Scaling limit of vicious walks and two-matrix model

TL;DR

This paper analyzes the diffusion scaling limit of vicious walkers, revealing a transition in particle distribution linked to eigenvalue distributions of Gaussian random matrices and connecting to two-matrix models.

Contribution

It establishes a connection between vicious walker models and eigenvalue distributions of Gaussian matrices, demonstrating a transition in distribution related to the two-matrix model.

Findings

Distribution transition from 0 to 1 ratio of t/T

Eigenvalue distribution corresponds to Gaussian random matrices

Contact interactions lead to eigenvalue-like correlations

Abstract

We consider the diffusion scaling limit of the one-dimensional vicious walker model of Fisher and derive a system of nonintersecting Brownian motions. The spatial distribution of $N$ particles is studied and it is described by use of the probability density function of eigenvalues of $N \times N$ Gaussian random matrices. The particle distribution depends on the ratio of the observation time $t$ and the time interval $T$ in which the nonintersecting condition is imposed. As $t/T$ is going on from 0 to 1, there occurs a transition of distribution, which is identified with the transition observed in the two-matrix model of Pandey and Mehta. Despite of the absence of matrix structure in the original vicious walker model, in the diffusion scaling limit, accumulation of contact repulsive interactions realizes the correlated distribution of eigenvalues in the multimatrix model as the particle distribution.