Dynamical Bulk Scaling limit of Gaussian Unitary Ensembles and Stochastic Differential Equation gaps

The distributions of $ N $-particle systems of Gaussian unitary ensembles converge to Sine$_2$ point processes under bulk-scaling limits. These scalings are parameterized by a macro-position $ \theta $ in the support of the semicircle distribution. The limits are always Sine$_{2}$ point processes and independent of the macro-position $ \theta $ up to the dilations of determinantal kernels. We prove a dynamical counter part of this fact. We prove that the solution of the $ N $-particle systems given by stochastic differential equations (SDEs) converges to the solution of the infinite-dimensional Dyson model. We prove the limit infinite-dimensional SDE (ISDE), referred to as Dyson's model, is independent of the macro-position $ \theta $, whereas the $ N $-particle SDEs depend on $ \theta $ and are different from the ISDE in the limit whenever $ \theta \not= 0 $.


Introduction
Gaussian unitary ensembles (GUE) are Gaussian ensembles defined on the space of random matrices M N (N ∈ N) with independent random variables, the matrices of which are Hermitian. By definition, M N = [M N i,j ] N i,j=1 is then an N × N matrix having the form where {ξ i , τ i,j , ζ i,j } ∞ i<j are i.i.d. Gaussian random variables with mean zero and unit variance. Then the eigenvalues λ 1 , . . . , λ N of M N are real and have distributionμ N such thať where x N = (x 1 , . . . , x N ) ∈ R N and Z N is a normalizing constant [1]. Wigner's celebrated semicircle law asserts that their empirical distributions converge in distribution to a semicircle distribution: We consider the scaling of the next order in such a way that the distribution is supported on the set of configurations. That is, let θ be the position of the macro scale given by and take the scaling x → y such that Let µ N θ be the point process for which the labeled density m N θ dx N is given by The position θ in (1.2) is called the bulk and the scaling in (1.3) the bulk scaling (of the point processes). It is well known that the rescaled point processes µ N θ satisfy lim N →∞ µ N θ = µ θ in distribution, (1.5) where µ θ is the determinantal point process with sine kernel K θ : By definition µ θ is the point process on R for which the m-point correlation function ρ m θ with respect to the Lebesgue measure is given by We hence see that the limit is universal in the sense that it is the Sine 2 point process and independent of the macro-position θ up to the dilation of determinantal kernels K θ . This may be regarded as a first step of the universality of the Sine 2 point process, which has been extensively studied (see [2] and references therein).
Once a static universality is established, then it is natural to enquire of its dynamical counter part. Indeed, we shall prove the dynamical version of (1.5) and present a phenomenon called stochastic-differential-equation (SDE) gaps for θ = 0. Two natural N -particle dynamics are known for GUE. One is Dyson's Brownian motion corresponding to time-inhomogeneous N -particle dynamics given by the time evolution of eigenvalues of time-dependent Hermitian random matrices M N (t) for which the coefficients are Brownian motions B i,j t [9]. The other is a diffusion process X θ,N = (X θ,N,i ) N i=1 given by the SDE such that for which has a unique strong solution for X θ,N 0 ∈ R N \N and X θ,N never hits N , where N = {x = (x k ) N k=1 ; x i = x j for some i = j} [4]. The derivation of (1.6) is as follows: Letμ N θ (dx N ) = m N θ (x N )dx N be the labeled symmetric distribution of µ N θ . Consider a Dirichlet form on L 2 (R N ,μ N θ ) such that Then using (1.4) and integration by parts, we specify the generator −A N of Eμ N θ on L 2 (R N ,μ N θ ) such that From this we deduce that the associated diffusion X θ,N is given by (1.6).
Taking the limit N → ∞ in (1.6), we intuitively obtain the infinite-dimensional SDE (ISDE) of which was introduced in [21] with θ = 0. For each θ, we have a unique, strong solution X θ of (1.7) such that X θ 0 = s for µ θ • l −1 -a.s. s, where l is a labeling map. Although only the θ = 0 ISDE of X 0 =: X = (X i ) i∈N is studied in [16,22], the general θ = 0 ISDE is nevertheless follows easily using the transformation be the associated delabeled process. Then X θ = {X θ t } takes µ θ as an invariant probability measure, and is not µ θ -symmetric for θ = 0.
