Infinite-dimensional stochastic differential equations related to random matrices

We solve infinite-dimensional stochastic differential equations (ISDEs) describing an infinite number of Brownian particles interacting via two-dimensional Coulomb potentials. The equilibrium states of the associated unlabeled stochastic dynamics are the Ginibre random point field and Dyson's measures, which appear in random matrix theory. To solve the ISDEs we establish an integration by parts formula for these measures. Because the long-range effect of two-dimensional Coulomb potentials is quite strong, the properties of Brownian particles interacting with two-dimensional Coulomb potentials are remarkably different from those of Brownian particles interacting with Ruelle's class interaction potentials. As an example, we prove that the interacting Brownian particles associated with the Ginibre random point field satisfy plural ISDEs.


Introduction
Consider infinitely many Brownian particles X = (X i ) i∈N moving in R d interacting via the two-dimensional (2D) Coulomb potentials Ψ β : Then the stochastic dynamics X = (X i ) i∈N is described by the following infinitedimensional stochastic differential equation (ISDE): Here {B i } i∈N is a sequence of independent copies of d-dimensional Brownian motions and X = (X i ) i∈N is a continuous (R d ) N -valued process. Physically this dynamics describes the motion of an infinite system of a onecomponent plasma in R d . If d = 2, so that the particles can be thought of as infinitely long parallel charged lines perpendicular to the confining plane [1]. Because the Coulomb interactions Ψ β are two-dimensional, the ISDE (1.2) is meaningful only for d = 1, 2.
The purpose of this paper is to solve the ISDE (1.2) by relating the system to random matrix theory. Namely, we consider the cases d = 2, β = 2 and d = 1, β = 1, 2, 4. These are related to Ginibre ensembles (d = 2, β = 2) and Gaussian random matrices called GOE, GUE, and GSE (d = 1, β = 1, 2, 4). The former is the thermodynamic limit of the distributions of eigen values of non Hermitian random Gaussian matrices, and the latter are those of orthogonal, unitary and symplectic random Gaussian matrices, respectively.
For a given interaction potential Φ, the study of ISDEs of this type was initiated by Lang [6], [7], and followed by Shiga [17], Fritz [3], Tanemura [21] and others. In these works Φ is assumed to be a Ruelle's class potential, that is, Φ is super stable and integrable at infinity. In addition, Φ is assumed to be C 3 0 ([6], [7]) or to decay exponentially at infinity. Hence, the polynomial decay potentials have been excluded even for Ruelle's category.
We develop a new approach to solve ISDEs of this kind for general potentials Φ. As an application we solve (1.2) with (d, β) as mentioned above. Our condition is easily checked for all Ruelle's class potentials with suitable smoothness outside the origin, so we give a new result even for this class.
All our conditions to solve ISDEs are stated in terms of geometric assumptions on the ISDEs. The first step is the existence of the equilibrium state of the dynamics given by the ISDE. In case of Ruelle's class potentials this step is trivial because the equilibrium states are Gibbs measures, whose existence is well established in [16], and the relationship between the candidate equilibrium states and the ISDE follows from the Dobrushin-Lanford-Ruelle equation (DLR  equation).
On the other hand, when Φ is a 2D Coulomb potential, the situation is drastically changed. Because of the unboundedness at infinity of 2D Coulomb potentials, we can no longer use the method in [16] for the construction of equilibrium states, and the DLR equation becomes meaningless. In the 2D Coulomb case, even the construction of infinite-volume measures for general β has not yet been established. Moreover, the lack of the DLR equation requires a new device for clarifying the connection between the candidates for equilibrium states and the ISDE (1.2). For the construction, we use a result from random matrix theory [8] and determinantal random point fields [19], [18]. To clarify the relation between the measures and the ISDEs, we establish the integration by parts formula for the candidates for the equilibrium states. Because the candidates for the equilibrium states are given by the correlation functions defined by the determinants of some kernels, such a formula is extremely non-trivial. The calculation of such an integration by parts formula for the measures appearing in random matrix theory is the heart of the present paper.
