On the Ergodicity of Interacting Particle Systems under Number Rigidity

In this paper, we provide relations among the following properties: (a) the tail triviality of a probability measure $\mu$ on the configuration space ${\boldsymbol\Upsilon}$; (b) the finiteness of the $L^2$-transportation-type distance $\bar{\mathsf d}_{{\boldsymbol\Upsilon}}$; (c) the irreducibility of $\mu$-symmetric Dirichlet forms on ${\boldsymbol\Upsilon}$. As an application, we obtain the ergodicity (i.e., the convergence to the equilibrium) of interacting infinite diffusions having logarithmic interaction arisen from determinantal/permanental point processes including $\mathrm{sine}_{2}$, $\mathrm{Airy}_{2}$, $\mathrm{Bessel}_{\alpha, 2}$ ($\alpha \ge 1$), and $\mathrm{Ginibre}$ point processes, in particular, the case of unlabelled Dyson Brownian motion is covered. For the proof, the number rigidity of point processes in the sense of Ghosh--Peres plays a key role.


Introduction
The ergodicity (i.e., the convergence to the equilibrium) of interacting particle systems is one of the significant hypothesis supporting the foundation of statistical physics.In this paper, we study the ergodicity in terms of the optimal transportation theory and of the theory of point processes.
Configuration spaces.The configuration space Υ = Υ(X) over a locally compact Polish space X is the set of all locally finite point measures on X: The space Υ is endowed with the vague topology τ v by the duality of compactly supported continuous functions on X, and with a Borel probability measure µ, understood as the law of a proper point process on X.
Interacting diffusions.A system of interacting many diffusions on the base space X can be thought of as a single diffusion on Υ, provided the system does not condense too much by itself in the sense that every compact set in X contains only finitely many particles throughout the time evolutions.There have been a large number of studies on a diffusion in Υ, in particular, on a system of infinite stochastic differential equations on R n , written 'formally' as whereby Φ is a free potential, Ψ is an interaction potential between particles, β > 0 is a constant called inverse temperature, and B k k∈N are independent Brownian motions on R n .One approach addressing a solution to (1.1) is to construct a µ-symmetric Dirichlet form E Υ,µ , F Υ,µ on L 2 (Υ, µ), where µ is a (quasi-) Gibbs measure corresponding to the potentials Φ and Ψ, see, e.g., [AKR98b,Yos96] for Ruelle class potentials; [Spo87, Osa96, Yoo05, Osa13, OT14, HO15, DS21, Suz22] for more general interactions including logarithmic potentials.The other approaches to tackle (1.1) have been also studied such as the construction of time correlation functions in [Dys62,NF98,KT10]; the construction of the unique strong solution to (1.1) in the case of the Dyson models in [Tsa16].We refer the readers to [Rö09,Osa21] and also [DS21,§1.6]for more complete accounts and references.
Up to now, there were only few known examples, where one could show the ergodicity of {S Υ,µ t } in the case of infinite particle diffusions: one is a a class of Ruelle-type Gibbs measures with a compactly supported interaction potential and a small activity constant z ([AKR98b, Cor.6.2]); the other is a labelled particle system corresponding to the sine 2 process, which has been recently addressed in [OT21] by relying upon the arguments of strong solutions to (1.1) developed in [OT20].
Optimal transport theory on Υ.If the base space X is equipped with a metric d, the configuration space Υ is equipped with the L 2 -transportation (called also: L 2 -Wasserstein, or L 2 -Monge-Kantorovich-Rubinstein) distance , where the infimum is taken over all measures q on X ×2 with marginals γ and η.As opposed to the case of the space of probability measures having finite second moment (i.e., the L 2 -Wasserstein space), the function d Υ cannot be a distance function because d Υ may attain the value +∞ (e.g., when the total masses of γ and η are different, or the tails of γ and η are not close enough), and this often happens in the sense that this occurs on sets of positive measure for any reasonable choice of a reference measure on Υ.It is, therefore, called extended distance.In this article, we use a variant of d Υ defined as if γ E c = η E c for some bounded set E , +∞ otherwise .
Recent studies have revealed that the L 2 -transportation distance is the right object to describe geometry, analysis and stochastic analysis in Υ such as the curvature bounds on Υ ([EH15, DS22,Suz22]), the consistency between metric measure geometry and Dirichlet forms ([RS99, DS21], characterisations of BV functions and sets of finite perimeters on Υ ( [BS21]) and the integral Varadhan short-time asymptotic ( [Zha01,DS22]).
