Characterizations of the hydrodynamic limit of the Dyson model

TL;DR

This paper studies the hydrodynamic limit of the Dyson model, showing convergence of empirical measures to a process characterized by the Green's function satisfying the complex Burgers equation, with detailed analysis of related equations.

Contribution

It provides a detailed characterization of the hydrodynamic limit of the Dyson model using Green's functions and PDE techniques, extending understanding of the model's macroscopic behavior.

Findings

Empirical measures converge to a unique process independent of ta

Green's function satisfies the complex Burgers equation in the limit

Characterization of limit processes for specific initial configurations

Abstract

Under appropriate conditions for the initial configuration, the empirical measure of the $N$-particle Dyson model with parameter $\beta \geq 1$ converges to a unique measure-valued process as $N$ goes to infinity, which is independent of $\beta$. The limit process is characterized by its Stieltjes transform called the Green's function. Since the Green's function satisfies the complex Burgers equation in the inviscid limit, this is called the hydrodynamic limit of the Dyson model. We review the relations among the hydrodynamic equation of the Green's function, the continuity equation of the probability density function, and the functional equation of the Green's function. The basic tools to prove the relations are the Hilbert transform, a special case of the Sokhotski-Plemelj theorem, and the method of characteristics for solving partial differential equations. For two special initial configurations, we demonstrate how to characterize the limit processes using these relations.