The ISDE (1.2) with d = β = 2 is the primary example of the present paper. In this case we have plural ISDEs representing the same diffusion (see Theorems 2.1 and 2.2). Except for the unboundedness at infinity, the 2D Coulomb potentials have rather simple structure; they yield only repulsive force. The property of the associated stochastic dynamics is however drastically changed from that of the stochastic dynamics given by Ruelle's class potentials. Indeed, we will prove in a forth coming paper that the tagged particles are sub-diffusive. This contrasts strikingly with the result of Ruelle's class potentials [11]. We conjecture that when d = 2, a phase transition occurs in β.
The ISDE (1.2) with d = 1 and β = 2 was first investigated by Spohn [20], and followed by [10], Nagao-Forrester [9], and Katori-Tanemura [5]. In these works, the dynamics was constructed by Dirichlet forms or in terms of spacetime correlation functions. The ISDE was only intuitively obtained by analogy with SDEs for finite particle approximations. In this sense the ISDE (1.2) has not yet been solved. We remark that the passage of the SDE representation from the finite particle systems to the infinite one is an extremely sensitive problem because of the long range nature of the 2D Coulomb potentials.
It is plausible that our method is applicable to other measures appearing in random matrix theory and determinantal random point fields. We do not pursue this here.
The organization of the paper is as follows: In Section 2 we set up the mathematics and state some of the main theorems. In Section 3 we prove Theorems 2.6 and 2.7. These theorems give a general theory for solving ISDEs with long range potentials. In Section 4 we prove Theorem 4.5, which gives a general procedure for the integration by parts formula. In Section 5 we give a sufficient condition in (4.30), which is a key to the integration by parts formula in Section 4. In Section 6 we establish the integration by parts formula for the Ginibre random point field, which corresponds to the case d = 2 and β = 2 in (1.2). In Section 7 we prove Theorems 2.1-2.3. In Section 8 we prove the integration by parts formula for Dyson's models and complete the proof of Theorems 2.4 and 2.5. These theorems correspond to the cases d = 1 and β = 1, 2, 4 in (1.2). In the Appendix we give the definition of the determinantal kernels of the case d = 1 and β = 1, 4.

Set up and main results
Let S = R d and S = {s = i δ si ; s(K) < ∞ for all compact sets K ⊂ S}, where δ a stands for the delta measure at a. We endow S with the vague topology, under which S is a Polish space. S is called the configuration space over S. We write s(x) = s({x}). Let By definition, S s.i. is the set of the configurations consisting of an infinite number of single point measures.
For an infinite or finite product S k of S we define the map u from S k to the set of measures on S by u((s j )) = k j=1 δ sj . We omit k from the notation. We consider the restriction of u on u −1 (S s.i. ). Let u path be the map from where X = {(X j t ) j }. We set X = u path (X).
A symmetric locally integrable function ρ n : S n → [0, ∞) is called the n-point correlation function of a probability measure µ on S w.r.t. the Lebesgue measure if ρ n satisfies for any sequence of disjoint bounded measurable subsets A 1 , . . . , A m ⊂ S and a sequence of natural numbers k 1 , . . . , k m satisfying k 1 + · · · + k m = n. It is known that under a mild condition {ρ n } n∈N determine the measure µ [19]. Let µ gin be the probability measure on the configuration space over S = R 2 whose n-point correlation function ρ n gin w.r.t. the Lebesgue measure is given by Here we identify R 2 as C by the obvious correspondence: It is known that µ gin (S s.i. ) = 1. Moreover, µ gin is translation and rotation invariant. µ gin is called the Ginibre random point field.
Theorem 2.1. There exists a set S gin such that µ gin (S gin ) = 1, S gin ⊂ S s.i. , (2.6) and that, for all s ∈ u −1 (S gin ), there exists an (R 2 ) N -valued continuous process X = (X i ) i∈N , and (R 2 ) N -valued Brownian motion B = (B i ) i∈N satisfying One specific aspect of the ISDE (2.7) is that its solution satisfies the second ISDE. Such a phenomenon never occurs in Ruelle's class potentials.
Theorem 2.2. The solution (X, B) in Theorem 2.1 satisfies To clarify the meaning of the ISDEs we define the measure µ 1 gin on S × S by where µ gin,x = µ gin (· − δ x |s(x) ≥ 1) is the Palm measure conditioned at x and ρ 1 gin is the 1-point correlation function of µ gin . Let b,b : (2.14) We will see in Lemma 7.2 (3) that these two series converge in L 2 loc (µ 1 gin ). We remark that neither of the series converges absolutely and, as a result, b =b.