Theory of point processes.A probability measure µ on Υ is said to be tail trivial (T) 2.6 if (see Dfn. 2.6) µ(A) ∈ {0, 1} whenever A is a set in the tail σ-algebra .
The tail triviality has been originally discussed in relation to phase transition of Gibbs states (i.e., non-uniqueness of Gibbs measures with a given potential) and it is equivalent to the extremality in the convex set of Gibbs measures with a given potential (see [Geo11,Cor. 7.4]).The tail triviality has been extended also to determinantal/permanental point processes by [Lyo03] and [ST03] independently.Since then, it has been further developed for a wider class of determinantal/permantental processes both in the continuous and discrete settings by various studies, see Example 2.7.A probability measure µ on Υ is said to be number rigid (Assumption (R) 2.8 ) if the following holds µ-almost surely for every bounded Borel set E: Namely, if two configuration γ and η coincide outside E, then the numbers of particles inside E for γ and η coincide.The study of this remarkable spatial correlation phenomenon has been initiated by [Gho12,Gho15,GP17] for sine 2 , Ginibre and GAF point processes and it has been further developed for other point processes, see Example 2.9 for further references.
Setting.In this article, we work in the following setting.Let X = R n be the ndimensional Euclidean space and d be the Euclidean distance on R n .Let (B r ) r∈N be a monotone increasing sequence of convex compact domains covering R n and m r be the Lebesgue measure restricted on B r .For E ⊂ R n , define the projection pr E : Υ ∋ γ → γ E := γ ⇂ E by the restriction of γ on E. For a Borel probability measure µ on Υ, define to be the regular conditional probability measure with respect to the σ-algebra σ(pr B c r ) conditioned to be η ∈ Υ. Define the push-forwarded measure and its restriction on We denote by π mr the Poisson measure on Υ(B r ) with intensity m r and by π k mr the restriction on Υ k (B r ).Let Γ Υ(Br) be the square field on Υ(B r ) defined as where ∇ ⊙k is the symmetric product of the gradient operator ∇ on R n .List of Assumptions.We say that µ satisfies • conditional closability (CC) 3.2 if the form )-closable on a certain core (see Dfn. 3.2) for µ-a.e.η and every r ∈ N. We denote its closure by D(E Υ(Br),µ η r ); for µ-a.e.η, r ∈ N, k ∈ K η r , where C η,k r is a constant depending on r, η, k.
Main result.We define the following function associated with the L 2 -transportationtype distance dΥ : We now state the main theorem, where we provide relations among the following three properties: Theorem I (Thm.4.6).Let µ be a Borel probability measure on Υ.Then, Suppose that µ satisfies (CAC ′ ) 3.1 and (CC) 3.2 , and F Υ,µ ⊂ D(E Υ,µ ) is any closed Markovian subspace.Then the following hold.
We therefore have the following relation between the tail triviality and the irreducibility.
• If (CI) 4.1 , (QR) 3.20 and (R) 2.8 hold, then Applications.The first application of Theorem I as well as Corollary I is to considerably enlarge the list of (long-range) interactions for which one can prove the ergodicity of infinite particle systems.As an illustration, we will prove in §6 that (E Υ,µ , F Υ,µ ) is irreducible (i.e., {S Υ,µ t } is ergodic) for all the measures µ belonging to sine 2 , Airy 2 , Bessel α,2 (α ≥ 1), and Ginibre point processes.In particular, the semigroup {S Υ,µ t } associated with unlabelled Dyson Brownian motion is covered.
The second application is to show the finiteness of the L 2 -transportation distance d Υ (A, B) as well as dΥ (A, B) between sets A, B ⊂ Υ.As both d Υ and dΥ take value +∞ on sets of positive measure, it is not straightforward to answer the following geometric question: when do d Υ (A, B) and dΥ (A, B) return a finite value?(Q) Theorem I tells us the finiteness of dΥ (A, B) (thus, also the finiteness of d Υ (A, B) as d Υ ≤ dΥ by definition) only by checking the positivity of measures µ(A)µ(B) > 0, due to the tail triviality (T) 2.6 and the number rigidity (R) 2.8 of µ.
