa26 , Chapter 4], these unbounded-range dep endencies complicate the picture. Since the Hamiltonian H , and th us also the rates, dep ends on particles at arbitrary distances, classical results regarding the finite-sp eed-of-propagation, see, e.g., Date : March 24, 2026. 2020 Mathematics Subje ct Classification. Primary 82C22; Secondary 60K35. Key wor ds and phr ases. In teracting particle systems, decay of correlations, quantitativ e approxima- tion, attractor, time-translation symmetry breaking. 1 RESTRICTION AND MIXING PROPER TIES 2 [ Mar99 , Lemma 3.2], do not apply . Therefore, we ha ve to pro ceed differently and rely on the analytic to ols that the general existence theory for in teracting particle systems, as laid out in [ Lig05 , Chapter I], pro vides. In this article, w e address three questions related to the b eha viour of in teracting particle systems with un b ounded interaction range: (Q1) Finite-volume approximation: If we only observ e the pro cess in a fixed finite v olume Λ ⋐ S until some time t > 0, how w ell can the true infinite-volume dy- namics ( S ( t )) t ≥ 0 b e approximated by a finite-v olume system which only p erforms up dates in a finite region Λ h := { x ∈ S : d ( x, Λ) ≤ h } . In particular, how large do w e need to choose h = h ( t ) > 0 to obtain a sufficien tly goo d appro ximation? (Q2) Spatial decay of correlations: If we observe the infinite-volume dynamics in t w o distant v olumes Λ 1 , Λ 2 ⋐ S , how strongly are these t wo parts of the system correlated at some finite time t ? In particular, how fast do dep endencies spread in the system? (Q3) Long-time behaviour: Last but not least, what can the appro ximation via finite systems tell us ab out the infinite-volume dynamics? In particular, is it p ossible to transp ort certain facts ab out the long-time b ehaviour of finite systems to sufficien tly well-behav ed infinite volume systems? While the classical T rotter–Kurtz theorem, see [ Lig05 , Theorem 2.12 and Corol- lary 3.10], establishes the qualitative con vergence of finite-volume restrictions to the infinite-v olume dynamics, such results are typically asymptotic in nature. In contrast, w e pro vide non-asymptotic quantitative estimates. These b ounds explicitly c haracterise the appro ximation error in terms of the time t and the speed of deca y of the in teractions. Structurally , the error b ounds we derive can b e viewed as classical sto chastic analogues of the Lieb–Robinson b ounds found in quan tum spin systems, see e.g., [ NS10 , LR72 ]. Just as Lieb–Robinson b ounds define a light c one within which information can prop- agate in a quantum lattice spin system, our estimates quantify the spatial spread of dep endencies in a classical in teracting particle system. By b ounding the influence on and of distant sites, we effectively establish a rigorous con trol on the speed of information propagation, ev en in the presence of p ossibly long-range interactions. As a primary application of our approximation result, w e inv estigate the long-time b eha viour of in teracting particle systems on Z . W e sho w that for in teracting particle systems with exp onentially decaying interaction strengths the attractor of the measure- v alued dynamics is equal to the set of measures which are stationary with resp ect to the dynamics. In particular, non-trivial time-p erio dic b eha viour is imp ossible in such sys- tems. This result extends previous works of Mountford [ Mou95 ] and Ramirez–V aradhan [ R V96 ] to systems with unbounded interaction range. This extension is particularly notew orth y b ecause it is kno wn that the conclusion of our theorem does not hold in dimensions d ≥ 3, see [ JK14 , JK25b ], highlighting a fundamen tal phase transition in the dynamics dictated by the underlying spatial geometry . Organisation of the manuscript. In Section 2 , w e introduce the precise framew ork in whic h we are working and setup the required notation b efore w e state our main results. W e then provide brief outline of the pro of strategy for one of our main results in the direction of ( Q3 ), namely the strong attractor property for in teracting particle systems on Z , in Section 3 . After this, we start with the main w ork and provide the pro ofs of the results related to questions ( Q1 ) and ( Q2 ). The pro of of the strong attractor prop ert y is then prov ed via t w o steps in Section 5 and Section 6 . W e end the pap er with a short outlo ok and some open problems in Section 7 . RESTRICTION AND MIXING PROPER TIES 3 2. Setting and main resul ts Let S b e a countably infinite set of sites, equipp ed with a metric d , and for q ∈ N define the pro duct space Ω := Ω S o := { 0 , . . . , q − 1 } S , whic h we will equip with the usual pro duct top ology and the corresp onding Borel sigma-algebra F . F or Λ ⊂ S let F Λ b e the sub-sigma-algebra of F that is generated by the op en sets in Ω Λ := { 1 , . . . , q } Λ . W e will use the shorthand notation Λ ⋐ S to signify that Λ is a finite subset of S . If Λ ⋐ S and f : Ω → R is F Λ -measurable, then w e will also say that f is Λ-lo cal. In the follo wing we will often denote for a given configuration ω ∈ Ω by ω Λ its pro jection to the v olume Λ ⊂ S and write ω Λ ω ∆ for the configuration on Λ ∪ ∆ comp osed of ω Λ and ω ∆ for disjoint Λ , ∆ ⊂ S . F or the sp ecial case Λ = { x } w e will also write x c = S \ { x } and ω x ω x c . The set of probabilit y measures on Ω will b e denoted by M 1 (Ω) and the space of con tinuous functions f : Ω → R 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 ⊂ S we will denote the corresp onding cylinder sets by [ η Λ ] = { ω : ω Λ ≡ η Λ } . Whenever we are taking the probabilit y of such a cylinder even t with respect to some measure ν ∈ M 1 (Ω), w e will omit the square brac k ets and simply write ν ( η Λ ). 2.1. In teracting particle system. W e will consider time-con tin uous Marko v dynamics on Ω, namely in teracting particle systems c haracterised b y time-homogeneous generators L with domain dom( L ) and its asso ciated Marko vian semigroup ( S ( t )) t ≥ 0 . F or in ter- acting particle systems we adopt the notation and exp osition of the standard reference [ Lig05 , Chapter 1]. In our setting the generator L is giv en via a collection of translation- in v ariant transition rates c ∆ ( η , ξ ∆ ), in finite volumes ∆ ⋐ S , which are contin uous in the starting configuration η ∈ Ω. These rates can b e interpreted as the infinitesimal rate at which the particles inside ∆ switc h from the configuration η ∆ to ξ ∆ , given that the rest of the system is currently in state η ∆ c . The full dynamics of the interacting particle system is then giv en as the sup erp osition of these local dynamics, i.e., L f ( η ) = X ∆ ⋐ S X ξ ∆ ∈ Ω ∆ c ∆ ( η , ξ ∆ )[ f ( ξ ∆ η ∆ c ) − f ( η )] . In [ Lig05 , Chapter 1] it is shown that the following t w o conditions are sufficien t to guaran tee w ell-definedness of the dynamics. (L1) The total rate at which the particle at a particular site changes its spin is uniformly b ounded, i.e., C 1 := sup x ∈ S X ∆ ∋ x X ξ ∆ ∈ Ω ∆ ∥ c ∆ ( · , ξ ∆ ) ∥ ∞ < ∞ (L2) and the total influence of a single coordinate on all other co ordinates is uniformly b ounded, i.e., M γ := sup x ∈ S X y  = x γ ( x, y ) := sup x ∈ S X y  = x X ∆ ∋ x X ξ ∆ δ y ( c ∆ ( · , ξ ∆ )) < ∞ , where δ x ( f ) := sup η ,ξ : η x c = ξ x c | f ( η ) − f ( ξ ) | is the oscillation of a function f : Ω → R at the site x . RESTRICTION AND MIXING PROPER TIES 4 Under these conditions one can then show that the op erator L , defined as ab o ve, is the generator of a w ell-defined Mark o v process and that a core of L is giv en b y D (Ω) := n f ∈ C (Ω) : X x ∈ S δ x ( f ) < ∞ o . F or x ∈ S and ∆ ⋐ S we introduce the short-hand notation δ x c ∆ = X ξ ∆ ∈ Ω ∆ δ x ( c ∆ ( · , ξ ∆ )) . This measures the influence of the x -co ordinate on the rate of c hange in the finite v olume ∆. Therefore the quantit y γ ( y , x ) := ( P ∆ ∋ y δ x c ∆ , if x  = y 0 , otherwise, can b e interpreted as the total influence of the x -co ordinate on the rate of change of the spin at site y ∈ S . No w consider the Banach space ℓ 1 ( S ) of all functions β : S → R such that ∥ β ∥ ℓ 1 := X x ∈ S | β ( x ) | < ∞ . Note that for all f ∈ D (Ω) w e ha ve δ · ( f ) ∈ ℓ 1 ( S ) and we define | | | f | | | := ∥ δ · ( f ) ∥ ℓ 1 . F or β ∈ ℓ 1 ( S ) we will denote the supp ort of β b y Λ β , i.e., Λ β := { x ∈ S : β ( x )  = 0 } . By a sligh t abuse of notation, we will also write Λ f for the support of δ · f for f ∈ C (Ω). This is the set of sites on whic h the observ able f dep ends. In particular, f is alwa ys F Λ f -measurable. 2.2. Main results. F or our results we will sometimes also mak e the following additional assumptions on the maximal size of the up date regions. (R1) The maximal up date size is b ounded, i.e., there is a constant L > 0 such that if diam(∆) ≥ L , then c ∆ ( · , · ) ≡ 0. Additionally , we w an t to quantify the rate at which γ ( x, y ) tends to zero as a function of the distance d ( x, y ) b et ween the sites. F or this, let ϱ : [0 , ∞ ) → (0 , ∞ ) b e a non- increasing function and consider the following assumptions. (R2) There exists a constant C ϱ ∈ (0 , ∞ ) suc h that the follo wing inequalit y holds ϱ ( d ( x, z )) ϱ ( d ( z , y )) ≤ C ϱ ϱ ( d ( x, y )) , x, y , z ∈ S. (R3) The transition rates, or rather their oscillations, satisfy the deca y condition C γ := sup x ∈ S X y ∈ S γ ( x, y ) ϱ ( d ( x, y )) < ∞ . (R4) W e ha v e ∥ ϱ ∥ := sup x ∈ S X y ∈ S ϱ ( d ( x, y )) < ∞ . In the case S = Z d equipp ed with the standard ℓ 1 -distance, the functions ϱ ( r ) = (1 + r ) − α satisfy the condition ( R2 ) ab o ve for an y α ≥ 1 and ( R4 ) for any α > d . Moreov er, for an y µ ≥ 0 and any ϱ satisfying the ab o v e conditions, the function ϱ µ ( r ) := e − µr ϱ ( r ) also satisfies the ab o ve conditions with ∥ ϱ µ ∥ ≤ ∥ ϱ ∥ and C ϱ µ ≤ C ϱ . RESTRICTION AND MIXING PROPER TIES 5 2.3. Restriction to finite v olumes. F or a fixed finite subset Λ ⋐ S and a length-scale h > 0 define the h -blow-up of Λ b y Λ h := { u ∈ S : dist( u, Λ) ≤ h } , where dist(∆ , Λ) := inf { d ( u, v ) : u ∈ ∆ , v ∈ Λ } , and the generator of the dynamics restricted to Λ h b y L h f ( η ) = X ∆ ⊂ Λ h X ξ ∆ ∈ Ω ∆ c ∆ ( η , ξ ∆ ) [ f ( ξ ∆ η ∆ c ) − f ( η )] , η ∈ Ω , f ∈ D (Ω) . So only the particles inside of the finite volume Λ h participate in the dynamics, whereas all other particles remain fixed. If w e just observe the dynamics in the smaller volume Λ ⊂ Λ h ⋐ S , we w an t to estimate the error we make b y considering L h instead of L . This also tells us how large we ha v e to c ho ose h , depending on the time window [0 , t ] and the volume Λ, to get a decent approximation of the infinite-v olume dynamics. T o measure the error, we c ho ose the total v ariation metric with resp ect to the sub-sigma- algebra F Λ , i.e., d TV , Λ ( ν, ρ ) := sup f Λ-local , ∥ f ∥ ∞ ≤ 1 | ν ( f ) − ρ ( f ) | , ν, ρ ∈ M 1 (Ω) . The following theorem provides a non-asymptotic estimate on the approximation error made by considering the restricted dynamics instead of the infinite-volume pro cess. Let us further note that all of the constan ts appearing in the error b ound are explicit. Theorem 2.1 (Restriction to finite volumes) . Assume that the c onditions ( L1 ) and ( R1 ) − ( R4 ) ar e satisfie d. L et Λ ⋐ S and h > 0 . Then, for al l Λ -lo c al functions f : Ω → R we have    S ( t ) f − S h ( t ) f    ≤ ∥ f ∥ ∞ C 1 C S,L C 2 ϱ C γ exp( C γ C ϱ t ) X x / ∈ Λ h − L ϱ (dist( x, Λ)) , (2.1) wher e the c onstant C S,L is a ge ometric quantity define d by C S,L := sup x ∈ S |{ ∆ ⋐ S : x ∈ ∆ and diam(∆) < L }| . In p articular, for any initial distribution µ ∈ M 1 (Ω) the total variation err or in Λ is b ounde d by d TV , Λ ( µS ( t ) , µS h ( t )) ≤ C 1 C S,L C 2 ϱ C γ exp( C γ C ϱ t ) X x / ∈ Λ h − L ϱ (dist( x, Λ)) . This shows that it suffices to estimate the tail sums for ϱ ( d ( · , · )) to get the distance at which one can truncate the pro cess. How ev er, note that if S = Z d and ϱ ( r ) = (1 + r ) − α , then w e only get a decaying right-hand side for α > d , whic h means that the dep endencies in the rates, i.e., γ ( x, y ), need to deca y faster than | x − y | − ( α + d ) . Therefore, our result cov ers systems with pow er-la w interactions, but only up un til a certain threshold, dep ending on the geometry of the underlying graph S . A similar error b ound can b e deriv ed for situations where h is not a fixed constan t, but also dep ends on time. This extension is done in Prop osition 5.1 and is crucial for the proof of one of our other main results, namely Theorem 2.5 . RESTRICTION AND MIXING PROPER TIES 6 2.4. Appro ximation of stationary measures. F or interacting particle systems in infinite v olume it is in general notoriously difficult to say an ything non-trivial ab out the stationary measures. F or certain classes, e.g., attractive spin systems, one can ap- pro ximate the stationary measures for the infinite-volume dynamics b y the stationary measures for finite-volume dynamics with sp ecific b oundary conditions, see , e.g., [ Lig05 , Theorem I I I.2.7]. The following result gives sufficient conditions for not necessarily attractiv e in teracting particle systems to enjo y a similar appro ximation property . Theorem 2.2 (Approximation of stationary measures) . Assume that the c onditions ( L1 ) and ( R1 ) − ( R4 ) ar e satisfie d. Assume further that ther e exists a deterministic c onfigur ation η and a non-incr e asing function F : [0 , ∞ ) → R with F ( t ) ↓ 0 as t ↑ ∞ such that for any h > 0 ther e exists µ h ∈ M 1 (Ω) such that    S h ( t ) f ( η ) − µ h ( f )    ≤ C ( f ) F ( t ) , ∀ f : Ω → R lo c al , wher e the c onstant C ( f ) is al lowe d to dep end on f but not on h . Then, the se quenc e ( µ h ) h ≥ 0 c onver ges to a limit µ ∗ in M 1 (Ω) and µ ∗ is stationary for the infinite-volume dynamics ( S ( t )) t ≥ 0 . Note that w e only need the uniformit y for one sp e cific initial (and in some sense b oundary) condition η and not for all η ∈ Ω. Therefore, this theorem may also be applied in lo w temp erature situations where con v ergence to equilibrium in finite v olumes is t ypically not