The precise meaning of the drift term in (1.7) is the substitution of X θ t = (X θ,i t ) i∈N for the function b(x, y) given by the conditional sum where y = i δ yi and µ [1] θ is the one-Campbell measure of µ θ (see (2.1)). We do this in such a way that b(X θ,i t , j =i δ X θ,j t ). Because µ θ is translation invariant, it can be easily checked that (1.8) is equivalent to (1.9): Let l N and l be labeling maps. We denote by l N,m and l m the first m-components of l N and l, respectively. We assume that, for each m ∈ N, and X = (X i ) i∈N be solutions of SDEs (1.6) and (1.11), respectively, such that (1.11) We now state the first main result of the present paper.
We next consider non-reversible initial distributions. Let X N = (X N,i ) N i=1 and Y θ = (Y θ,i ) i∈N be solutions of (1.13) and (1.14), respectively, such that (1.14) Note that X N = X 0,N and that X N is not reversible with respect to µ N θ • l −1 N for any θ = 0. We remark that the delabeld process Y θ = { i∈N δ Y θ,i t } of Y θ has invariant probability measure µ θ and is not symmetric with respect to µ θ for θ = 0. We state the second main theorem.
• We refer to the second claim in Theorem 1.1, and (1.15) as the SDE gaps. The convergence in (1.15) of Theorem 1.2 resembles the "Propagation of Chaos" in the sense that the limit equation (1.14) depends on the initial distribution, although it is a linear equation. Because the logarithmic potential is by its nature long-ranged, the effect of initial distributions µ N θ still remains in the limit ISDE, and the rigidity of the Sine 2 point process makes the residual effect a non-random drift term θdt.
• Let S θ be a Borel set such that µ θ (S θ ) = 1, where − √ 2 < θ < √ 2. In [7], the first author proves that one can choose S θ such that S θ ∩ S θ ′ = ∅ if θ = θ ′ and that for each s ∈ S θ (1.11) has a strong solution X such that X = l(s) and that This implies that the state space of solutions of (1.11) can be decomposed into uncountable disjoint components. We conjecture that the component S θ is ergodic for each θ ∈ (− √ 2, √ 2).
• For θ = 0, the convergence (1.12) is also proved in [15]. The proof in [15] is algebraic and valid only for dimension d = 1 and inverse temperature β = 2 with the logarithmic potential. It relies on an explicit calculation of the space-time correlation functions, the strong Markov property of the stochastic dynamics given by the algebraic construction, the identity of the associated Dirichlet forms constructed by two completely different methods, and the uniqueness of solutions of ISDE (1.7). Although one may prove (1.10) for θ = 0 using the algebraic method in [15], this requires a lot of work as mentioned above. We remark that, as a corollary and an application, Theorem 1.1 proves the weak convergence of finite-dimensional distributions explicitly given by the spacetime correlation functions. We refer to [5,15] for the representation of these correlation functions.
• Tsai proves the pathwise uniqueness and the existence of strong solutions of for general β ∈ [1, ∞) in [22]. The proof uses the classical stochastic analysis and crucially depends on a specific monotonicity of SDEs (1.16). For β = 1, 4, we have a good control of the correlation functions as for β = 2. Hence our method can be applied to β = 1, 4 and the same result as for β = 2 in Theorem 1.1 holds. We shall return to this point in future.
The key point of the proof of Theorem 1.1 is to prove the convergence of the drift coefficient b N (x, y) of the N -particle system to the drift coefficient b(x, y) of the limit ISDE even if θ = 0. That is, as Note that support of the coefficients b N (x, y) and b(x, y) are mutually disjoint, and that the sum in b N is not neutral for any θ = 0. We shall prove uniform bounds of the tail of the coefficients using fine estimates of the correlation functions, and cancel out the deviation of the sum in b N with θ.
Because of rigidity of the Sine 2 point process, we justify this cancellation not only for static but also dynamical instances. The organization of the paper is as follows: In Section 2, we prepare general theories for interacting Brownian motion in infinite dimensions. In Section 3, we quote estimates for the oscillator wave functions and determinantal kernels. In Section 4, we prove key estimates (2.21)-(2.24). In Section 5, we complete the proof of Theorem 1.1. In Section 6, we prove Theorem 1.2.

Preliminaries from general theory
In this section we present the general theory described in [11,12,16,8] in a reduced form sufficient for the current purpose. In particular, we take the space where particles move in R rather than R d as in the cited articles.