Comparisons with [AKR98b].For a class of Gibbs measures or measures satisfying a certain integration-by-parts formula (denoted by (IbP1) and (IbP2) in [AKR98b, Thm.6.2, 6.5]), relations between the ergodicity and the extremality of these measures have been studied.We compare our result with theirs in the following three points: • Choice of a core.They studied Dirichlet forms whose core is cylinder functions while our Dirichlet forms have a flexibility for the choice of a core, which for instance allows not only cylinder functions, but also local functions as well as Lipschitz functions.This broadens the score of applications significantly as cores of Dirichlet forms corresponding to long-range interactions constructed so far (e.g., [Spo87, Osa96, Osa13, OT14, HO15, DS21, Suz22]) are covered by our setting, but not necessarily covered by the setting of cylinder functions.• Extremality vs. Tail-triviality.They proved that the extremality of a class of Gibbs measures implies the ergodicity.The concept of the extremality is equivalent to the tail triviality when Gibbs measures are considered (see [Geo11,Cor. 7.4]).However, the extremality is not necessarily defined beyond Gibbs measures nor beyond measures satisfying (IbP1) and (IbP2), and many point processes coming from random matrix theory are not always described as Gibbs measures nor (IbP1) and (IbP2), rather they are described by determinantal or permanental structures or by a scaling limit of eigenvalue distributions of random matrices.In contrast, the tail triviality is a concept that can be defined for arbitrary point processes, because of which Theorem I can be applied also to the latter cases.• Maximal domain vs. Rademacher-type property.They proved that the irreducibility of the maximal Dirichlet form implies the extremality of Gibbs measures, which corresponds to (c) =⇒ (a) in Thereom I. We however only assume the Rademacher-type property (Rad dΥ ,µ ) 3.21 of our Dirichlet form, whose domain is in general smaller than the maximal form.As the irreducibility of a larger domain is a stronger statement, Theorem I proved the extremality of Gibbs measures (as well as the tail triviality of general measures) under a weaker assumption.
Geometry and statistical physcis.We would like to draw the reader's attention that the relations between (b) and (c) in Theorem I provides a relation between the ergodicity of interacting diffusion processes and a quantitative information of the optimal transport distance, where the ergodicity is a statistical-physical concept, while the finiteness of the L 2 -transportation distance between µ-positive sets is a purely geometric concept of the extended metric measure space (Υ, dΥ , µ).
We close this introduction by providing an outlook on further improvements.The number rigidity (R) 2.8 requires a strong spatial correlation to µ, which is, however, not a necessary condition for the ergodicity.Indeed, [AKR98a, Thm.4.3] proved the ergodicity for the Poisson measures, which obviously do not posses the number rigidity (R) 2.8 since the laws of the Poisson point processes inside and outside bounded sets are independent.A challenging question is whether we can prove the ergodicity of Dirichlet forms for general tail trivial invariant measures without (R) 2.8 .
Organisation of the paper.In §2, we introduce necessary concepts and recall results used for the arguments in later sections.In §3, we construct Dirichlet forms on Υ.In §4, we prove the main results.In §5, we give sufficient conditions to verify the main assumptions of Theorem I.In §6, we confirm that Theorem I can be applied to sine 2 , Airy 2 , Bessel α,2 (α ≥ 1), and Ginibre point processes.
Data Availability Statement.Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Let (X, τ ) be a topological space with σ-finite Borel measure ν.Throughout this article, we shall use the following symbols and phrases: (a) L p (ν) (1 ≤ p ≤ ∞) for the space of ν-equivalence classes of real-valued functions u so that |u| p is ν-integrable when 1 ≤ p < ∞, and u is ν-essentially bounded when p = ∞; The L p (ν)-norm is denoted by u p L p (ν) := u p p := X |u| p dν for 1 ≤ p < ∞, and u L ∞ (ν) := u ∞ = esssup X u; When p = 2, the inner-product is denoted by (u, v) L 2 (ν) := (u, v) 2 := X uv dν; (b) L p s (ν ⊗n ) := {u ∈ L p (ν ⊗n ) : u is symmetric} where u is said to be symmetric if u(x 1 , . . ., x k ) = u(x σ(1) , . . ., x σ(k) ) for every element σ ∈ S(k) in the k-symmetric group; (c) B(X, τ ) for the Borel σ-algebra; B(X, τ ) ν for the completion of B(X, τ ) with respect to ν; B(X, τ ) * be the universal σ-algebra, i.e., the intersection of B(X) ρ among all Borel probability measures ρ on X (we do not specify the topology and simply write B(X), B(X) ν , B(X) * where the topology is clear from the context); Measurable functions with respect to B(X), B(X) ν , B(X) * are called Borel measurable, ν-measurable, universally measurable respectively.(d) C b (X) for the space of τ -continuous bounded functions on X; if X is locally compact, C 0 (X) denotes the space of τ -continuous and compactly supported functions on X; C ∞ 0 (R n ) for the space of compactly supported smooth functions on the n-dimensional Euclidean space R n ; (e) 1 A for the indicator function on A, i.e., 1 A (x) = 1 if and only if x ∈ A and 1 A (x) = 0 otherwise; δ x for the Dirac measure at x, i.e., δ x (A) = 1 if and only if and ∪ r∈N B r = X; If B r possesses a certain property P for every r ∈ N (e.g., B r is compact, convex, or domain), we call it P exhaustion (e.g., compact exhaustion, compact convex exhaustion, domain exhaustion).
