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 $1\le i<j\le n$, 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, $u(x,y)dxdy={\lim }_{n\to \infty }\frac{1}{n}{\sum }_{i=1}^n{\delta }_{(i,{\pi }_i)}$. 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.

14 pages, several important references added

References in corpus (4)

• The mean field Ising model trough interpolating techniques
• A mechanical approach to mean field spin models
• Quantum Crystals and Spin Chains
• Path integral representations for the spin-pinned quantum XXZ chain

Cited by in corpus (24)

• $q$-exchangeability via quasi-invariance
• Estimation in exponential families on permutations
• Splitting groups with cubic Cayley graphs of connectivity two
• The feasible region for consecutive patterns of permutations is a cycle polytope
• Triangulations of simplices with vanishing local h-polynomial
• Shock dynamics of phase diagrams
• Fixed points and cycle structure of random permutations
• Notes on the p-spin glass studied via Hamilton-Jacobi and Smooth-Cavity techniques
• Local convergence for permutations and local limits for uniform $\rho$-avoiding permutations with $|\rho |=3$
• Square permutations are typically rectangular
• Mallows permutations as stable matchings
• Geometry of Permutation Limits
• The skew Brownian permuton: a new universality class for random constrained permutations
• Critical parameter of random loop model on trees
• Mallows Permutations and Finite Dependence
• The Limit of the Empirical Measure of the Product of Two Independent Mallows Permutations
• The Length of the Longest Common Subsequence of Two Independent Mallows Permutations
• Binary search trees of permuton samples
• Lengths of Monotone Subsequences in a Mallows Permutation
• The Top-K Tau-Path Screen for Monotone Association
• The Length of the Longest Increasing Subsequence of a Random Mallows Permutation
• A Phase Transition in Arrow's Theorem
• Arcsine laws for random walks generated from random permutations with applications to genomics
• Limit Theorems for the Length of the Longest Common Subsequence of Mallows Permutations