The generalised sine random point field arises from the scaling limit at the origin of the eigenvalues of the generalised Gaussian ensembles. We solve an infinite-dimensional stochastic differential equation (ISDE) describing an infinite number of interacting Brownian particles which is reversible with respect to the generalised sine random point field. Moreover, finite particle approximation of the ISDE is shown, that is, a solution to the ISDE is approximated by solutions to finite-dimensional SDEs describing finite-particle systems related to the generalised Gaussian ensembles.

Consider the unitary ensembles of random matrices whose density is given by

Note that

To focus on fluctuation of the generalised Gaussian ensembles around the origin, we take a scaling. We set

As with other random matrix models, the universality of the limit kernel has been studied. Let

It is natural to try to study stochastic dynamics with infinitely many particles related to the universal random point field

Taking

It is important to point out that (

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

We use the Dirichlet form approach to construct the unique strong solution to (

This paper is organised as follows. In Section

Let

For each

We define an unlabelling map

Recall that

The solutions obtained in Theorem

Five assumptions

Let

Theorem

We set

A random point field

From Theorem

This corollary immediately follows from Theorem

We remark that each particle does not hit the origin. Let

From (

We next prepare some quantities to introduce the logarithmic derivative, which is a crucial quantity for the representation of ISDE. For a random point field

A function

An

The space of test functions is not

The next claim is the key theorem in the present paper.

This subsection is devoted to the general framework – the Dirichlet form approach for infinite particle systems. We first introduce Dirichlet forms describing

For

Let

For each

There exists the logarithmic derivative

For each

There exists

From

Recall that

We can naturally lift each

From

We shall apply Lemma

It is easy to see that

We derive Theorem

Assumption

Combining these with Lemma

With the above argument, Theorem

For

The orthogonal polynomials

This lemma makes an expression of the one-correlation function of

Noting that

From (

Because of Lemma

Remark that several

With a computation similar to that used in [

For

From (

To show (ii), let

From (

Using the results in Section

Fix a constant

We prove for the case

Next, we shall show (ii). Using (

Inequality (

The logarithmic derivative

For each

Note that

Let

For each

It holds that

It holds that

For each

Then, we obtain an explicit expression of the logarithmic derivative of

Let

From (

This lemma follows from [

We begin by rewriting the conditions in Lemma

Recall that

We shall prove (

From (

Our proof starts with the observation that (

Applying the Schwartz inequality to (

From

Let

Equation (

By the same argument as above, (

From (

Next we prove (

We next consider the integration on

Lastly, we consider the case

We estimate three terms in (

The first term in (

For the second term in (

Lastly, the third term in (

It is easy to see

To show Theorem

There exists a sequence of random point fields

For each

Let

There exists

We first check

Recall

Taking

Following [

Recall that

Let

Let

For

We formulate strong solution to (

We say that

SDE (

Then, we introduce the IFC condition for

For each

We next introduce several conditions for the strong uniqueness of solutions to ISDEs. See [

Let