2.2.Dirichlet form.We refer the reader to [MR90,BH91] for this subsection.Throughout this article, a Hilbert space always means a separable Hilbert space with inner product (•, •) H taking value in R.
Dirichlet form.Given a bilinear form (Q, D(Q)) on a Hilbert space H, we write Let (X, Σ, ν) be a σ-finite measure space.A symmetric Dirichlet form on L 2 (ν) is a non-negative definite densely defined closed symmetric bilinear form (Q, D(Q)) on L 2 (ν) satisfying the Markov property Throughout this article, Dirichlet form always means symmetric Dirichlet form.A subspace F ⊂ D(Q) is called Markovian subspace if (2.1) holds for every u ∈ F. If not otherwise stated, D(Q) is always regarded as a Hilbert space with norm To distinguish Dirichlet forms defined in different base spaces with different reference measures, we write Q X,ν to specify the base space X and the reference measure ν.We denote the extended domain of D(Q) by D(Q) e defined as This v is denoted by Γ(u).The square field Γ can be uniquely extended as an operator on Resolvent, semigroup and generator.We refer the reader to [MR90, Chap.I, Sec.2] for this paragraph.Let (Q, D(Q)) be a symmetric closed form on a Hilbert space H.The infinitesimal generator (A, D(A)) corresponding to (Q, D(Q)) is the unique densely defined closed operator on H satisfying the following integration-by-parts formula: The resolvent operator {R α } α>0 is the unique bounded linear operator on H satisfying The semigroup {T t } t>0 is the unique bounded linear operator on H satisfying Quasi-notion.Let (X, τ ) be a Polish space and ν be a σ-finite Borel measure on X and (Q, D(Q)) be a Dirichlet form on L 2 (ν).For any A ∈ B(X), define depending on x ∈ X holds quasi-everywhere (in short: q.e.) if there exists a polar set N so that (p x ) holds for every x ∈ X \ N .For a closed nest is quasi-regular if the following conditions hold: (QR1) there exists a compact nest (A n ) n∈N ; (QR2) there exists a dense subspace D ⊂ D(Q) so that every u ∈ D has a quasicontinuous ν-version ũ; (QR3) there exists {u n : n ∈ N} ⊂ D(Q) and a polar set N ⊂ X so that every u n has a quasi-continuous ν-version ũn and {ũ n : n ∈ N} separates points in X \ N .
is called an extended distance if it is symmetric, satisfying the triangle inequality and not vanishing outside the diagonal in X ×2 , i.e. d(x, y) = 0 iff x = y; a distance if it is finite, i.e., d(x, y) < ∞ for every x, y ∈ X.A space X equipped with an extended metric d is called an extended metric space (X, d).Let ν be a measure on X on a σ-algebra Σ. Define the latter of which is well-defined whenever inf y∈B d(•, y) is ν-measurable (i.e., Σ ν -measurable).