. Then (2.7) and (2.11) can be rewritten as follows: A diffusion with state space S 0 is a family of continuous stochastic processes with the strong Markov property starting from each point of the state space S 0 . So far, the unlabeled dynamics are known to be S-valued diffusions. We refine this as follows: Theorem 2.3. Let P s be the distribution of the fully labeled dynamics X given by Theorem 2.1. Then {P s } s∈u −1 (Sgin) is a diffusion with state space u −1 (S gin ).
The second example is Dyson's model. Let S = R and let µ dys,β (β = 1, 2, 4) be the probability measure on S whose n-point correlation function ρ n β is given by Here we take K 2 (x) = sin(πx)/πx. The definition of K β for β = 1, 4 is given in the Appendix. We use quaternions to denote the kernel K β for β = 1, 4. The precise meaning of the determinant of (2.17) for β = 1, 4 is given by (9.3). The kernel K 2 is called the sine kernel. We remark that K 2 (t) = 1 2π |k|≤π e ikt dk and 0 ≤ K 2 ≤ Id as an operator on L 2 (R).
(2.21) Theorem 2.5. Let S dys,β = u −1 (S dys,β ). Let P s be the distribution of X given by Theorem 2.4. Then {P s } s∈S dys,β is a diffusion with state space S dys,β .
To solve the infinite-dimensional SDEs above, we prepare a general theory. Let σ : S × S → R d 2 and b : S × S → R d be measurable functions. Let a = σσ t . We assume for each (x, y) Here c 1 is a positive constant independent of (x, y). For (X i ) i∈N we set X i * t = j =i,j∈N δ X j t as before. Then the ISDEs we study are of the form: Letσ(x, (y j ) j∈N ) be the function defined on S × S N being symmetric in (y j ) j∈N for each x and satisfyingσ(x, (y j ) j∈N ) = σ(x, j∈N δ yj ). We setb similarly. Then we can rewrite (2.23) as (2.24): Then intuitively the generator L of the diffusion given by (2.24) is Here s i = (s i1 , . . . , s id ) ∈ S ≡ R d . Our strategy for solving ISDE (2.23) and (2.24) is to use a geometric property behind the ISDE (2.23). We first consider an invariant probability measure µ of the unlabeled dynamics associated with (2.23). Namely, we consider a probability measure µ whose log derivative d µ satisfies b(x, y) = ∇ x a(x, y) + a(x, y)d µ (x, y). Here, to be more precise, d µ is the log derivative of the measure µ 1 given by (2.26), and the definition of d µ is given by (2.32).
Note that for a given pair (a, µ), b is uniquely determined. We construct the unlabeled diffusion associated with (a, µ) by using the Dirichlet space given by (a, µ) and prove that the labeled process consisting of each component of the unlabeled diffusion satisfies (2.23) and (2.24).
If there were a Dirichlet space associated with the fully labeled diffusion X = (X i ) i∈N , we could use the Ito formula for each component X i and X i X j , and prove that X satisfies (2.25) since all coordinate functions x i , x i x j (i, j ∈ N) would be in the domain of the Dirichlet space locally. We emphasize that there exist no Dirichlet spaces associated with the fully labeled diffusion X. Instead we introduce an infinite sequence of Dirichlet spaces associated with the klabeled process {((X 1 t , . . . , X k t , j>k δ X j t ))} for all k = 0, 1, . . .. This sequence of k-labeled processes has consistency and satisfies the ISDEs (2.23) and (2.24).
Let µ be a probability measure on (S, B(S)). Let ρ k be the k-point correlation function of µ w.r.t. the Lebesgue measure. Let µ k be the measure on S k × S defined by (2.26) Here x = (x 1 , . . . , x k ) ∈ S k and dx = dx 1 · · · dx k . Moreover µ x is the Palm measure conditioned at x = (x 1 , . . . , x k ) defined by We now introduce Dirichlet forms describing the k-labeled dynamics. For a subset A ⊂ S we define the map π A : S → S by π A (s) = s(A ∩ ·). We say a function f : Let D • be the set of all local, smooth functions on S with compact support.
