Phase Uniqueness for the Mallows Measure on Permutations

TL;DR

This paper proves that the Mallows measure on permutations exhibits phase uniqueness, meaning no phase transition occurs, by analyzing its large deviation properties as a mean-field model.

Contribution

It establishes the absence of phase transition for the Mallows measure using large deviation principles, extending understanding of its statistical mechanics properties.

Findings

Proves phase uniqueness for the Mallows measure.

Shows no phase transition occurs in the model.

Uses large deviation principles to characterize equilibrium.

Abstract

For a positive number $q$ the Mallows measure on the symmetric group is the probability measure on $S_n$ such that $P_{n,q}(\pi)$ is proportional to $q$-to-the-power-$\mathrm{inv}(\pi)$ where $\mathrm{inv}(\pi)$ equals the number of inversions: $\mathrm{inv}(\pi)$ equals the number of pairs $i<j$ such that $\pi_i>\pi_j$. One may consider this as a mean-field model from statistical mechanics. The weak large deviation principle may replace the Gibbs variational principle for characterizing equilibrium measures. In this sense, we prove absence of phase transition, i.e., phase uniqueness.