The band structure of a model of spatial random permutation

TL;DR

This paper analyzes the band structure of a lattice permutation model with Euclidean displacement energy, revealing how mean displacement scales with temperature and system size, and introduces a novel connection to Gaussian fields.

Contribution

It establishes the band structure of the model as system size grows and temperature decreases, using a novel approach linking permutations to Gaussian fields.

Findings

Mean displacement scales as min{1/β, N}

Precise results in one dimension with constants and convergence rates

Asymptotics for permanents of KMS matrices

Abstract

We study a random permutation of a lattice box in which each permutation is given a Boltzmann weight with energy equal to the total Euclidean displacement. Our main result establishes the band structure of the model as the box-size $N$ tends to infinity and the inverse temperature~$\beta$ tends to zero; in particular, we show that the mean displacement is of order $\min \{ 1/\beta, N\}$. In one dimension our results are more precise, specifying leading-order constants and giving bounds on the rates of convergence. Our proofs exploit a connection, via matrix permanents, between random permutations and Gaussian fields; although this connection is well-known in other settings, to the best of our knowledge its application to the study of random permutations is novel. As a byproduct of our analysis, we also provide asymptotics for the permanents of Kac-Murdock-Szego (KMS) matrices.