Random walks and random permutations

TL;DR

This paper establishes a link between vicious walkers' random turns model and random permutations, revealing that the maximum displacement distribution aligns with the GUE eigenvalues at the soft edge, highlighting universal scaling behaviors.

Contribution

It connects the random turns model of vicious walkers with random permutations and eigenvalue distributions, providing new insights into their asymptotic behavior.

Findings

Maximum mean displacement scales as (2t)^{1/2}

Distribution matches the scaled eigenvalues at the GUE soft edge

Standard deviation scales as the mean displacement to the 1/3 power

Abstract

A connection is made between the random turns model of vicious walkers and random permutations indexed by their increasing subsequences. Consequently the scaled distribution of the maximum displacements in a particular asymmeteric version of the model can be determined to be the same as the scaled distribution of the eigenvalues at the soft edge of the GUE. The scaling of the distribution gives the maximum mean displacement $\mu$ after $t$ time steps as $\mu = (2t)^{1/2}$ with standard deviation proportional to $\mu^{1/3}$. The exponent 1/3 is typical of a large class of two-dimensional growth problems.