uniform in the system size. 2.5. Spatial decay of correlations. In equilibrium statistical mechanics, the decay of correlations b et w een spins at distan t sites is one of the k ey c haracteristics. In the situation we are interested in, one can ask very similar questions but additionally has to consider ho w dep endencies ma y spread in time due to the interactions in the transition rates. W e first give a general estimate on the spatial decay of correlations for fixed times t ≥ 0 and then sho w ho w this can b e used to obtain information about some stationary measures. The follo wing estimate extends and generalises [ Lig05 , Prop osition I.4.18] to systems with unbounded range of in teraction and possibly non-translation-in v arian t transition rates. Theorem 2.3 (Spatial deca y of correlations) . Assume that ( L1 ) and ( R1 ) − ( R3 ) hold. Then, for al l f , g ∈ D (Ω) and t ≥ 0 we have ∥ S ( t )[ f g ] − [ S ( t ) f ][ S ( t ) g ] ∥ ∞ ≤ C 1 ϱ ( L ) C 2 ϱ C γ | | | f | | || | | g | | | exp(2 C ϱ C γ t ) ϱ (dist(Λ f , Λ g )) . Dep ending on the deca y of ϱ ( · ), this giv es us an upp er b ound on how far correlations b et w een distant sites ha ve spread due to the in teractions up until time t . On S = Z d for an y rate α > 0 and ϱ ( r ) = exp( − αr ) this tells us that the sp eed at which information is b eing propagated through the system is linear. If ϱ decays lik e a pow er la w, this theorem only provides an exp onential b ound on this sp eed, which is not exp ected to b e optimal. 2.6. Quan titativ e decay of correlations for limiting stationary measures. The non-asymptotic b ound in Theorem 2.3 holds rather generally and already tells us some- thing ab out how fast dep endencies spread in the system. Ho w ev er, the right-hand side still dep ends on t and it do es in particular not giv e us information ab out the deca y of correlations for stationary measures µ that can b e obtained as limits for some fixed initial configuration. RESTRICTION AND MIXING PROPER TIES 7 Under the additional assumption of rapid conv ergence to equilibrium for some fixed initial condition, we can remo v e this inhomogeneit y and also obtain that, in this case, the limiting measure also satisfies a quantitativ e decay of correlations estimate. Theorem 2.4 (Spatial mixing for limiting stationary measures) . Assume that ( L1 ) and ( R1 ) − ( R3 ) hold and that for some η ∈ Ω , µ ∈ M 1 (Ω) , and c onstants ˆ K , ˆ α > 0 , | S ( t ) f ( η ) − µ ( f ) | ≤ ˆ K e − ˆ αt | | | f | | | , ∀ f ∈ D (Ω) , t ≥ 0 . (2.2) Then, ther e exist K , α > 0 such that for that p articular η and al l t ≥ 0 we have | S ( t )( f g )( η ) − [ S ( t ) f ( η )][ S ( t ) g ( η )] | ≤ K | | | f | | || | | g | | | ϱ (dist(Λ f , Λ g )) α for al l f , g ∈ D (Ω) with dist(Λ f , Λ g ) > L C ϱ . In p articular, we get the fol lowing b ound on the c orr elations of µ : | µ ( f g ) − µ ( f ) µ ( g ) | ≤ K | | | f | | || | | g | | | ϱ (dist(Λ f , Λ g )) α for al l f , g ∈ D (Ω) with dist(Λ f , Λ g ) > L C ϱ . Note that the theorem ab o v e do es not assume uniqueness of the stationary distribution but just the rapid conv ergence for one particular initial configuration. In particular, this assumption can also b e justified in the phase-co existence regime. This builds a bridge from temp oral mixing to spatial mixing without requiring finite-range or exp onen tially deca ying in teractions and also w orks for ϱ ( · ) that b eha v e lik e a p o w er law. Since w e do not require assumption ( R4 ) for Theorem 2.3 and Theorem 2.4 they even apply to interacting particle systems on Z d with long-range interactions as long as γ ( x, y ) ≲ | x − y | − α for α > d . 2.7. Absence of time-translation symmetry breaking in one dimension. His- torically , in the literature on interacting particle systems the most attention has b een paid to the set of time-stationary me asur es for the dynamics whic h is given by S = { ν ∈ M 1 (Ω) : ν S ( t ) = ν ∀ t ≥ 0 } . Ho w ever, if one is in terested in the long-term b ehaviour of an in teracting particle system, a more natural and ric her ob ject to study is the so-called attr actor of the measure-v alued dynamics, whic h is defined as A =  ν ∈ M 1 (Ω) : ∃ ν 0 ∈ M 1 (Ω) and t n ↑ ∞ such that lim n →∞ ν t n = ν  , where the con v ergence is in the w eak sense. In other w ords, A is the set of all accumula- tion points of the measure-v alued dynamics induced b y L . In the language of dynamical systems this is the ω -limit set and it enco des (most of ) the dynamically relev ant informa- tion ab out the long-time b eha viour of the system. In particular, it is the natural ob ject to consider for studying the phenomenon of sp ontane ous symmetry br e aking in the con- text of in teracting particle systems. F or an in teracting particle system we say that a symmetry of the transition rates is sp ontaneously brok en if there is an element of the at- tractor A that do es not satisfy the symmetry . F or example, for the Ising mo del Glaub er dynamics on S = Z d , tw o ob vious symmetries are the inv ariance under global spin-flip, i.e., c x ( η , − η x ) = c x ( − η , η x ) and under translations, i.e., c x ( η , − η x ) = c 0 ( τ x η , − ( τ x η ) 0 ). It is well known, that both of these symmetries can be sp on taneously broken in infinite v olume, at least in higher dimensions. Indeed, as shown by P eierls for the spin-flip sym- metry in dimensions d ≥ 2 and by Dobrushin for the spatial translation symmetry in d ≥ 3. RESTRICTION AND MIXING PROPER TIES 8 Another ob vious and hence often ov erlo ok ed symmetry is that the transition rates do not dep end on time, i.e., the generator is autonomous, and is th us in v ariant under time- shifts. Of course, trivially any stationary measure is also inv ariant under time shifts and one needs to b e a bit more careful when defining the notion of time-translation symmetry breaking. This can b e done in the language of the attractor. F or this, we first introduce another subset of the attractor, namely the measures which lie on a stationary orbit , i.e., O := { ν ∈ M 1 (Ω) : ∃ t > 0 suc h that ν S ( t ) = ν } . It is clear b y the definition that S ⊂ O ⊂ A . W e will sa y that (str ong) time-tr anslation symmetry br e aking o ccurs if S ⊊ O , in other words, there exists a non-trivial time- p eriodic orbit ( µ s ) s ∈ [0 ,τ ] in the space M 1 (Ω) of probabilit y measures on Ω. This is similar to the crystallisation phase transition, where the contin uous-translation symme- try by any vector in R d is sp ontaneously reduced to a discrete translation symmetry . Additionally , w e sa y that we ak time-tr anslation symmetry br e aking o ccurs if S ⊊ A . It is obvious that the strong notion of time-translation symmetry breaking implies the w eak one. The con v erse is not clear and w e exp ect that generally the w eak notion do es not imply the strong one. The conditions under whic h in teracting particle sys- tems can exhibit stable time-p eriodic b eha viour, thereby spontaneously breaking the time-translation symmetry , hav e b een extensively discussed in the physics literature [ GMSB93 , BGH + 90 , CM92 ]. As it turns out, the possibility or non-possibility of sp on- taneous time-translation symmetry breaking seems to dep end hea vily on the dimension of the underlying graph. Non-rigorous argumen ts [ GMSB93 ] and extensive numerical studies [ AFM + 25b , AFM + 25a , GB24 ] suggest that in teracting particle systems with short-range in teractions can exhibit strong time-translation symmetry breaking in di- mensions d ≥ 3 but cannot pro duce stable time-p erio dic b eha viour in d = 1 , 2. By combining the ab o v e results on information propagation and approximation prop- erties with the proof strategy of [ R V96 ], w e obtain the following generalisation of Moun t- ford’s theorem for in teracting particle systems on S = Z with p ossibly un bounded range. Theorem 2.5 (Absence of time-translation symmetry breaking) . L et S = Z b e e quipp e d with the ℓ 1 -metric and assume that L is the gener ator of an inter acting p article system that satisfies assumptions ( L1 ) and ( R1 ) − ( R3 ) with ϱ ( r ) = exp( − r α ) for some α > 0 . Denote the asso ciate d Markov semigr oup by ( S ( t )) t ≥ 0 . Then, we have A = O = S . In p articular, such systems c annot exhibit we ak or str ong time-tr anslation symmetry br e aking and | S | = 1 implies that the inter acting p article system is er go dic. In other words, every weak limit p oin t of the dynamics is a stationary measure. In d ≥ 3 there are kno wn counterexamples that satisfy the regularit y assumptions of our theorem but exhibit strong time-translation symmetry breaking, whereas in d = 2 the situation is unclear but b eliev ed to b e similar to the one-dimensional case. If one drops the short-range assumption, there are also coun terexamples in d = 1 , 2 that exhibit strong time-translation symmetry breaking, see [ JK25b ]. Let us note that w e do not require an y shift-inv ariance or reversibilit y . 3. Proof stra tegy for the absence of time-transla tion symmetr y breaking Let us briefly comment on the strategy for proving Theorem 2.5 that was first used in this con text in [ R V96 ] and how it relates to the error estimates for restricting in teracting particle systems to finite v olumes. F or this, consider a c on tin uous-time Marko v c hain RESTRICTION AND MIXING PROPER TIES 9 with generator L on a finite space X . Denote the asso ciated semigroup by ( P ( t )) t ≥ 0 . T o sho w that A = S it suffices to show that for any initial distribution µ and any τ > 0 w e ha v e lim sup t →∞ d TV ( µP ( t ) , µP ( t + τ )) = 0 . Indeed, by the triangle inequality , if this holds, then any p ossible limit p oin t of the measure-v alued dynamics induced by ( P ( t )) t ≥ 0 on M 1 (Ω) has to b e a stationary measure for the Marko v chain, and hence A = S . Now note that one can use Pinsk er’s inequality to b ound the total v ariation distance b et ween µP ( t ) and µP ( t + τ ) in terms of the relative en trop y , i.e., d TV ( µP ( t ) , µP ( t + τ )) ≤ r 1 2 H ( µP ( t + τ ) | µP ( t )) . Additionally , instead of shifting time b y τ , we can also interpret µP ( t + τ ) as the distribu- tion of a suitably sp ed-up pro cess with generator L ′ at time t , i.e, µP ( t + τ ) = µP ′ ( t ) for L ′ = (1 + τ /t ) L and P ′ ( t ) = e tL ′ . No w, by a Girsanov-t yp e form ula for con tin uous-time Mark o v c hains on finite state spaces, see, e.g., [ KL99 , Prop osition 2.6. in App endix 1], one c hec ks that the en tropic cost of this speed-up can b e b ounded b y H ( µP ( t + τ ) | µP ( t )) = H ( µP ′ ( t ) | µP ( t )) ≤ H ( P ′ [0 ,t ] | P [0 ,t ] ) ≤ c t  τ t  2 = c τ 2 t , (3.1) where P (resp ectiv ely P ′ ) denotes the law of the whole pro cess with initial distribution µ and generator L (respectively L ′ ) and c := max x ∈X X y  = x L ( x, y ) . Since the right-hand side of ( 3.1 ) go es to 0 as t tends to infinit y , w e are done. Ho w ever, via the use of the Girsano v formula, this proof heavily relies on the fact that the sp ed-up process with generator L ′ is absolutely contin uous with resp ect to the original pro cess with generator L . This is in general only the case if L has uniformly b ounded total jump rate, i.e., if c < ∞ . F or in teracting particle systems this is essentially nev er the case and we therefore hav e to pro ceed a bit differently b y first restricting the dynamics to finite volumes Λ h ⋐ Z and considering the appro ximation error made by this restriction. If w e are interested in controlling the total v ariation error up until time t > 0, then Theorem 2.1 suggests that, even in the finite-range case, the best w e can do is scaling h ( t ) ∼ t . In this case, the corresp onding total jump rate c ( h ( t )) gro ws lik e t d and ev en in the case d = 1 we only get an O (1) upp er b ound from ( 3.1 ). But there is still another screw one can turn to make the argument w ork. Instead of considering a constan t sp eed-up λ ≡ (1 + τ /t ) one can also p erform a time-dep enden t sp eed-up λ ( · ) and optimise o v er all the admissible options. This is just enough to mak e the argumen t w ork in d = 1 under suitable conditions on the decay of the in teraction strength. T o rigorously carry out this argument in detail, we first derive a time-dep enden t generalisation and refinement of the restriction estimate in Section 5 . W e then estimate the en tropic cost of the optimal time-dep enden t sp eed-up and put all the ingredients together in Section 6 . Before w e get in to this business we first provide the proofs of the general restriction estimate in Theorem 2.1 and the decay of correlation prop erties in Theorem 2.3 and Theorem 2.4 . RESTRICTION AND MIXING PROPER TIES 10 4. Restriction proper ty and deca y of correla tions The follo wing lemma is a consequence of the general existence theory dev elop ed by Liggett and for example con tained in [ Lig05 , Theorem I.3.9]. Lemma 4. 1. Assume that ( L1 ) and ( L2 ) hold. Then, for f ∈ D (Ω) and t ≥ 0 we have the fol lowing r e gularity estimate for al l x ∈ S δ x ( S ( t ) f ) ≤ exp( t Γ)[ δ · ( f )]( x ) , wher e Γ : ℓ 1 ( S ) → ℓ 1 ( S ) , define d by Γ β ( x ) := X y ∈ S γ ( y , x ) β ( y ) , x ∈ S, is a p ositive and b ounde d line ar op er ator on ℓ 1 ( S ) with op er ator norm ∥ Γ ∥ op = M γ . F or estimating the sp eed at whic h distant parts of the system get correlated by the dynamics, the following estimate from [ Lig05 , Prop osition I.4.4] is quite useful. Lemma 4.2. Assume that ( L1 ) and ( L2 ) hold. Then, for any f , g ∈ D (Ω) and t ≥ 0 we have ∥ S ( t )[ f g ] − [ S ( t ) f ][ S ( t ) g ] ∥ ∞ ≤ X x,y ∈ S h X ∆ ∋ x,y X ξ ∆ ∥ c ∆ ( · , ξ ∆ ) ∥ ∞ i Z t 0  e s Γ δ · f  ( x )  e s Γ δ · g  ( y )d s. In the situations w e are interested in, we hav e c ∆ ≡ 0 for diam(∆) ≥ L for some constan t L > 0, so one typically has the sligh tly less sharp but simpler estimate ∥ S ( t )[ f g ] − [ S ( t ) f ][ S ( t ) g ] ∥ ∞ ≤ C 1 X x,y ∈ S : d ( x,y ) ≤ L Z t 0  e s Γ δ · f  ( x )  e s Γ δ · g  ( y )d s. (4.1) These estimates already tell us that, to obtain upp er b ounds on ho w fast information spreads in interacting particle systems, it is generally a goo d idea to study the action of the op erator Γ and the asso ciated semigroup (exp( t Γ)) t ≥ 0 on ℓ 1 ( S ). They will indeed pla y a key role in the rest of this section. 