Consider the unitary ensembles of random matrices whose density is given by (1.1) 1 Z | det M | 2 α e − Tr M ∗ M d M for α > − 1 2 , -\frac{1}{2},\]]]> where d M is the usual flat Lebesgue measure on the space of N × N Hermite matrices, and Z is the normalising constant. Hereafter, by abuse of notation, we use the same letter Z for normalising constants of several ensembles. Let x N = ( x 1 , … , x N ) ∈ R N be the eigenvalues of an N × N matrix M. When M is distributed as (1.1), the probability density function of its eigenvalues is written as the following m G , α N , which is called generalised Gaussian ensembles [20]: (1.2) m G , α N ( d x N ) = 1 Z ∏ 1 ≤ i < j ≤ N | x i − x j | 2 ∏ 1 ≤ k ≤ N | x k | 2 α e − x k 2 d x N . Then, m G , α N naturally gives a random point field μ G , α N in the sense that the labelled density of μ G , α N with respect to the Lebesgue measure is given by m G , α N .

Note that μ G , α N is a determinantal random point field. Let { p α n } n ∈ N be the monic orthogonal polynomials with respect to | x | 2 α e − x 2 d x, and we set h n = ∫ R ( p α n ( x ) ) 2 | x | 2 α e − x 2 d x. Let K G , α N : R × R → R be the determinantal kernel defined as (1.3) K G , α N ( x , y ) = | x | α | y | α e − x 2 2 − y 2 2 ∑ k = 0 N − 1 p α k ( x ) p α k ( y ) h k . Then, for each n, the n-correlation function ρ G , α N , n of μ G , α N is given by ρ G , α N , n ( x 1 , … , x n ) = det 1 ≤ i , j ≤ n [ K G , α N ( x i , x j ) ] . From the Cristoffel–Darboux formula [6, Proposition 5.1.3], we have K G , α N ( x , y ) = | x | α | y | α e − x 2 2 − y 2 2 h N − 1 p α N ( x ) p α N − 1 ( y ) − p α N − 1 ( x ) p α N ( y ) x − y .

To focus on fluctuation of the generalised Gaussian ensembles around the origin, we take a scaling. We set (1.4) K α N ( x , y ) = 1 N K G , α N ( x N , y N ) . Let μ α N be the determinantal random point field with kernel K α N . Clearly, the labelled density m α N of μ α N is written in the form (1.5) m α N ( d x N ) = 1 Z ∏ 1 ≤ i < j ≤ N | x i − x j | 2 ∏ 1 ≤ k ≤ N | x k | 2 α e − x k 2 N d x N . Then, the scaled kernel K α N has a nontrivial limit (1.6) lim N → ∞ K α N ( x , y ) = K α ( x , y ) compact uniformly . Here, the limit kernel K α ( x , y ) : R × R → R is given by, for x ≠ y, (1.7) K α ( x , y ) = x | x − 1 y | J α + 1 2 ( 2 | x | ) J α − 1 2 ( 2 | y | ) − y | x y − 1 | J α − 1 2 ( 2 | x | ) J α + 1 2 ( 2 | y | ) 2 ( x − y ) , and, for x = y, (1.8) K α ( x , x ) = | x | 2 { J α + 1 2 2 ( 2 | x | ) + J α − 1 2 2 ( 2 | x | ) − J α + 3 2 ( 2 | x | ) J α − 1 2 ( 2 | x | ) − J α + 1 2 ( 2 | x | ) J α − 3 2 ( 2 | x | ) } , where J ν denotes the Bessel function of the first kind of order ν. Remark that (1.6) yields weak convergence of random point fields. Let μ α be the determinantal random point field whose kernel is K α . More precisely, the n-correlation function ρ α n of μ α is given by (1.9) ρ α n ( x 1 , … , x n ) = det 1 ≤ i , j ≤ n [ K α ( x i , x j ) ] . Then, we obtain lim N → ∞ μ α N = μ α weakly . The kernel K α is called the generalised sine kernel of order α [1]: when α = 0, K α becomes the classical sine kernel sin ( x − y ) / ( x − y ). Note that, for x , y > 00$]]> or α ∈ N, the kernel (1.7) takes a simpler form K α ( x , y ) = x y J α + 1 2 ( 2 x ) J α − 1 2 ( 2 y ) − J α − 1 2 ( 2 x ) J α + 1 2 ( 2 y ) 2 ( x − y ) .

As with other random matrix models, the universality of the limit kernel has been studied. Let V : R → R be a function that increases fast enough at infinity, for instance, an even degree polynomial with a positive leading coefficient. Then, we consider the following m V , α N , which is a generalisation of (1.2): m V , α N ( d x N ) = 1 Z ∏ 1 ≤ i < j ≤ N | x i − x j | 2 ∏ 1 ≤ k ≤ N | x k | 2 α e − V ( x k ) d x N . Then, a natural problem is the universality of random matrices. More precisely, it is of interest to show that, for a potential V of a quite wide class, local fluctuation at the origin for m V , α N yields the universal random point field μ α under suitable scaling. Akemann et al. showed the universality under the assumptions that α is a nonnegative integer, V is an even degree polynomial, and recurrence coefficients of associated orthogonal polynomials satisfy certain condition [1]. The assumption that α is a nonnegative integer was excepted by Kanzieper and Freilikher [10]. Moreover, the restriction that V is even degree polynomial was removed by Kuijlaars and Vanlessen. They showed universality for real analytic potential with mild assumptions [15].