Lipschitz algebra. A function
The smallest constant L satisfying (2.5) is called the (global) Lipschitz constant of u, denoted by Lip d (u).For any non-empty set A ⊂ X, define Lip(A, d), resp.Lip b (A, d) for the family of all d-Lipschitz functions, resp.bounded d-Lipschitz functions on A. For simplicity of notation, we omit specifying the base space X and simply write Lip(d For a measure ν on X defined on a σ-algebra Σ and a topology τ on X, we define respectively Let ν be a finite measure on X and let (Q, D(Q)) be a local Dirichlet form on L 2 (ν) having a square filed Γ Q .We say that Rademacher-type property holds for Lip b (d, ν) 2.4.Configuration space.A configuration on a locally compact Polish space X is an N 0 -valued Radon measure γ on X, which can be expressed by γ = N i=1 δ x i for N ∈ N 0 , where x i ∈ X for every i and γ ≡ 0 when N = 0.The configuration space Υ = Υ(X) is the space of all configurations over X.The space Υ is equipped with the vague topology τ v , i.e., the topology generated by the duality of the space C 0 (X) of continuous functions with compact support.We write the restriction γ A := γ ⇂ A for A ∈ B(X) and the restriction map is denoted by The N -particle configuration space is denoted by Let S k be the k-symmetric group.It can be readily seen that the k-particle configuration space Υ k is isomorphic to the quotient space X ×k /S k : The associated projection map from X ×k to the quotient space X ×k /S k is denoted by P k .For η ∈ Υ and E ∈ B(X), we define Conditional probability.For a Borel probability measure µ on Υ and E ∈ B(X), denotes the regular conditional probability of µ conditioned to be η ∈ Υ with respect to the σ-algebra generated by the projection map γ ∈ Υ → pr E (γ) = γ E ∈ Υ(E) (see e.g., [DS21,Dfn. 3.32]).Let µ η E be the probability measure on Υ(E) defined as and its restriction on the k-particle configuration space ) is a probability measure on Υ whose support is contained in Υ η E while µ η E is a probability measure on Υ(E).We may identify the two of them without loss of information in the sense that Namely, the projection map pr E is bijective with the inverse map pr −1 E defined as pr −1 E (γ) := γ+η, and both pr E and pr −1 E are measure-preserving between the two measures µ( For a measurable function u : Υ → R, E ∈ B(X) and η ∈ Υ, we define By the property of the conditional probability, it is straightforward to see that for every u ∈ L 1 (µ), as By applying the disintegration formula (2.12) to u = 1 Ω , we obtain Poisson measure.Let (X, τ, ν) be a locally compact Polish space with Radon measure ν satisfying ν(X) < ∞.The Poisson measure π ν on Υ(X) with intensity ν is defined in terms of the symmetric tensor measures {ν ⊙k : k ∈ N} as follows: In the case that ν is σ-finite, take an exhaustion (B r ) r∈N so that ν(B r ) < ∞ for every r ∈ N. The Poisson (random) measure π ν with intensity ν is the unique probability measure on Υ satisfying The measure π ν does not depend on the choice of (B r ) r∈N .
L 2 -transportation distance.Let (X, d) be a locally compact complete separable metric space.For i = 1, 2 let proj i : X ×2 → X denote the projection to the i th coordinate for i = 1, 2. For γ, η ∈ Υ, let Cpl(γ, η) be the set of all couplings of γ and η, i.e., Here M (X ×2 ) denotes the space of all Radon measures on X ×2 .The L 2 -transportation extended distance on Υ(X) is We refer the reader to e.g., [DS21, Prop.4.27, 4.29, Thm.4.37, Prop.5.12] and [RS99, Lem.4.1, 4.2] for details regarding the L 2 -transportation extended distance d Υ and examples of d Υ -Lipschitz functions.It is important to note that d Υ is an extended distance, attaining the value +∞ and d Υ is lower semi-continuous with respect to the product vague topology τ ×2 v but never τ ×2 v -continuous.
We introduce a variant of the L 2 -transportation extended distance, called L 2 -transportationtype extended distance dΥ defined as where (B r ) r∈N is a compact exhaustion.The definition (2.19) does not depend on the choice of an exhaustion.By definition, d Υ ≤ dΥ on Υ and d Υ = dΥ on Υ(E) for every compact subset E ⊂ X.In particular, we have It can be readily seen readily that Proof.According to (2.19), we can write where The following universal measurability of the distance function from a set will be used in Thm.4.6.
Lemma 2.4.Let u ∈ Lip(Υ, dΥ ) and E ⊂ X be a Polish subset.Then, Remark 2.5.By the same proof, one can replace dΥ with d Υ in the statement of Lem.2.4 and obtain (2.29) The tail set T (Ξ) of Ξ does not depend on the choice of the exhaustion (B r ).It can be readily shown that T (Ξ) ∈ T (Υ) and Ξ ⊂ T (Ξ).

Definition 2.6 (Tail triviality). A Borel probability measure
Example 2.7.The tail triviality has been verified for a wide class of point processes.