4.1. Propagation of b ounds. Recall that we assume that the coefficients of Γ satisfy the bounds C γ = sup x ∈ S X y ∈ S γ ( x, y ) ϱ ( d ( x, y )) < ∞ and C ϱ = sup x,y ,z ∈ S ϱ ( d ( x, z )) ϱ ( d ( z , y )) ϱ ( d ( x, y )) < ∞ . Let us see what this tells us ab out the action of the semigroup (exp( t Γ)) t ≥ 0 on ℓ 1 ( S ). By boundedness of Γ we can expand exp( t Γ) for any t ≥ 0 in to e t Γ β ( x ) = X n ≥ 0 t n n ! [Γ n β ]( x ) X y ∈ Z d β ( y ) X n ≥ 0 t n n ! γ ( n ) ( y , x ) =: X y ∈ Z d β ( y ) γ t ( y , x ) . Our first step is to show how the b ounds on γ = γ 0 propagate to later times t > 0. Lemma 4.3. Under the assumptions ( R2 ) − ( R3 ) we have for any t ≥ 0 sup x ∈ S X y ∈ S γ t ( x, y ) ϱ ( d ( x, y )) ≤ C − 1 ϱ exp( C γ C ϱ t ) . RESTRICTION AND MIXING PROPER TIES 11 Pr o of. W e can first use an induction argumen t to show that, for any n ∈ N , one can express the co efficien ts γ ( n ) ( u, v ) of the iterated op erator Γ n b y γ ( n ) ( u, v ) = X u 1 ∈ S · · · X u n − 1 ∈ S γ ( u, u 1 ) · · · γ ( u n − 1 , v ) . F or fixed x ∈ S and n ∈ N the assumption ( R2 ) for ϱ ( d ( · , · )) implies that for any x ∈ S X y ∈ S γ ( n ) ( x, y ) ϱ ( d ( x, y )) ≤ C n − 1 ϱ X y ∈ S X x 1 ∈ S · · · X x n − 1 ∈ S γ ( x, x 1 ) ϱ ( d ( x, x 1 )) · · · γ ( x n − 1 , y ) ϱ ( d ( x n − 1 , y )) . So b y applying assumption ( R3 ) n times w e ha v e sup x ∈ S X y ∈ S γ ( n ) ( y , x ) ϱ ( d ( y , x )) ≤ C n γ C n − 1 ϱ . Th us for t ≥ 0 w e obtain sup x ∈ S X y ∈ S γ t ( x, y ) ϱ ( d ( x, y )) ≤ C − 1 ϱ ∞ X n =0 t n n ! C n γ C n ϱ ≤ C − 1 ϱ exp( C γ C ϱ t ) , as desired. □ The bound in Lemma 4.3 directly implies the following quantitativ e bound on the spatial deca y of e t Γ β for compactly supported β ∈ ℓ 1 ( S ). Lemma 4.4. Assume that ( R2 ) − ( R3 ) hold. If β ∈ ℓ 1 ( S ) has c omp act supp ort Λ β ⋐ S , then e t Γ β ( x ) ≤ ∥ β ∥ ∞ C − 1 ϱ exp( C γ C ϱ t ) ϱ (dist( x, Λ β )) . This in p articular applies to β = ( δ x f ) x ∈ S for lo c al observables f : Ω → R that only dep end on some finite volume Λ f ⋐ S . 4.2. Restriction to finite v olumes. W e proceed with the error b ound for appro xi- mating the infinite-volume dynamics b y restrictions to finite v olumes. Pr o of of The or em 2.1 . By Duhamel’s formula we hav e S h ( t ) f ( η ) − S ( t ) f ( η ) = Z t 0  S h ( s )( L h − L ) S ( t − s )  f ( η )d s. No w for any g ∈ D (Ω) we can use a te lescoping trick to obtain the uniform estimate   ( L h − L ) g ( η )   ≤ X ∆ ⊂ Λ h X ξ ∆ | c ∆ ( η , ξ ∆ ) [ g ( ξ ∆ η ∆ c ) − g ( η )] | ≤ C 1 C S,L X x / ∈ Λ h − L δ x g , where C S,L uniformly b ounds the n umber of up date regions ∆ in which a particular site x is included. F or Λ-lo cal observ ables f : Ω → R we can combine Lemma 4.1 and the quan titativ e estimate from Lemma 4.4 to get δ x  S ( t − s ) f  ≤  e ( t − s )Γ δ · f  ( x ) ≤ ∥ f ∥ ∞ C − 1 ϱ exp( C γ C ϱ ( t − s )) ϱ (dist( x, Λ)) . After using that   S h ( s ) h   ∞ ≤ ∥ h ∥ ∞ for an y h ∈ C (Ω), w e can combine this with an application of a telescoping estimate from ab o ve to the function g = S ( t − s ) f to see RESTRICTION AND MIXING PROPER TIES 12 that    S h ( t ) f − S ( t ) f    ∞ ≤ Z t 0     S h ( s )( L h − L ) S ( t − s )  f    ∞ d s ≤ Z t 0    ( L h − L ) S ( t − s ) f    ∞ d s ≤ ∥ f ∥ ∞ C 1 C S,L C − 1 ϱ Z t 0 exp( C γ C ϱ ( t − s ))d s X x / ∈ Λ h − L ϱ (dist( x, Λ)) = ∥ f ∥ ∞ C 1 C S,L C 2 ϱ C γ exp( C γ C ϱ t ) X x / ∈ Λ h − L ϱ (dist( x, Λ)) . This finishes the pro of. □ 4.3. Appro ximation of stationary measures. W e no w apply the results of Theo- rem 2.1 to sho w that one can appro ximate the stationary measures of the infinite-v olume dynamics via the stationary measures of the restricted dynamics. Pr o of of The or em 2.2 . W e first sho w that the sequence ( µ h ) h ≥ 0 con v erges to a limit. F or this, note that the compactness of M 1 (Ω) implies the existence of limit p oin ts and w e only hav e to sho w uniqueness. F or this, it suffices to show that for an y fixed Λ-local observ able f : Ω → R the sequence ( µ h ( f )) h ≥ 0 is a Cauch y sequence. T o this end, note that for any t ≥ 0 and h, k > 0 we can use Theorem 2.1 to get    µ h ( f ) − µ k ( f )    ≤    µ h ( f ) − S h ( t ) f ( η )    +    S h ( t ) f ( η ) − S ( t ) f ( η )    +    S ( t ) f ( η ) − S k ( t ) f ( η )    +    S k ( t ) f ( η ) − µ k ( f )    ≤ 2 C ( f ) F ( t ) + c | Λ | ∥ f ∥ ∞ exp( ct ) (Φ ϱ,S ( h − L ) + Φ ϱ,S ( k − L )) for some constant c > 0, where we use the notation Φ ϱ,S ( r ) := sup x ∈ S X y ∈ S : d ( y ,x ) >r ϱ ( d ( y , x )) . F or ε > 0 w e can first c ho ose t > 0 sufficien tly large to mak e the first term smaller than ε/ 2 and then use assumption ( R4 ) to choose h ( ε ) > 0 sufficiently large to make the second term smaller than ε/ 2 for all k ≥ h ≥ h ( ε ). Thus, lim h →∞ µ h = µ ∗ exists. It remains to sho w that the limiting measure µ ∗ is stationary for the infinite-volume dynamics. F or this, let f : Ω → R b e a Λ-lo cal observ able and note that for any t ≥ 0 and h > 0 w e hav e   µ ∗ S ( t )[ f ] − µ ∗ [ f ]   ≤   µ ∗ S ( t )[ f ] − µ h S ( t )[ f ]   +   µ h S ( t )[ f ] − µ h S h ( t )[ f ]   +   µ h S h ( t )[ f ] − µ h [ f ]   +   µ h [ f ] − µ ∗ [ f ]   . By stationarit y of µ h with respect to ( S h ( t )) t ≥ 0 the third term v anishes and w e only ha v e to estimate the remaining three. Note that by w eak con v ergence of ( µ h ) h ≥ 0 to µ ∗ the first and the fourth term go to zero as h tends to infinity and the second term can again be b ounded by inv oking Theorem 2.1 to get   µ h S ( t )[ f ] − µ h S h ( t )[ f ]   ≤ c | Λ | ∥ f ∥ ∞ exp( ct )Φ ϱ,S ( h − L ) . Since t is arbitrary but fixed, we can again use ( R4 ) to c hoose h sufficien tly large to mak e the right side arbitrarily small. This sho ws that µ ∗ is indeed a stationary measure for the unrestricted dynamic. □ RESTRICTION AND MIXING PROPER TIES 13 4.4. Spatial deca y of correlations. Let us no w turn our atten tion tow ards the bounds on the correlations at time t . Pr o of of The or em 2.3 . By applying the estimate in Lemma 4.2 and assumptions ( L1 ) and ( R1 ) we get ∥ S ( t )[ f g ] − [ S ( t ) f ][ S ( t ) g ] ∥ ∞ ≤ C 1 X x,y ∈ S : d ( x,y ) ≤ L Z t 0  e s Γ δ · f  ( x )  e s Γ δ · g  ( y )d s and it th us again reduces our problem to understanding the behaviour of the Γ-operator. In the notation of the previous section, w e can write  e s Γ δ · h  ( z ) = X u ∈ Z d γ s ( u, z ) δ u h, for an y h ∈ D (Ω), z ∈ Z d and s ≥ 0. By non-negativity we can exchange the order of summation and integration to get ∥ S ( t )[ f g ] − [ S ( t ) f ][ S ( t ) g ] ∥ ∞ ≤ C 1 X u,v ∈ S ( δ u f )( δ v g ) X x,y ∈ S : d ( x,y ) ≤ L Z t 0 γ s ( u, x ) γ s ( v , y )d s. By definition of | | |·| | | w e ha ve | | | h | | | = P x ∈ Z d δ x h , so it suffices to sho w that for any s ≥ 0 and u, v ∈ Z d w e ha v e X x,y ∈ S : d ( x,y ) ≤ L γ s ( u, x ) γ s ( v , y ) ≤ ϱ ( d ( u, v )) ϱ ( L ) C ϱ exp(2 C ϱ C γ s ) . (4.2) Indeed, if