Moments of vicious walkers and M\"obius graph expansions

TL;DR

This paper analyzes the transition of vicious walkers' spatial distribution from GUE to GOE statistics over time, using graphical expansions involving ribbon and M"obius graphs to characterize the underlying topological changes.

Contribution

It introduces a graphical expansion formula for moments of vicious walkers, capturing the GUE-to-GOE transition through M"obius graph contributions and topological surface classification.

Findings

GUE-to-GOE transition characterized by graph topology changes

Closed-form expressions for M"obius expansion coefficients

Connection between random matrix statistics and topological graph theory

Abstract

A system of Brownian motions in one-dimension all started from the origin and conditioned never to collide with each other in a given finite time-interval $(0, T]$ is studied. The spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue-statistics of random Hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the Gaussian unitary ensemble (GUE) statistics to the Gaussian orthogonal ensemble (GOE) statistics as the time $t$ is going on from 0 to $T$. In the present paper, we characterize this GUE-to-GOE transition by presenting the graphical expansion formula for the moments of positions of vicious walkers. In the GUE limit $t \to 0$, only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus. Following the time evolution of the vicious walkers, however, the graphs with twisted ribbons, called M\"obius graphs, increase their contribution to our expansion formula, and we have to deal with the topology of non-orientable surfaces. Application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the M\"obius expansion using the Stirling numbers of the first kind.