2.6.Number-rigidity.The following definition of the number rigidity on the configuration space Υ over a locally compact Polish space X is an adaptation of the number rigidity introduced by Ghosh-Peres [GP17] originally in the setting of the configuration space over the complex plane.

Construction of Dirichlet forms
In this section, we construct a Dirichlet form on Υ = Υ(R n ).Let (B r ) r∈N be a compact convex domain exhaustion in R n .We first construct a Dirichlet form on Υ(B r ) called conditioned form with invariant measure µ η Br .We then lift it onto Υ, which is called truncated form, whose gradient operator is truncated on B r .Finally we take the monotone limit of the truncated forms as r → ∞ and construct the limit Dirichlet form on Υ.
Notation.Hereinafter, we use the following notation.
• m, m r for the Lebesgue measure on R n and its restriction on B r respectively; • d(x, y) := |x − y| for the Euclidean distance in R n ; • µ η r := µ η Br for a probability measure µ on Υ, defined in (2.9); • u η r := u η Br for a function u : Υ → R, defined in (2.11).
3.1.Conditioned Dirichlet forms on Υ(B r ).Let W 1,2 s (m ⊗k r ) be the space of m ⊗k rclasses of (1, 2)-Sobolev and symmetric functions on the product space B ×k r , i.e., where ∇ ⊗k denotes the weak derivative on (R n ) ×k : ∇ ⊗k u := (∂ 1 u, . . ., ∂ k u).As the space W 1,2 s (m ⊗k r ) consists of symmetric functions, the projection s (m ⊗k r ) and the resulting quotient space is denoted by W 1,2 (m ⊙k r ): where ∇ ⊙k is the quotient operator of the weak gradient operator ∇ ⊗k through the projection P k and m ⊙k r is the symmetric product measure defined as Definition 3.1 (Conditional absolute continuity).A Borel probability measure µ on Υ is conditionally absolutely continuous (to π m ) if We say that µ satisfies (CAC ′ ) 3.1 if Let us define the following algebra The quadratic functional associated with µ η,k r is denoted by |∇ ⊙k u| 2 dµ η,k r , (3.2a) Definition 3.2 (Conditional closability).Let µ be a Borel probability measure on Υ satisfying (CAC) 3.1 .We say that µ satisfies the conditional closability (CC) 3.2 if the form is closable on L 2 Υ(B r ), µ η r for every r ∈ N and µ-a.e.η ∈ Υ.

Truncated Dirichlet forms.
In this subsection, we construct the truncated Dirichlet form on Υ.We start this section by giving an operator mapping functions on Υ to functions on R n .Definition 3.5 ([MR00, Lem.1.2], see also [DS21, Lem.2.16]).For u : Υ → R, define U γ,x (u) : R n → R by We now define a square field operator on Υ truncated up to particles inside B r .Definition 3.6 (Truncated square field on Υ).The following operator is called the truncated square field Γ Υ r whenever ∇U γ,x (u)| Br makes sense m r -a.e. for u : Υ → R: Thanks to Lem.A.1, Formula (3.7) is well-defined for µ-a.e.γ.Indeed, as the weak gradient ∇U γ,x (u) is well-defined pointwise on a measurable set Σ ⊂ B r with m r (Σ c ) = 0, by applying Lem.A.1, Formula (3.7) is well-defined on a set Ω(r) of µ-full measure.
Based on the truncated square field Γ Υ r , we introduce the truncated form on Υ defined on a certain core.Definition 3.7 (Core).Let {C r } r∈N be a sequence of algebras of µ-classes of measurable functions so that C r ⊃ C r ′ for r ≤ r ′ and the following hold for every r ∈ N: is well-defined µ-a.e. for every u ∈ C r ; (c) the following integral is well-defined and finite for every u ∈ C r : Example 3.8.We have several choices of {C r } r∈N .In each of the following examples, we take a certain common core C and take C r = C for every r > 0.
where γ(f ) := R n f dγ.We say that µ satisfies (m µ ) if the intensity measure m µ is locally finite intensity, viz.
where σ(pr Br ) is the σ-algebra generated by the map pr Br .Define Assume (m µ ), (CAC) 3.1 and (CC) 3.2 , and take C r = C = C 1 b,loc (Υ) for every r ∈ N. Then all the conditions of Dfn.3.7 are satisfied.
The following proposition relates the two square fields Γ Υ r and Γ Υ(Br ) .