we hav e ( 4.2 ) w e obtain X u,v ∈ S ( δ u f )( δ v g ) X x,y ∈ S : d ( x,y ) ≤ L Z t 0 γ s ( u, x ) γ s ( v , y )d s ≤ 1 ϱ ( L ) C 2 ϱ C γ X u,v ∈ S ( δ u f )( δ v g ) Z t 0 C ϱ C γ e 2 C ϱ C γ s ϱ ( d ( u, v ))d s = 1 ϱ ( L ) C 2 ϱ C γ | | | f | | || | | g | | | e 2 C ϱ C γ t ϱ (dist(Λ f , Λ g )) . T o sho w ( 4.2 ), note that for fixed u, v ∈ S and L, s > 0 w e get X x,y ∈ S : d ( x,y ) ≤ L γ s ( u, x ) γ s ( v , y ) ϱ ( d ( u, v )) ≤ C ϱ X x,y ∈ S : d ( x,y ) ≤ L γ s ( u, x ) γ s ( v , y ) ϱ ( d ( u, x )) ϱ ( d ( x, y )) ϱ ( d ( v , y )) ≤ C ϱ ϱ ( L ) X x ∈ S γ s ( u, x ) ϱ ( d ( u, x )) X y ∈ S γ s ( v , y ) ϱ ( d ( v , y )) ≤ e 2 C ϱ C γ s ϱ ( L ) C ϱ . Rearranging this yields the claimed estimate ( 4.2 ) and w e are done. □ No w we can pro ceed to use the b ound on the sp eed at whic h information spreads as stated in Theorem 2.3 to derive some information ab out the limiting measures. Pr o of of The or em 2.4 . First note that since adding a constant to f or g has no effect on b oth the left and the righ t side of the inequalit y , we can assume without loss of generalit y RESTRICTION AND MIXING PROPER TIES 14 that ν ( f ) = ν ( g ) = 0. This in particular implies that f and g cannot b e iden tically equal to some non-zero constan t and hence ∥ f ∥ ∞ ≤ | | | f | | | , ∥ g ∥ ∞ ≤ | | | g | | | , and | | | f g | | | ≤ 2 | | | f | | || | | g | | | . (4.3) F or 0 < s < t w e can write   S ( t )( f g )( η ) − [ S ( t ) f ( η )][ S ( t ) g ( η )]   ≤   S ( s )( f g )( η ) − [ S ( s ) f ( η )][ S ( s ) g ( η )]   +   S ( t )( f g )( η ) − S ( s )( f g )( η )   +   S ( t ) f ( η )     S ( t ) g ( η ) − S ( s ) g ( η )   +   S ( s ) g ( η )     S ( t ) f ( η ) − S ( s ) f ( η )   . F or the last three terms w e can use ( 2.2 ), while the first term will b e estimated using Theorem 2.3 . T ogether with ( 4.3 ) this yields   S ( s )( f g )( η ) − [ S ( s ) f ( η )][ S ( s ) g ( η )]   ≤ C | | | f | | || | | g | | | h e C ϱ C γ s ϱ (dist(Λ f , Λ g ) ϱ ( L ) C ϱ + 8e − δ s i , where C := max { C 1 , ˆ K } . Now, w e still hav e freedom in choosing s appropriately to optimise the b ound o v er 0 ≤ s ≤ t . A brief calculation yields that the optimal s ∗ is s ∗ = 1 C ϱ C γ + δ log  8 δ ϱ ( L ) C γ ϱ (dist(Λ f , Λ g ))  . So for t ≥ s ∗ w e can plug this in to obtain the desired b ound with constants given by K := C ( C ϱ C γ + δ )( C 2 ϱ C γ ϱ ( L )) − δ C ϱ C γ + δ (8 /δ ) C ϱ C γ C ϱ C γ + δ and α := δ / ( C ϱ C γ + δ ) . F or t < s ∗ , the b ound follo ws directly from Theorem 2.3 . □ 5. The strong a ttra ctor proper ty F or reasons that will b ecome clear later, see Lemma 6.3 , we will need the following extension of Theorem 2.1 to time-dep endent restrictions. W e will only make use of this result for interacting particle systems on Z but state and pro v e it for arbitrary dimensions d ∈ N . Let Λ ⋐ Z d , fix a non-decreasing function h : [0 , ∞ ) → (0 , ∞ ) and define time-dependent generators b y L h, Λ s f ( η ) = X ∆ ⊂ Λ h ( s ) X ξ ∆ ∈ Ω ∆ c ∆ ( η , ξ ∆ )[ f ( ξ ∆ η ∆ c ) − f ( η )] , η ∈ Ω , f ∈ D (Ω) . The asso ciated flow on C (Ω) will b e denoted by ( S h, Λ s,t ) 0 ≤ s ≤ t and we will also use the notation S h, Λ ( t ) := S h, Λ 0 ,t . Prop osition 5.1 (Refined restriction estimate) . Assume that the c onditions ( L1 ) and ( R1 ) − ( R3 ) ar e satisfie d for ϱ ( r ) = exp( − αr ) for some α > 0 . L et Λ ⋐ Z d and c onsider the time-dep endent r estrictions ( L h, Λ s ) s ≥ 0 for h : [0 , ∞ ) → (0 , ∞ ) define d by h ( s ) = 2 C ϱ C γ α ( t − s ) + L + k . Then, for any k > ( d − 1) /α , ther e exists a c onstant C = C ( d, α, C 1 , C γ , C ϱ , L ) > 0 such that for any initial distribution µ ∈ M 1 (Ω) the total variation err or in Λ is b ounde d as d TV , Λ  µS ( t ) , µS h ( t )  ≤ C | Λ | e − αk k d − 1 . RESTRICTION AND MIXING PROPER TIES 15 Pr o of. By Duhamel’s form ula w e hav e S h, Λ 0 ,t f ( η ) − S t f ( η ) = Z t 0  S h 0 ,s ( L h, Λ s − L ) S t − s  f ( η )d s. No w for any g ∈ D (Ω) we can use a te lescoping trick to obtain the uniform estimate   ( L h, Λ s − L ) g ( η )   ≤ X ∆ ⊂ Λ h ( s ) X ξ ∆ ∈ Ω ∆ | c ∆ ( η , ξ ∆ ) [ g ( ξ ∆ η ∆ c ) − g ( η )] | ≤ C ( d, L, C 1 ) X x / ∈ Λ h ( s ) − L δ x g . After using that for an y h ∈ C (Ω) we hav e ∥ S h, Λ 0 ,s h ∥ ∞ ≤ ∥ h ∥ ∞ , one can apply the ab o ve estimate to the function g = S t − s f to see that sup η   S h, Λ 0 ,t f ( η ) − S t f ( η )   ≤ Z t 0 sup η    S h, Λ 0 ,s ( L h, Λ s − L ) S t − s  f ( η )   d s ≤ Z t 0 sup η    ( L h, Λ s − L ) S t − s  f ( η )   d s ≤ C ( d, L, C 1 ) Z t 0 X x / ∈ Λ h ( s ) − L δ x ( S t − s f )d s. Here we can no w use that, in the case where the dep endence of the transition rates deca ys exponentially , a combination of Lemma 4.1 and Lemma 4.4 yields δ x ( S t − s f ) ≤ exp(( t − s )Γ)[ δ · f ]( x ) ≤ ∥ f ∥ ∞ C − 1 ϱ exp  C γ C ϱ ( t − s ) − α · dist( x, Λ)  and hence Z t 0 X x / ∈ Λ h ( s ) − L δ x ( S t − s f ) ds ≤ ∥ f ∥ ∞ C − 1 ϱ Z t 0 X x / ∈ Λ h ( s ) − L exp  C γ C ϱ ( t − s ) − α · dist( x, Λ)  d s. W e first upper b ound the sum b y using that there are O ( r d − 1 ) points at distance equal to r for an y giv en site x ∈ Z d , i.e., Z t 0 X x / ∈ Λ h ( s ) − L exp  C γ C ϱ ( t − s ) − α · dist( x, Λ)  d s ≤ C ( d ) | Λ | Z t 0 exp  C γ C ϱ ( t − s )  X r ≥ h ( s ) − L exp( − αr ) r d − 1 d s. No w we can use that the function r 7→ exp( − αr ) r d − 1 is non-increasing on the interv al (( d − 1) /α , ∞ ), so by c ho osing k sufficien tly large w e can estimate the sum b y an in tegral o v er a slightly larger domain to get Z t 0 X x / ∈ Λ h ( s ) − L exp  C γ C ϱ ( t − s ) − α · dist( x, Λ)  d s ≤ C ( d ) Z t 0 exp  C γ C ϱ ( t − s )  Z h ( s ) − L − 1 exp( − αr ) r d − 1 d r d s ≤ C ( d, α ) Z t 0 exp  C γ C ϱ ( t − s )  Z ∞ α ( h ( s ) − L − 1) u d − 1 exp( − u )d u d s, where w e applied a c hange of v ariable u = α − 1 r in the inner integral to get the last inequalit y . Note that the undefined constants ma y v ary from line to line. By using the RESTRICTION AND MIXING PROPER TIES 16 recursion formula for the upp er incomplete Gamma function, see Lemma 5.2 b elo w for details, this can in turn b e estimated as Z t 0 exp  C γ C ϱ ( t − s )  Z ∞ α ( h ( s ) − L − 1) u d − 1 exp( − u )d u d s ≤ C ( d, α ) Z t 0 exp  C γ C ϱ ( t − s ) − α ( h ( s ) − L − 1)  α ( h ( s ) − L − 1)  d − 1 d s. No w, since we chose h ( s ) = 2 C γ C ϱ α ( t − s ) + L + 1 + k , where k > 0 is some constan t that is assumed to b e sufficien tly large to make use of the previously men tioned monotonicity , w e get Z t 0 exp  C γ C ϱ ( t − s ) − α ( h ( s ) − L − 1)  α ( h ( s ) − L − 1)  d − 1 d s ≤ C ( α, d, C γ , C ϱ )e − αk k d − 1 Z t 0 exp  − C γ C ϱ ( t − s )  ( t − s ) d − 1 d s ≤ C ( α, d, C γ , C ϱ )e − αk k d − 1 . Putting ev erything together yields the claimed upp er bound. □ In the abov e proof w e used the following elementary estimate for the Gamma function. Lemma 5.2. F or d ∈ N and x > 0 , the upp er inc omplete Gamma functions define d by Γ( d, x ) = Z ∞ x r d − 1 e − r d r , satisfies the upp er b ound Γ( d, x ) ≤ e( d − 1)! e − x x d − 1 . Pr o of. Via integration-b y-parts one