Vicious walkers and random contraction matrices

TL;DR

This paper establishes a connection between a deformed ensemble of random contraction matrices and Fisher's vicious walker model, revealing that the moment generating function relates to the partition function of the walkers with mutual attraction.

Contribution

It introduces a novel link between truncated random unitary matrices and vicious walker models, providing an exact solution for the generating function of the matrix trace.

Findings

The moment generating function corresponds to the partition function of the vicious walker model.

The model describes mutually attracting particles in the vicious walker framework.

A new exactly solvable connection between random matrix theory and statistical mechanics is demonstrated.

Abstract

The ensemble $\CUE^{(q)}$ of truncated random unitary matrices is a deformation of the usual Circular Unitary Ensemble depending on a discrete non-negative parameter $q.$ $\CUE^{(q)}$ is an exactly solved model of random contraction matrices originally introduced in the context of scattering theory. In this article, we exhibit a connection between $\CUE^{(q)}$ and Fisher's random-turns vicious walker model from statistical mechanics. In particular, we show that the moment generating function of the trace of a random matrix from $\CUE^{(q)}$ is a generating series for the partition function of Fisher's model, when the walkers are assumed to represent mutually attracting particles.