µ-reversible diffusions
Let S r = {s ∈ R ; |s| < r}. The configuration space S over R is a Polish space equipped with the vague topology such that Each element s ∈ S is called a configuration regarded as countable delabeled particles. A probability measure µ on (S, B(S)) is called a point process (a random point field).
A locally integrable symmetric function ρ n : R n → [0, ∞) is called the n-point correlation function of µ with respect to the Lebesgue measure if ρ n satisfies for any sequence of disjoint bounded measurable subsets A 1 , . . . , A m ⊂ R and a sequence of natural numbers k 1 , . . . , k m satisfying k 1 + · · · + k m = n. Here we assume that s Let Φ : R → R and Ψ : R 2 → R ∪ {∞} be measurable functions called free and interaction potentials, respectively. Let H r be the Hamiltonian on S r given by For each m, r ∈ N and µ-a.s. ξ ∈ S, let µ m r,ξ denote the regular conditional probability such that Let Λ r denote the Poisson point process with intensity being a Lebesgue measure on S r . We , [13]). A point process µ is said to be a (Φ, Ψ)-quasi Gibbs measure if its regular conditional probabilities µ m r,ξ satisfy, for any r, m ∈ N and µ-a.s. ξ, Here c 1 is a positive constant depending on r, m, ξ.
The significance of the quasi-Gibbs property is to guarantee the existence of µ-reversible diffusion process {P s } on S given by the natural Dirichlet form associated with µ, in analogy with distorted Brownian motion in finite-dimensions.
To introduce the Dirichlet form, we provide some notations. We say a function Let D be the standard square field on S such that for f, g ∈ D • and s = i δ si it as a function of s. We set L 2 (µ) = L 2 (S, µ) and let We quote: 12]). Assume that µ is a (Φ, Ψ)-quasi Gibbs measure with upper semicontinuous (Φ, Ψ). Assume that the correlation functions {ρ n } are locally bounded for all n ∈ N.

Infinite-dimensional SDEs
Suppose that diffusion {P s } in Lemma 2.1 is collision-free and that each tagged particle does not explode. Then we can construct labeled dynamics X = (X i ) i∈Z by introducing the initial labeling l = (l i ) i∈Z such that Indeed, once the label l is given at time zero, then each particle retains the tag for all time because of the collision-free and explosion-free property.
To specify the ISDEs satisfied by X above, we introduce the notion of the logarithmic derivative of µ, which was introduced in [11].
A point process µ x is called the reduced Palm measure of µ conditioned at x ∈ R if µ x is the regular conditional probability defined as Under these assumptions, we obtain the following: 11]). Assume that X = (X i ) i∈N is the collision-free and explosion-free. Then X is a solution of the following ISDE:

Finite-particle approximations
Let µ be a point process with correlaton functions {ρ n } n∈N . Let {µ N } N ∈N be a sequence of point processes on R such that µ N ({s(R) = N }) = 1. We assume: (A1) Each µ N has correlation functions {ρ N,n } n∈N satisfying, for each r ∈ N, where 0 < c 2 (r) < ∞ and 0 < c 3 (r) < 1 are constants independent of n ∈ N.
It is known that (2.3) and (2.4) imply the weak convergence of {µ N } to µ [12, Lemma A.1]. As in Section 1, let l and l N be labels of µ and µ N , respectively. We assume: We shall later take µ N •l −1 N as an initial distribution of labeled finite particle system. Therefore, (A2) means the convergence of the initial distribution of the labeled dynamics.
For a labeled process with these coefficients such that for We assume: s. s for each N . This solution does not explode.
Let u, u N , w : R → R and g : R 2 → R be measurable functions. We set where y = i δ yi and χ r ∈ C ∞ 0 (R) is a cut-off function such that 0 ≤ χ r ≤ 1, χ r (x) = 0 for |x| ≥ r + 1, and χ r (x) = 1 for |x| ≤ r. We assume the following.
Furthermore, we assume that Furthermore, u N and ∇u N converge uniformly to u and ∇u, respectively, on each compact set in R.
where ρ N,1 x is a one-correlation function of the reduced Palm measure µ N x .
(3) There exists a continuous function w : R → R such that for each R ∈ N Let p be such that 1 < p <p. Assume (A1) and (A4). Then [11,Theorem 45] deduces that the logarithmic derivative d µ of µ exists in L p loc (µ [1] ) and is given by Here g(x, y) = lim r→∞ g r (x, y) and the convergence of lim g r takes place in L p loc (µ [1] ). Taking (2.11) into account, we introduce the ISDE of X = (X i ) i∈N : Under the assumptions of Lemma 2.2, ISDE (2.12) with X 0 = s has a solution for µ • l −1 -a.s. s. Moreover, the associated delabeled diffusion X = {X t } is µ-reversible, where X t = i∈N δ X i t for X t = (X i t ) i∈N . As for uniqueness, we recall the notion of µ-absolute continuity solution introduced in [16]. Let X = (X i ) i∈N be a family of solution of (2.12) satisfying X 0 = s for µ • l −1 -a.s. s. Let µ t be the distribution of the delabeled process X t = i∈N δ X i t at time t with initial distribution µ. That is, µ t is given by We say that X satisfies the µ-absolute continuity condition if µ t ≺ µ for all t ≥ 0, (2.13) where µ t ≺ µ means that µ t is absolutely continuous with respect to µ. If X is µ-reversible, then (2.13) is satisfied. We say ISDE (2.12) has µ-uniqueness in law of solutions if X and X ′ are solutions with the same initial distributions satisfying the µ-absolute continuity condition, then they are equivalent in law. We assume: (A5) ISDE (2.12) has µ-uniqueness in law of solutions.
It is proved in [16] that ISDE (2.2) has a µ-pathwise unique strong solution if µ is tail trivial, the logarithmic derivative d µ has a sort of off-diagonal smoothness, and the one-correlation function has sub-exponential growth at infinity. This results implies µ-uniqueness in law. We refer to If, in addition, we assume (A5), then for any m ∈ N lim N →∞ (X N,1 , . . . , X N,m ) = (X 1 , . . . , X m ).
weakly in C([0, ∞), R m ). Here X N = (X N,i ) N i=1 and X = (X i ) i∈N as before.

Reduction of Theorem 1.1 to (2.10)
In this subsection, we deduce Theorem 1.1 from Lemma 2.3 by assuming (2.10). We take µ N θ and µ θ as in Section 1. Then the logarithmic derivative d µ N θ of µ N θ is given by where y = i δ yi . From (2.14), we take coefficients in (A4) as follows:  [12]. The assumptions in Lemma 2.2 are checked in [11]. (A1) is well known. (A2) is assumed by (1.10). (A3) is obvious as the interaction is smooth outside the origin, and the capacity of the colliding set {x i = x j for some i = j} is zero (see [10,4]). Furthermore, the one-correlation functions are bounded, which guarantees explosion-free of tagged particles. We take functions in (A4) as (2.15) and (2.16). These satisfy (2.8), (2.9), and (1) of (A4). (2.10) is satisfied by assumption. It is known that µ θ is tail trivial [14]. Then (A5) follows from tail triviality of µ θ and [16, Theorem 3.1]. All the assumptions in Lemma 2.3 are thus satisfied, and hence yields (1.12).

A sufficient condition for (2.10)
The most crucial step to apply Lemma 2.3 is to check (2.10). Indeed, it only remains to prove (2.10) for Theorem 1.1. We quote then a sufficient condition for (2.10) in terms of correlation functions from [11]. Lemma 2.6 below is a special case of [11,Lemma 53].
Let µ N θ,x be the reduced Palm measure of µ N θ conditioned at x. We denote the supremum norm in x over S R by · R . Let E · and Var · denote the expectation and variance with resoect to ·, respectively.   [11], respectively. We note that in [11] we use 1 Sr (x) instead of χ r (x). This slight modification yields no difficulty. Multiplying w r (x, y) by a half, we give a sufficient condition of (2.
Proof. Lemma 2.6 follows immediately from a standard calculation of correlation functions and the definitions of w r and χ r .

Subsidiary estimates
Keeping Lemma 2.6 in mind, our task is to prove (2.21)-(2.24). To control the correlation functions in Lemma 2.6 we prepare in this section estimates of the oscillator wave functions and determinantal kernels. We shall use these estimates in Section 4.