Furthermore, if Lip b ( dΥ , µ) ⊂ C r , then Proof.Although the idea of the proof is similar to [Suz22, Prop.4.7], the core chosen there is different from the core C r here.We therefore give the proof below for the sake of completeness.
We first prove (3.12).As the second line of (3.12) is an immediate consequence of the first line and the disintegration formula (2.12), we only give the proof of the first line of (3.12).Let u ∈ C r .Then, the RHS of (3.12) is well-defined on a measurable set Ω of µ-full measure by (b) in Dfn.3.7.Let Ω η r be the section as defined in (2.13), which is of µ η r -full measure for µ-a.e.η ∈ Ω by (2.14).As µ η r is absolutely continuous with respect to the Poisson measure π mr by (CAC) 3.1 and the Poisson measure does not have multiple points almost everywhere, we may assume that every γ ∈ Ω η r does not have multiple points, i.e., γ({x}) ∈ {0, 1} for every where the first equality is the definition of the square field Γ Υ r ; the third equality holds as u η r 1 X\{x} •γ does not depend on the variable denoted as • on which the weak gradient ∇ operates; the fourth equality followed from the definition of the symmetric gradient operator ∇ ⊙k , for which we used the fact that γ ∈ Ω η r does not have multiple points.As this argument holds for arbitrary k ∈ N 0 , (3.12) has been shown.The local property follows immediately by (3.12) and the local property of (E )).The Markov property of (E Υ,µ r , C r ) follows by (c).We now show the closability.Noting that E Υ(Br),µ η r is closable for µ-a.e.η by (CC) 3.2 , the superposition form ( ĒΥ,µ )) is Markovian as well.We now prove (3.13).By the Rademacher-type property of E Υ(Br),µ k,η r , we have that In view of the relation between Γ Υ r and Γ Υ(Br) in (3.12) and the Lipschitz contraction (2.26) of the operator (•) η r , we concluded (3.13).The proof is complete.
It is known that ( ĒΥ,µ } t>0 associated with the form E Υ(Br),µ η r .The following proposition shows that the semigroup (resp.resolvent) corresponding to the superposition form is identified with the superposition of the semigroup (resp.resolvent), which has been proved by [Del21] in a general framework.Remark 3.12.The proof of [Del21, (iii) Prop.2.13] has been given in terms of direct integral.As the measure µ η r can be identified to the conditional probability µ(• | • B c r = η B c r ) by a bi-measure-preserving isomorphism as remarked in (2.10), our setting can be identified with a particular case of direct integrals in [Del21].
As the former form is constructed as the smallest closed extension of (E Υ,µ r , C r ), it is clear by definition that Assumption 3.13.We call (D) 3.13 if for µ-a.e.γ ∈ Υ, every t > 0. ) and the square field Γ Υ r are monotone increasing as r ↑ ∞, viz., )) and C s ⊂ C r by Dfn.3.7, it suffices to check Γ Υ r (u) ≤ Γ Υ s (u) on C s , which is a immediate consequence of the definition (3.7).The proof is complete.Definition 3.17 (Monotone limit form).The form (E Υ,µ , D(E Υ,µ )) is defined as the monotone limit: The form (E Υ,µ , D(E Υ,µ )) is a Dirichlet form on L 2 (µ) as it is the monotone limit of Dirichlet forms (e.g., by [MR90, Exercise 3.9]).Note that the limit form does not depend on the choice of the exhaustion (B r ) r∈N .The square field Γ Υ is defined as the monotone limit of Γ Υ r as well: We now show that the form (E Υ,µ , D(E Υ,µ )) is a local Dirichlet form on L 2 (µ) and satisfies the Rademacher-type property with respect to the L 2 -transportation-type distance dΥ .Proposition 3.18.Assume (CAC) 3.1 and (CC) 3.2 .The form Proof.The local property of (E Υ,µ , D(E Υ,µ )) follows from (3.20).We show the Rademachertype property.Since Γ Υ is the limit square field of Γ Υ r as in (3.20), it suffices to show Γ Υ r (u) ≤ Lipd Υ (u) 2 , u ∈ Lip( dΥ , µ) r > 0 , which has been already proven in Prop.3.9.We verified (Rad dΥ ,µ ) 3.21 .The proof is complete.
3.5.Quasi-regularity.In this subsection, we discuss a sufficient condition for the quasiregularity.
