Thermodynamic Limit for the Mallows Model on \(S_n\)

Abstract

The Mallows model on \(S_n\) is a probability distribution on permutations, \(q^{d(\pi,e)}/P_n(q)\), where \(d(\pi,e)\) is the distance between \(\pi\) and the identity element, relative to the Coxeter generators. Equivalently, it is the number of inversions: pairs \((i,j)\) where \(1\leq i<j\leq n\), but \(\pi_i>\pi_j\). Analyzing the normalization \(P_n(q)\), Diaconis and Ram calculated the mean and variance of \(d(\pi,e)\) in the Mallows model, which suggests the appropriate \(n \to \infty\) limit has \(q_n\) scaling as \(1-\beta/n\). We calculate the distribution of the empirical measure in this limit, \(u(x,y) dx dy = \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n} \delta_{(i,\pi_i)}\). Treating it as a mean-field problem, analogous to the Curie-Weiss model, the self-consistent mean-field equations are \(\frac{\partial^2}{\partial x \partial y} \ln u(x,y) = 2 \beta u(x,y)\), which is an integrable PDE, known as the hyperbolic Liouville equation. The explicit solution also gives a new proof of formulas for the blocking measures in the weakly asymmetric exclusion process, and the ground state of the \(\mathcal{U}_q(\mathfrak{sl}_2)\)-symmetric XXZ ferromagnet.