. . , s id ) ∈ S, and s = (s i ). For given f and g in D • , it is easy to see that the right-hand side of (2.28) depends only When k = 0, we take D a,0 = D a , µ 0 = µ, and E a,0 = E a . We set L 2 (µ) = L 2 (S, µ) and L 2 (µ k ) = L 2 (S k × S, µ k ) and so on. We assume that there exists a probability measure µ on S with correlation Here is quasi-regular and that the associated diffusion (P k , X k ) exists. These diffusions have consistency in the sense of (3.6) and (3.7) (see [14]). We remark that Cap µ in (A.4) is the capacity of the Dirichlet space (E a,0 , D a,0 , L 2 (µ)). We call d µ the log derivative of µ.

35)
and that, for all s ∈ u −1 (S 0 ), there exists an S N -valued continuous process The assumption b ∈ L 2 loc (µ 1 ) in (2.33) is used to ensure that N = {N t } is a continuous additive functional locally of zero energy in the sense of [2], and that for any ϕ ∈ D a,1 such that ϕ = 0 on O c n . Here E 1 (x,y) denotes the expectation w.r.t. the diffusion measure starting at (x, y). Indeed, the property b ∈ L 2 loc (µ 1 ) is used only for this. So we can relax the assumption that b ∈ L 2 loc (µ 1 ). This fact will be used for Dyson's model with β = 1 because b ∈ L p loc (µ 1 ) for any 1 ≤ p < 2, but b ∈ L 2 loc (µ 1 ) in this case. (2) The solutions obtained in [3], [6], [7], [17], [21] for Ruelle's class interaction potentials are strong solutions in the sense that they are functionals of given Brownian motions. The strong Markov property of the solutions was however not proved in these works except [3].
In this section we prove Theorems 2.6 and 2.7. We assume (A.1)-(A.5) throughout this section. Let (E a,k , D a,k ) be the closure of (E a,k , Proof. (1) follows from Lemma 2.3 in [14]. (2) follows from (1) and Dirichlet form theory.
Let l : S s.i. → S N be a measurable map such that u • l is the identity map. We represent this map by l(s) = (s 1 , . . . , ), where s = ∞ i=1 δ si . The map l means the label of the originally unlabeled particle s and is called a label. So there are infinitely many labels l satisfying the above mentioned condition. Moreover, it is easy to see that u −1 (S s.i. ) = ∪ l l(S s.i. ), where the union is taken over all labels.
. For a given label l as above let l k : S s.i. → S k s.i. be the map defined by Note that u • l k is the identity map.
One can extend l naturally as the map from Then there exists a setS satisfying Here X t = i∈N δ X i t . Moreover, for all k ∈ N and any label l Proof. This lemma is immediate from Theorems 2.4 and 2.5 in [14].
Proof. We recall that {P s } s∈S is a diffusion with state spaceS by Lemma 3.1 and Lemma 3.2. Since P s (l path (C([0, ∞); S s.i. ) s )) = 1 and for any labels l andl satisfying l(s) =l(s) = s, we deduce that P s depends only on P s and the value of the label l at s. Hence the strong Markov property follows from that of {P s }. The continuity of the sample paths is clear by construction.
Let a = [a mn ] and b be as in (2.22) and (2.23), respectively.
Proof. For a diffusion process (P, {X t }) with state space S and a continuous function becomes an additive functional (AF). An AF of this type is called a Dirichlet process. It is worthwhile to note that one can apply the Fukushima decomposition for Dirichlet processes if f is locally in the domain of the Dirichlet form associated with the diffusion. We note that The process X i t − X 0 0 is an AF of the unlabeled diffusion (P, X). However, X i t − X 0 0 is not a Dirichlet process of (P, X). Indeed, we can not identify the position of the ith particle without tracing all of the trajectory of the unlabeled process X = {X u } until u ≤ t. On the other hand, one can regard X i t − X 0 0 as a Dirichlet process of the labeled process (P, X) since the coordinate function x i is a function of the state space S N of (P, X). However, there is no Dirichlet form associated with the labeled process (P, X). Taking these into account, we consider the k-labeled process ((X 1 t , . . . , X k t , ∞ l=k+1 δ X l t )). Here k is taken such that i, j ≤ k. We note that the k-labeled process is associated with the Dirichlet space (E a,k , D a,k , L 2 (µ k )).