Determinantal kernels of N-particle systems
We recall the definition of determinantal point processes. Let K : R 2 → C be a measurable kernel. A probability measure µ on S is called a determinantal point process with kernel K if, for each n, its n-point correlation function is given by If K is an Hermitian symmetric and of locally trace class such that 0 ≤ Spec(K) ≤ 1, then there exists a unique determinantal point process with kernel K [19,20]. The distribution of the delabeled eigenvalues of GUE associated with (1.1) is a determinantal point process with kernel K N such that (3.5) The Christoffel-Darboux formula and a simple calculation yield the following.
From the scaling (1.3), µ N θ is a determinantal point process with kernel From (3.7) and (3.8) we then clearly see that From (3.6) we deduce Using the Schwartz inequality to (3.5) we see from (3.6) and (3.8) that From here on, we assume We set The next lemma will be used in Section 4. (2) Assume (3.12). Then there exists a constant c 6 such that From this and (3.10), we see that with a simple calculation Combining this with (3.2) we obtain From this and (3.11) we deduce (3.14). From Lemma 3.1 and (3.17), we see that We deduce (3.15) from this and (3.11). Taking a constant c 5 in (3.14) and (3.15) in common completes the proof of (1). Claim (3.16) follows from Lemma 3.1, (3.6), and (3.8).
4 Proof of (2.21)-(2.24) As we see in Section 2, the point of the proof of Theorem 1.1 is to check conditions (2.21)-(2.24) in Lemma 2.6. The purpose of this section is to prove these equations. We recall a property of the reduced Palm measures of determinantal point processes.

Lemma 4.1 ([19]
). Let µ be a determinantal point process with kernel K. Assume that K(x, y) = K(y, x) and 0 ≤ Spec(K) ≤ 1. Then the reduced Palm measure µ x is a determinantal point process with kernel K x given by Let K N θ be the determinantal kernel of µ N θ given by (3.7). Let µ N θ,x be as in Lemma 2.6. Recall that K N θ (y, z) = K N θ (z, y) by definition. Then from this, (3.7), and (4.1), µ N θ,x is a determinantal point process with kernel .
Let B N and U N be as in Lemma 3.3. Decompose U N into U N 1 and U N 2 such that Then clearly We begin by the integral outside U N 1 .
Proof. From (3.11) and (4.12) we deduce that as N → ∞ From (3.16) and (4.14) for each N ∈ N and r > 0 Hence (4.15) follows from (4.17) and (4.18). This completes the proof of (4.15). We next prove (4.16). From (3.11), (4.12), and (4.14) we see for each r > 0 lim sup From (3.16) and (4.14) we see that for each N ∈ N and r > 0 Combining ( Proof. Note that L N (y, y) ≤ c 5 on U N by (3.15). Then by the definition of T N r,∞ (x), (4.23) By (3.14) we see L N (y, y) ≤ c 5 N 1 3 on R. Recall that |B N | = 4N α by construction. Furthermore, Here we used α < −1/2. We thus obtain (4.21) from (4.23) and (4.24). We proceed with the proof of (4.22). We first consider the integral away from the diagonal line. By (3.16) and the Schwartz inequality, we see that The last line follows from a straightforward calculation. Indeed, first integrating z over {|y − z| ≥ 1 N }, and then integrating y over T N r,∞ (x), we obtain the inequality in the last line. We therefore see that We next consider the integral near the diagonal. From (3.15), we see that From (4.25) and (4.26), we have We next consider the integral on B N × B N . Let Then, we deduce from (3.14) and the definition of B N given by (3.13) that Here we used |B N | = 4N α for the inequality and α < −1/2 for the last equality.
We finally consider the case U N × B N . Then a similar argument yields Collecting ( Next, using the Schwartz inequality, we have for the second term Applying (4.22) and (4.15) to the last line, we obtain lim r→∞ lim sup We finally estimate the third term. From (4.16), as N → ∞, we have → 0 by (4.16).
From (4.30), (4.31), and (4.32) we obtain (4.11). This completes the proof.  with the same initial condition as X N . Let P θ,N and Q θ,N be the distributions of X N and Y θ,N on C([0, T ]; R N ), respectively. Then applying the Girsanov theorem [3, pp.190-195] to (6.2), we see that dQ θ,N dP θ,N (W) = exp{ where we write W = (W i ) ∈ C([0, T ]; R N ) and {B i } N i=1 under P θ,N are independent copies of Brownian motions starting at the origin.