A random walk on the permutation group, some formal long-time asymptotic relations

TL;DR

This paper investigates the long-time behavior of a random walk on the permutation group of lattice vertices, exploring asymptotic relations and conjectures related to independent Gaussian diffusion of vertices, with applications to statistical mechanics.

Contribution

It provides formal asymptotic results for the heat equation on the permutation group, connecting random walks to Gaussian diffusion conjectures in the context of the Heisenberg model.

Findings

Long-time distribution approximates independent Gaussian distributions for each vertex

Formal asymptotic relations support the conjecture of vertex independence

Insights applicable to models of ferromagnetism in statistical mechanics

Abstract

We consider the group of permutations of the vertices of a lattice. A random walk is generated by unit steps that each interchange two nearest neighbor vertices of the lattice. We study the heat equation on the permutation group, using the Laplacian associated to the random walk. At t=0 we take as initial conditions a probability distribution concentrated at the identity. A natural conjecture for the probability distribution at long times is that it is 'approximately' a product of Gaussian distributions for each vertex. That is, each vertex diffuses independently of the others. We obtain some formal asymptotic results in this direction. The problem arises in certain ways of treating the Heisenberg model of ferromagnetism in statistical mechanics.