Proposition 3.21 (smaller domain).Assume (CAC) 3.1 and (CC) 3.2 .Let (C r ) r∈N be a sequence of algebras in Dfn.3.7.Then, the form (E Υ,µ , C) defined as ) is a local Dirichlet form on L 2 (µ).Furthermore, if either of the following holds for every r ∈ N Proof.As C ⊂ D(E Υ,µ ) by definition, the closability of C follows by the closedness of D(E Υ,µ ) proven in Prop.3.18.The local property of F Υ,µ is inherited from D(E Υ,µ ).The Markov property of (E Υ,µ , C) follows by the Markov property of (E Υ,µ r , C r ) and (3.22).As the Markov property is inherited to the closure by e.g., [FOT11, Thm.3.1.1],we concluded that F Υ,µ is Markovian.The rest of the arguments follows by the same proofs as in Prop.3.18.
Corollary 3.22.Assume (CAC) 3.1 and (CC) 3.2 .If C r = C (r ∈ N) is either one of the following: Remark 3.23 (A different core).Another sufficient condition for (QR) 3.20 has been studied in [Osa96, Thm.1] by taking a core C r = D ∞ in Dfn.3.7, where D ∞ is a space of smooth local functions (see, [Osa96,(0.3)])and take the domain to be the closure of D ∞ .We note that functions in the core C in (3.23) are not necessarily local functions.The domain F Υ,µ defined as the closure of C in Cor.3.22 is therefore not necessarily the same as the domain constructed as the closure of D ∞ in [Osa96].
• If (CI) 4.1 , (QR) 3.20 and (R) 2.8 hold, then Remark 4.8.We proved the implication (c) =⇒ (b) in Thm.4.3 under (D) 3.13 with the domain D(E Υ,µ ).The same implication was proved in Cor.4.7 under a different assumption (Rad dΥ ,µ ) 3.21 with a smaller domain F Υ,µ .The assumption (D) 3.13 is a condition for the truncated forms E Υ,µ r while (Rad dΥ ,µ ) 3.21 is a condition for (E Υ,µ , F Υ,µ ).We do not have a simple comparison of these two different conditions: as the irreducibility with a smaller domain is a weaker statement than that with a larger domain, Cor.4.7 looks providing the tail-triviality under a weaker assumption than Thm.4.3.However, we do not know whether (Rad dΥ ,µ ) 3.21 is weaker than (D) 3.13 .For the verification, Cor.4.7 is more convenient as will be seen in Section 5.

Examples
Based on verifying the sufficient conditions provided in the previous section, we provide several examples to which our main results (Theorems 4.3, 4.6) can apply.In the following, we discuss four classes of examples: sine 2 , Airy 2 , Bessel α,2 (α ≥ 1), and Ginibre point processes.They belong to the class of quasi-Gibbs measures as explained below, in particular, (CAC ′ ) 3.1 holds true.As all the examples discussed in the following are determinantal point processes, the tail triviality (T) 2.6 is a consequence of e.g., [Lyo18, Theorem 2.1] (see Example 2.9 for more complete references).
Thus, we only discuss the interaction potentials Ψ for these cases below.
Thus the same arguments as in Example 6.1 apply to (CC) 3.2 , (CI) 4.1 and (QR) 3.20 .The number rigidity (R) 2.8 has been proved by [Buf16].Assumption (CC) 3.2 follows from (5.2).Assumptions (CI) 4.1 can be verified immediately by Proposition 5.6 by the same argument in Example 6.1 for Ψ.For Φ, it suffices to take F := {0} in (i) in Assumption 5.5, with which Φ belongs to L ∞ loc (R \ F, m).The number rigidity (R) 2.8 has been proved by [Buf16].A Markovian subspace F Υ,µ having the quasiregularity (QR) 3.20 and (Rad dΥ ,µ ) 3.21 has been constructed by Cor.5.7.We remark that Ω n := Ω n,εn .By the upper semi-continuity of probability measures regarding the limit superior of sets, we obtain The proof is complete.
CI) 4.1 where C η,k r is a constant depending on r, η and k.Remark 4.2.(a) In terms of the corresponding diffusion process, Assumption (CI) 4.1 can be understood as the ergodicity of the interacting finite particles in Υ(B r ) conditioned to be η B c r outside B r .(b) Assumption (CI) 4.1 can be verified for a wide class of invariant measures µ such asGibbs measures including Ruelle measures, and determinantal/permanental point processes including sine β , Airy β , Bessel α,β , Ginibre, which will be discussed in §6.
(i) for every Borel measurable µ-integrable function u, it holds P µ -a.s. that lim