Abstract
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.
Highlights
Consider the unitary ensembles of random matrices whose density is given by 1 |detM|2αe−TrM∗M dM for α > − 1, Z (1.1)where dM is the usual flat Lebesgue measure on the space of N ×N Hermite matrices, and Z is the normalising constant
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
We showed that solutions to infinite-dimensional stochastic differential equation (ISDE) satisfy (IFC) under mild assumptions, and if a solution is associated with a Dirichlet form, we can check the five assumptions [13, 14]
Summary
Consider the unitary ensembles of random matrices whose density is given by 1 |detM|2αe−TrM∗M dM for α > − 1 , Z (1.1). 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. 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. 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]. Provided suitable labelling, the approximation of the logarithmic derivatives imply that the first m-particles of (XN,i)1≤i≤N converge to that of (Xi)i∈N weakly in the path space.
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