obtains the recurrence relation Γ( n + 1 , x ) = n · Γ( n, x ) + x n e − x and using this inductiv ely yields the explicit form ula Γ( d, x ) = ( d − 1)!e − x d − 1 X n =0 x n n ! , whic h directly yields the claimed estimate. □ 6. Rela tive entropy, speed-up, and time-shift F or t w o probabilit y la ws P , Q on a measurable space ( X , X ) w e define the r elative entr opy of P with r esp e ct to Q by H ( P | Q ) = ( R X log(d P / d Q )d P , if P ≪ Q , ∞ , otherwise. W e begin b y stating the follo wing Girsano v-type form ula, whic h we will use to compare the sped-up pro cess with the original dynamics. RESTRICTION AND MIXING PROPER TIES 17 Lemma 6.1 (Girsanov formula) . Consider a c ontinuous-time Markov chain with time- inhomo gene ous gener ator ( L s ) s ≥ 0 on a finite state sp ac e X and let ( ˆ L s ) s ≥ 0 b e the gen- er ator of another c ontinuous-time Markov chain such that for every s ≥ 0 the tr ansition r ates of L s and ˆ L s satisfy the c ondition L s ( x, y ) = 0 ⇒ ˆ L s ( x, y ) = 0 , ∀ x, y ∈ X . A dditional ly, assume that the set D of disc ontinuity p oints of the set of tr ansition r ates { L · ( x, y ) : x, y ∈ X } ∪ { ˆ L · ( x, y ) : x, y ∈ X } has zer o L eb esgue me asur e. Denote the induc e d p ath me asur es on the sp ac e of X -value d c´ ad l´ ag p aths X ([0 , t ]) up to time t > 0 by Q x r esp e ctively ˆ Q x , wh er e the initial c ondition X (0) = x is deterministic. Then, the fol lowing Girsanov-typ e formula holds d Q x d ˆ Q x  X ([0 , t ])  = exp  − Z t 0 δ s ( X ( s )) + X s ∈ [0 ,t ] : X ( s − )  = X ( s ) X y  = X ( s ) log L s ( X ( s − ) ,X ( s )) ˆ L s ( X ( s − ) ,X ( s ))  d s, wher e δ s ( x ) := X y  = x  L s ( x, y ) − ˆ L s ( x, y )  . See for example [ KL99 , Prop osition 2.6. in App endix 1] for a pro of of the Girsanov form ula for con tinuous-time Marko v c hains in the time-homogeneous case. The extension to inhomogeneous transition rates follo ws along similar lines but is somewhat tedious, so w e omit it here. W e can no w use this to obtain b ounds for the relativ e en trop y on path space. Lemma 6.2 (En tropic cost of sp eed-up) . L et X b e a finite set and ( L s ) s ≥ 0 gener ators of a time-inhomo gene ous c ontinuous-time Markov chain ( X ( t )) t ≥ 0 . Define the maximal r ate at time s via c ( s ) := max x ∈X X y  = x L s ( x, y ) . L et t, τ > 0 and λ : [0 , t ] → (0 , ∞ ) a b ounde d and me asur able function. Denote by ( L λ s ) s ≥ 0 the gener ators of the sp e d-up pr o c ess, i.e., L λ s f ( x ) = X y ∈X L s ( x, y )[ f ( y ) − f ( x )] , x ∈ X , f : X → R . We assume that the set D ⊂ [0 , ∞ ) of disc ontinuity p oints of the set of jumps r ates { L · ( x, y ) : x, y ∈ X } and λ ( · ) has zer o L eb esgue me asur e. F or some fixe d initial distribu- tion ρ ∈ M 1 ( X ) let P r esp e ctively P λ denote the law of the asso ciate d Markov pr o c ess with gener ator L r esp e ctively L λ . Then, we have H ( P λ | P ) ≤ Z t 0 c ( s )  λ ( s ) − 1  2 d s. (6.1) Pr o of. W e start by calculating the Radon–Nik o dym density via the Girsano v formula d P λ d P ( X ([0 , t ])) = exp  X s ∈ [0 ,t ] : X s −  = X s log λ ( s ) − Z t 0  λ ( s ) − 1  X y  = X s L ( X s , y )d s  . RESTRICTION AND MIXING PROPER TIES 18 By definition of the relativ e entrop y and c ( · ) this implies H ( P λ | P ) = Z log  d P λ d P ( X ([0 , t ]))  P λ (d X [0 , t ]) = Z t 0  Z X y  = X s L ( X s , y ) P λ (d X ([0 , t ]))   λ ( s ) log  λ ( s )  −  λ ( s ) − 1  d s ≤ Z t 0 c ( s )  λ ( s ) log  λ ( s )  −  λ ( s ) − 1  d s ≤ Z t 0 c ( s )  λ ( s ) − 1  2 d s, where w e used that log x ≤ x − 1 for all x > 0 in the last step. □ No w if c ( · ) gro ws linearly , then b y choosing a constant sp eed-up function λ ≡ (1 + τ /t ) w e w ould only get H ( P λ | P ) ≲ τ 2 , whic h do es not allo w us to conclude an ything ab out the t ↑ ∞ limit. But we can still optimise ov er the sp eed-up to bring us back in to the game. Lemma 6.3 (Minimal cost) . Denote the set of admissible sp e e d-ups by H t,τ := n λ ∈ L 1 ([0 , t ]) : Z t 0  λ ( s ) − 1  d s = τ o . Then, for any f : [0 , t ] → (0 , ∞ ) such that f , f − 1 ∈ L 1 ([0 , t ]) we have inf λ ∈ H t,τ Z t 0 f ( s )  λ ( s ) − 1  2 d s = τ 2  Z t 0 1 f ( s ) d s  − 1 and the infimum is attaine d at λ ∗ ( s ) = 1 + τ  Z t 0 f ( s ) f ( r ) d r  − 1 , s ∈ [0 , t ] . Pr o of. By using the formalism of conv ex optimisation with constraints, the problem can b e b oiled do wn to determining the critical p oin ts of the Lagrangian L ( f , γ ) = Z t 0 f ( s )  λ ( s ) − 1  2 d s − γ  Z t 0  f ( s ) − 1  d s − τ  , γ ∈ R , λ ∈ H t,τ . It therefore suffices to determine λ ( · ) such that 2 Z t 0 f ( s )( λ ( s ) − 1)d s − γ t = 0 . An elemen tary calculation shows that this can b e done by choosing λ ( s ) = 1 + γ / (2 f ( s )) . It remains to determine the correct v alue for the Lagrange multiplier γ to mak e sure the constrain t R t 0 ( λ ( s ) − 1)d s = τ is satisfied. This yields the equation γ = 2 τ  Z t 0 1 f ( s ) d s  − 1 and plugging this in gives precisely the claimed form ula for the minimiser and the min- im um. □ RESTRICTION AND MIXING PROPER TIES 19 Remark 6.4. The optimality statement in L emma 6.3 dir e ctly tel ls us that this appr o ach c an only work if the maximal r ate c ( s ) of le aving a p articular state at time s do es not gr ow to o fast, sinc e we ne e d that lim t →∞ Z t 0 1 c ( s ) d s = ∞ . This rules out an applic ation of this metho d for dimensions d > 1 wher e one should exp e ct c ( s ) ∼ s d . One c ould however extend the r esult to gr aphs whose volume gr ows just a tiny bit faster than Z so that the inte gr al stil l blows up as t → ∞ . Let us finally put everything together and provide the pro of of the strong attractor prop ert y in dimension one. Pr o of of The or em 2.5 . F or ev ery Λ ⋐ Z , the triangle inequality implies d TV , Λ ( µ t , µ t + τ ) ≤ d TV , Λ ( µ t , µ h t ) + d TV , Λ ( µ h t , µ h t + τ ) + d TV , Λ ( µ h t + τ , µ t + τ ) . The first and the third term can b e estimated by using Prop osition 5.1 and Pinsker’s inequalit y allo ws us to b ound the second term via d TV , Λ ( µ h t , µ h t + τ ) ≤ r 1 2 H ( µ h,λ t | µ h t ) . So b y plugging in the corresp onding estimates w e get d TV , Λ ( µ t , µ t + τ ) ≤ 2 C | Λ | e − αk k d − 1 + τ √ 2  Z t 0 1 2 h ( s ) d s  − 1 / 2 , where h ( s ) = cs + L + k for some fixed c > 0. By first sending t to infinit y we see that for an y sufficiently large k > 0 lim sup t →∞ d TV , Λ ( µ t , µ t + τ ) ≤ 2 C | Λ | e − αk k d − 1 =: F ( k ) , but since k is arbitrary and F ( k ) → 0 as k ↑ ∞ , the limit must b e equal to 0. □ 7. Outlook F rom the results in [ JK25a , K¨ o26 ] and the long-range construction in [ JK25b ], w e exp ect that for S = Z , ev en for interacting particle systems with γ ( x, y ) ≤ | x − y | − α and α > 2, time-p erio dic b eha viour should b e imp ossible, y et the metho d used in this article fails in this regime. This limitation is mainly due to the fact that the relativ e-en tropy b ound in Lemma 6.2 and Lemma 6.3 requires a linear growth of the region in whic h the particles participate in the