It is natural to try to study stochastic dynamics with infinitely many particles related to the universal random point field μ α . More specifically, our purpose is to construct reversible diffusion whose equilibrium measure is μ α , and describe its infinite-dimensional stochastic differential equation (ISDE). In order to derive such an ISDE, we consider a stochastic process related to μ α N , which is given by (1.10) d X t N , i = d B t i + { α X t N , i − X t N , i N + ∑ j ≠ i N 1 X t N , i − X t N , j } d t , 1 ≤ i ≤ N , where ( B i ) i = 1 N is the N-dimensional Brownian motion. We derive (1.10) as follows. Consider a Dirichlet form E ( f , g ) = ∫ R N 1 2 ∑ i = 1 N ∂ f ∂ x i ∂ g ∂ x i m α N ( d x N ) on L 2 ( R N , m α N ( d x N ) ). Then, by using integration by parts and (1.5), the generator − A N of E on L 2 ( R N , m α N ( d x N ) ) is given by A N = 1 2 Δ + ∑ i = 1 N { α x i − x i N + ∑ j ≠ i N 1 x i − x j } ∂ ∂ x i , which corresponds to (1.10).

Taking N to infinity in (1.10), it is expected that the limit ISDE is given by (1.11) d X t i = d B t i + { α X t i + lim r → ∞ ∑ | X t i − X t j | < r , j ≠ i 1 X t i − X t j } d t , i ∈ N . Here, ( B i ) i ∈ N is the collection of independent copies of the one-dimensional Brownian motion. Clearly, (1.11) becomes the Dyson model in the case α = 0. Our purpose is to construct a solution ( X i ) i ∈ N to (1.11).

It is important to point out that (1.11) is essentially different from the Bessel interacting ISDE, and deriving (1.11) from (1.10) is nontrivial for the long-range effect of logarithmic interaction. Honda and Osada derived the following Bessel interacting ISDE for α > 1 / 21/2$]]> [9]: (1.12) d X t i = d B t i + { α X t i + ∑ j ∈ N , j ≠ i 1 X t i − X t j } d t , i ∈ N . Note that the (unlabelled) solution to (1.12) is reversible with respect to the Bessel random point field. Seemingly, our model (1.11) resembles (1.12). However, the sum of the drift term in (1.12) converges absolutely unlike (1.11). The Bessel random point field is a random point field on the positive line, and its one-correlation function, which stands for the mean density of particles, decreases with order x − 1 / 2 as x → ∞; therefore, the sum of the interaction term in (1.12) converges absolutely. In contrast, the generalised sine random point field is on R, and its one-correlation function is of constant order as | x | → ∞; this implies that the drift term in (1.11) does not converge absolutely. Thus, we need careful computation to show the conditional convergence of the drift term as opposed to (1.12), and (1.11) is more difficult to study than (1.12) in this respect.

We refer to historical remarks on interacting Brownian motions with infinitely many particles. For a free potential Φ and an interaction potential Ψ, interacting Brownian motions in infinite-dimensions are given by the ISDE of the form (1.13) d X t i = d B t i − β 2 ∇ x Φ ( X t i ) d t − β 2 ∑ j ≠ i ∞ ∇ x Ψ ( X t i , X t j ) d t , i ∈ N . Here, ( B i ) i ∈ N is the collection of independent copies of d-dimensional Brownian motion and β > 00$]]> is the inverse temperature. Lang derived general solutions to ISDE (1.13) under the condition Φ = 0 and Ψ ∈ C 0 3 ( R d ) [16, 17]. Fritz explicitly described the set of starting points for up to four dimensions [7], and Tanemura solved equations for hardcore Brownian balls [31]. These results were achieved through Itô’s method, and required the coefficients to be smooth and have compact support. These conditions exclude physically interesting examples of long-range interaction potentials, such as the Lennard-Jones 6-12 potential and Riesz potentials. In particular, the logarithmic potential, which appears in random matrix theory, is also excluded. Osada established a Dirichlet form approach to solve ISDEs with a long-range potential, which can be applied to the logarithmic interacting particle systems [22, 24–27]. Osada and Tanemura also showed strong uniqueness of solutions [29]. Furthermore, Tsai constructed nonequilibrium solutions to the Dyson model for 1 ≤ β < ∞ [32].

We use the Dirichlet form approach to construct the unique strong solution to (1.11). More precisely, we construct a reversible diffusion on the configuration space with respect to μ α , and show that the process satisfies (1.11). To show ISDEs by the Dirichlet form approach, expression of the logarithmic derivatives is crucial, because the logarithmic derivatives correspond the drift terms of ISDEs. Bufetov, Dymov, and Osada introduced a method to compute the logarithmic derivatives for determinantal random point fields [2]. Since μ α is determinantal, their result seems to be applicable to our case. In spite of this, we use finite-particle approximation method to find the logarithmic derivative of μ α , which was introduced in [25]. This is because finite-particle approximation of the logarithmic derivative implies that of dynamics [12]. More accurately, let ( X N , i ) 1 ≤ i ≤ N and ( X i ) i ∈ N be the solutions to (1.10) and (1.11), respectively. Then, provided suitable labelling, the approximation of the logarithmic derivatives imply that the first m-particles of ( X N , i ) 1 ≤ i ≤ N converge to that of ( X i ) i ∈ N weakly in the path space. We will show such convergence in Theorem 2.4. This is an advantage of our method.

This paper is organised as follows. In Section 2, main results are shown. In Section 3, we prepare estimates of determinantal kernels. In Section 4 and Section 5, the logarithmic derivative and quasi-Gibbs property of μ α are shown, respectively. In Section 6, the strong uniqueness of our solution is proved.