Applying [2, Theorem 5.5.1] to the function x i = x i ⊗ 1 ∈ R d and taking Lemma 3.2 into account we deduce that there exists a set S k 0 ⊂S satisfying Cap µ (S\S k 0 ) = 0 and, for each Here M [x i ] is a martingale AF (MAF), locally of finite energy, and N [x i ] is a continuous AF (CAF) locally of zero energy. By a straightforward calculation we deduce that for any . This, combined with the relation (M i , P k l k (s) ) = (M i , P s ) given by Lemma 3.2, yields that (M i , P s ) is a continuous local martingale. As for the quadratic variation of M i , we note that .
(3.12) Proof of Theorem 2.6. For s ∈ u −1 (S 0 ) let P s as in Lemma 3.4. Let B = (B i ) i∈N be defined by (3.14) Then B i are d-dimensional continuous local martingales. By (3.8) and (3.14) Hence we deduce that P s (X t ∈ u −1 (S 0 ) for all t) = 1 for each s ∈ u −1 (S 0 ). So we conclude that {P s } s∈u −1 (S0) is a diffusion with state space u −1 (S 0 ).

Log derivative of random point fields.
Let µ be a probability measure on S with locally bounded n-point correlation function ρ n for each n ∈ N. Let µ 1 be the measure defined by (2.26) with k = 1. In this section we present a sufficient condition for the existence of the log derivative d µ in L p loc (µ 1 ) with 1 < p (Theorem 4.3) and its explicit representation (Theorem 4.5). We shall apply these to the Ginibre random point field and Dyson's model in the subsequent sections.
We set S r = {x ∈ S ; |s| < r}. Let {µ N } be a sequence of probability measures on S. We assume that their n-point correlation functions {ρ N,n } satisfy for each r ∈ N lim N →∞ ρ N,n (x) = ρ n (x) uniformly on S n r , where 0 < c 4 (r) < ∞ and 0 < c 5 (r) < 1 are constants independent of n ∈ N. We remark that (4.2) and (4.4) imply {µ N } N ∈N converge weakly to µ. Let µ N x be the Palm measure conditioned at x as before. Let ρ N,n x (resp. σ N,n x,r ) be the n-point correlation (resp. density) function of µ N x . Let µ N,1 be the measure defined by (2.26) with n = 1. Then we deduce that Here f ∈ C ∞ 0 (S)⊗D • and f (x, ·) is σ[π Sr ]-measurable for each x ∈ S. Moreover, f n (x, y) is the function on S × S n r being symmetric in y = (y 1 , . . . , y n ) for each x and f (x, y) =f n (x, y) when y(S r ) = n and y = n i=1 δ yi . We set dy = dy 1 · · · dy n . It is easy to see that ρ N,n x (y) = ρ N,1+n (x, y)/ρ N,1 (x), ρ n x (y) = ρ 1+n (x, y)/ρ 1 (x). (4.7) Here ρ n x is the n-point correlation function of µ x . , and the assumption that ρ n are locally bounded.
Let B(S r ) be the Borel σ-field of S r . We regard B(S r ) as a subset of B(S) in an obvious manner and denote it by the same symbol B(S r ). Let ̟ s : S × S → S × S such that ̟ s (x, y) = (x, |x−yi|<s δ yi ), where y = i δ yi . Let Set c 6 (r, N ) = µ N,1 (S r × S). Then by (4.2) sup N c 6 (r, N ) < ∞ for each r ∈ N. Without loss of generality, we can assume that c 6 > 0 for all r, N . So letμ N,1 r be the probability measure defined byμ N,1 r (·) = µ N,1 (· ∩ S r × S)/c 6 .