dynamics. How ev er, if the interaction strength only decays lik e a p o w er law, the error b ound in Theorem 2.1 do es not decay if h ( t ) ∼ ct for some constan t c > 0. Unfortunately , there is little hop e that refining the metho d based on an analysis of the op erator Γ can yield results in the p ow er-law regime. Indeed, let us assume for a momen t that the rates are translation inv arian t. Then, we hav e γ ( x, y ) = ˆ γ ( x − y ) for some function ˆ γ : Z d → [0 , ∞ ) and denoting the operator norm of Γ by M w e can write e t Γ = e tM e tQ , where Q is the generator of a random walk on Z d with transition rates giv en b y ( ˆ γ ( x )) x ∈ Z d . The first factor alw a ys giv es us exp onen tial gro wth in t , but heat- k ernel b ounds for random walks with heavy-tailed jump k ernel tell us that the second part cannot comp ensate this exp onen tial growth by decaying sufficiently fast in space. This back of the env elope calculation suggests that one cannot use the strategy in this man uscript to extend Theorem 2.5 to regimes with p o w er-la w decay . W e do ho wev er RESTRICTION AND MIXING PROPER TIES 20 b eliev e, that the result is still true for suc h systems, at least when α > 2. F or α ∈ (1 , 2) there are counterexamples, see [ JK25b ]. A setting that can b e used as a test case and where one has some more to ols av ailable is if one considers interacting particle systems with ”random-range” interactions. As an example, consider the transition rates as in ( 1.1 ) but for ev ery update, first sample the range of the Hamiltonian from some radius distribution µ ∈ M 1 ([0 , ∞ )) and then use the truncated Hamiltonian to resample the spin. One can then first extend the graphical represen tation in [ Swa26 , Chapter 4] to this setting and use a discrete version of the mo del in [ Dei03 ] and ideas from [ GM08 ] and [ CGGK93 ] to prov e that information spreads at linear sp eed as long as the radius distribution satisfies µ ( r ) ∼ r − α for α > 2 d + 1. Note that this is the threshold at which the sp eed in long-range first passage p ercolation c hanges from linear to polynomial, see [ CSD16 ], it is therefore not so surprising that α > 2 d + 1 is the b est one can do. The strategy describ ed ab o v e will b e carried out in [ JK26 ]. A cknowledgments BJ and JK received supp ort by the Leibniz Asso ciation within the Leibniz Junior Re- searc h Group on Pr ob abilistic Metho ds for Dynamic Communic ation Networks as part of the Leibniz Comp etition (gran t no. J105/2020). BJ is also funded b y the Deutsche F orsch ungsgemeinsc haft (DFG, German Researc h F oundation) under Germany’s Excel- lence Strategy – The Berlin Mathematics Researc h Cen ter MA TH+ (EX C-2046/1, EX C- 2046/2, pro ject ID: 390685689) through the pro jects EF45-3 on Data T r ansmission in Dynamic al R andom Networks and EF-MA-Sys-2 on Information Flow & Emer gent Be- havior in Complex Networks , as well as the SPP2265 Pro ject P27 Gibbs p oint pr o c esses in r andom envir onment . References [AFM + 25a] Y. Avni, M. F ruchart, D. Martin, D. Seara, and V. Vitelli. Dynamical phase transitions in the nonrecipro cal Ising mo del. Physic al Review E , 111(3):034124, 2025. [AFM + 25b] Y. Avni, M. F ruchart, D. Martin, D. Seara, and V. Vitelli. Nonreciprocal Ising mo del. Physic al R eview L etters , 134(11):117103, 2025. [BGH + 90] C. H. Bennett, G. Grinstein, Y. He, C. Jay aprak ash, and D. Muk amel. Stability of temp o- rally p eriodic states of classical man y-bo dy systems. Physic al R eview A , 41(4):1932–1935, 1990. [CGGK93] J. T. Co x, A. Gandolfi, P . S. Griffin, and H. Kesten. Greedy lattice animals I: Upper bounds. The Annals of Applie d Prob ability , 3(4), 1993. [CM92] H. Chat´ e and P . Manneville. Collective b eha viors in spatially extended systems with lo cal in teractions and synchronous up dating. Pr o gr ess of The or etical Physics , 87(1):1–60, 1992. [CSD16] S. Chatterjee and P . S. Dey . Multiple phase transitions in long-range first-passage p ercola- tion on square lattices. Communic ations on Pur e and Applie d Mathematics , 69(2):203–256, 2016. [Dei03] M. Deijfen. Asymptotic shap e in a contin uum growth mo del. A dvanc es in Applie d Pr ob abil- ity , 35(2):303–318, 2003. [GB24] L. Guislain and E. Bertin. Collectiv e oscillations in a three-dimensional spin mo del with non- recipro cal interactions. Journal of Statistic al Me chanics: The ory and Exp eriment , 2024(9), 2024. [GM08] J.-B. Gou´ er ´ e and R. Marchand. Con tin uous first-passage percolation and contin uous greedy paths mo del: Linear growth. The Annals of Applie d Pr ob ability , 18(6), 2008. [GMSB93] G. Grinstein, D. Muk amel, R. Seidin, and C. H. Bennett. T emp orally p erio dic phases and kinetic roughening. Physic al R eview L etters , 70(23):3607–3610, 1993. [JK14] B. Jahnel and C. K ¨ ulske. A class of nonergo dic interacting particle systems with unique in v arian t measure. The Annals of Applie d Pr ob ability , 24(6), 2014. RESTRICTION AND MIXING PROPER TIES 21 [JK25a] B. Jahnel and J. K¨ oppl. On the long-time behaviour of reversible in teracting particle systems in one and t wo dimensions. Pr ob ability and Mathematic al Physics , 6(2):479–503, 2025. [JK25b] B. Jahnel and J. K¨ oppl. Time-p eriodic b eha viour in one- and tw o-dimensional interacting particle systems. Annales Henri Poinc ar´ e , 2025. [JK26] B. Jahnel and J. K¨ oppl. Interacting particle systems with random range: Restriction esti- mates and long-time b ehaviour. 2026+. In preparation. [KL99] C. Kipnis and C. Landim. Sc aling Limits of Interacting Particle Systems , volume 320 of Grund lehr en der Mathematischen Wissenschaften . Springer Berlin Heidelb erg, Berlin, Hei- delb erg, 1999. [K¨ o26] J. K¨ oppl. On the absence of time-translation symmetry breaking in some non-reversible in teracting particle systems, 2026. [Lig05] T. M. Liggett. Inter acting Particle Systems . Classics in Mathematics. Springer Berlin Hei- delb erg, Berlin, Heidelb erg, 2005. [LR72] E. H. Lieb and D. W. Robinson. The finite group velocity of quantum spin systems. Com- munic ations in Mathematic al Physics , 28(3):251–257, 1972. [Mar99] F. Martinelli. Lectures on Glaub er Dynamics for Discrete Spin Models. In P . Bernard, editor, L e ctur es on Pr ob ability Theory and Statistics , volume 1717, pages 93–191. Springer Berlin Heidelb erg, Berlin, Heidelb erg, 1999. Series Title: Lecture Notes in Mathematics. [Mou95] T. S. Mountford. A coupling of infinite particle systems. Kyoto Journal of Mathematics , 35(1), 1995. [NS10] B. Nac htergaele and R. Sims. Lieb–Robinson b ounds in quantum man y-bo dy physics, 2010. [R V96] A. F. Ramirez and S. R. S. V aradhan. Relativ e entrop y and mixing properties of interacting particle systems. Kyoto Journal of Mathematics , 36(4), 1996. [Sw a26] J. M. Sw art. A Course in Inter acting Particle Systems . Cambridge Universit y Press, 2026. Technische Universit ¨ at Braunschweig & Weierstrass Institute, Berlin, Germany. Email addr ess : benedikt.jahnel@tu-braunschweig.de Weierstrass Institute, Berlin, Germany. Email addr ess : koeppl@wias-berlin.de