We assume that each µ N has a log derivative d N = d N (x, y) such that d N − u N ∈ L p loc (µ N,1 ) for some 1 < p < ∞, where u N = u N (x) is a distribution on S. We note that u N is supposed to be independent of y ∈ S. Letd N s ∈ L p loc (µ N,1 ) be such that for all r ∈ N for each r. We remark that the second equality in (4.12) comes from the fact that u N is independent of y.   For (x, y) ∈ S r × S we write y = i δ yi and y = (y i ). We set x,r+s (y) < a t , y(S r+s ) = m} (4. 18) and S m t similarly as S N,n t by replacing ρ N,1 (x)σ N,n x,r+s (y) by ρ 1 (x)σ m x,r+s (y). Here {a t } t∈N is an increasing sequence of positive numbers such that lim t→∞ a t = ∞ and that for each m, r, s, t, ∈ N By applying the Hölder inequality to |d N s | p and by using (4.17) we have Hence we can rewrite the limit points asd t,n s = 1 T n td s for any s, t, n ∈ N. By construction lim n→∞ lim t→∞d t,n s =d s µ 1 -a.s.. Then by Fatou's lemma and (4.25) we obtain the first inequality in (4.16). The second one is immediate from the Hölder inequality.
By (4.9) in Lemma 4.1 we see that ∇ϕdµ 1 = lim N →∞ ∇ϕdµ N,1 . By definition, we have − ∇ϕ dµ N,1 = d N ϕ dµ N,1 . Hence we deduce that where y = i δ yi . We assume that . Assume (4.28)-(4.30). Then the log derivative d µ exists in L p loc (µ 1 ) and is given by The convergence lim g s takes place in L p loc (µ 1 ). Proof. By (4.27) and (4.28) we see that d N − u N = g N s + w N s . Then (4.13) follows from (4.29) and (4.30 This combined with d µ = u +d = u + lim sds implies (4.31). Because the convergence of limd s takes place in L p loc (µ 1 ), so does the convergence of lim g s .

Sufficient conditions for (4.30)
The purpose of this section is to give sufficient conditions for g s ∈ L 2 loc (µ 1 ) and (4.30) in terms of correlation functions. Lemma 5.1. Let g s (x, y) = 1 Ss (x − y)g(x, y). Then g s ∈ L 2 loc (µ 1 ) follows from Here · denotes the standard inner product of R d .
Let w N s (x, y) be as in (4.27). Let µ N and µ N x be as in Section 4 with n-point correlation functions ρ N,n and ρ N,n x , respectively.
We give a sufficient condition of (5.2)-(5.5) in terms of correlation functions.
Proof. This lemma is clear from the standard calculation of correlation functions combined with (4.27).
6 Log derivative of the Ginibre random point field.
In this section we calculate the log derivative d µgin of the Ginibre random point field µ gin . Let µ N gin be the probability measure on S whose n-point correlation function ρ N,n gin is given by Here x n = (x 1 , . . . , x n ) and K N gin is the kernel defined by We easily see that By (6.1) and (6.2) the 1-point correlation function ρ N,1 gin is given by Moreover, it holds that ρ N,n gin = 0 if n ≥ N + 1, and that for 2 ≤ n ≤ N ρ N,n gin (x n ) = Note that µ N gin ({s(S) = N }) = 1. So by (6.5) with n = N we deduce that Let µ 1 gin be the measure defined by (2.26) for µ gin . Theorem 6.1. The log derivative d µgin ∈ L p loc (µ 1 gin ) exists for any 1 ≤ p < 2 and is given by The convergence of the series in the right-hand side takes place in L p loc (µ 1 gin ). To prove Theorem 6.1 we use Theorem 4.5. So we check all the conditions in Theorem 4.5. For this purpose we first prepare several lemmas.
It is known that the Palm measure conditioned at x of determinantal random point fields with kernel K is again a determinantal random point field with kernel K x (y, z) = K(y, z) − {K(y, x)K(x, z)/K(x, x)} (see [18,Theorem 1.7]). Applying this to µ N,1 gin we deduce that the kernel K N gin,x of the Palm measure µ N gin,x is then given by .

Proof. By [15, Theorem 1.3] we see that Var
Hence lim s→∞ h rs /s = 0 in L 2 (µ gin ) compact uniformly in x ∈ S r . From this, combined with (6.21), we deduce that g r∞ := lim s→∞ g rs converges in L 2 (µ gin ) compact uniformly in x ∈ S r . So by (7.3) we obtain (7.7). We have thus proved (1). By (6.9) and (6.1) and a similar representation of correlation functions of µ N gin,x we deduce that the first statement of (2) follows from that of (1). Since g r ,g r ∈ L 2 (µ gin,x ), the second follows from (7.7). So we obtain (2).
(A.1) and (A.5) are clear from (2.4) and (2.5). (A.2) follows from Theorem 6.1 and Lemma 7.2 (3). In [13,Theorem 2.6] we proved that the closability in (A.3) holds for k = 0. Indeed, we proved that µ gin is a quasi-Gibbs measure in the sense of [13, Definition 2.1] and deduced the closability for k = 0 from this. The closability for general k ∈ N also follows in a similar fashion from the quasi-Gibbs property of µ gin . Since the kernel K gin is locally Lipschitz continuous, (A.4) immediately follows from [ In this section we prove Theorems 2.4 and 2.5 by using Theorems 2.6 and 2.7. So we take µ = µ dys,β and prove that µ dys,β satisfies (A.1)-(A.5). Proof. Since the correlation functions {ρ n β } of µ dys,β have the expression (2.17) and the kernels K β are bounded, (A.1) and (A.5) are clear.
In [13,Theorem 2.5] we proved that the closability in (A.3) holds for k = 0. Indeed, we proved that µ dys,β is a quasi-Gibbs measure and deduced the closability for k = 0 from this. The closability for general k ≥ 1 also follows from the quasi-Gibbs property of µ dys,β in a similar fashion. Since the kernel We take µ N in (A.2) to be the probability measure µ N β on S whose n-point correlation function ρ N,n β is given by where x = (x i ). It is well known [8] that µ N β (s(R) = N ) = 1 and that We can regard R N as a torus and µ N β to be a translation invariant probability measure on the configuration space on the torus R N . The image measure of µ N β under the map ω N (x) = e 2πix/N gives the distributions of the eigenvalues of the random matrices called circular ensembles [8]. We can rewrite (8.2) as Taking (8.2) into consideration we set Proof. We use Theorem 4.5 to prove Theorem 8.2. So we check the conditions of Theorem 4.5. We take u(x) = w(x) = 0, u N (x) = δ −N/2 (x) − δ N/2 (x), where δ ±N/2 (x) are delta measures, and v N (x, y) = v(x, y) = 0. We set g N as (8.4) and g(x, y) = 2/(x − y). The conditions (4.1) and (4.2) follow from (8.1) and the definition of K N β . (4.14) and (4.28) are clear. For β = 2, 4, the condition (4.29) withp = 2 follows from (2.17), g(x, y) = 2/(x − y), (8.4), and Lemma 5.1. For β = 1 one can check that (4.29) with 1 <p < 2 holds by the Hölder inequality in addition to the above.
We next prove (4.30). For this it is sufficient to check (5.7)-(5.10) by Lemma 5.3. Let µ N β,x be the Palm measure of µ N β conditioned at x ∈ R N and let ρ N,n β,x be its n-point correlation function. Then µ N β,x has a determinantal structure with kernel When β = 2, (8.8) follows from [18,Theorem 1.7]. When β = 1, 4, one can also check (8.8). By (9.2) and (9.4)-(9.6) we easily see that K N β (x, x) = 1 RN (x). Hence (8.8) implies that for x ∈ R N and y, z ∈ R K N β,x (y, z) = K N β (y, z) − K N β (y, x)K N β (x, z). Proof. Let g s (x, y) = |x−yi|<s 2/(x − y i ). Then by Theorem 8.2 it is sufficient for Lemma 8.3 to prove g s converge in L 2 loc (µ 1 dys,β ). Let µ x be the Palm measure of µ dys,β conditioned at x. Then since µ dys,β are translation invariant, it is enough to show that g s (x, y) converge in L 2 (µ x ) for each x.

Appendix.
We begin by defining K β for β = 1, 4. For this purpose, we recall the standard quaternion notation for 2 × 2 matrices (see [8,Ch. 2.4]), A quaternion q is represented by q = q (0) 1 + q (1) e 1 + q (2) e 2 + q (3) e 3 , where q (i) are complex numbers. There is a natural identification between the 2 × 2 complex matrices and the quaternions given by We denote by Θ( a b c d ) the quaternion defined by the right hand side of (9.2).