On the partition function of a class of Mallows model

TL;DR

This paper proves Pal's conjecture by providing a complete proof for the exact asymptotic behavior of the partition function in a specific class of Mallows models, linking it to Fredholm determinants.

Contribution

It offers a full proof of Pal's conjecture, establishing the precise limit of the scaled partition function in the Mallows model class.

Findings

Confirmed Pal's conjecture on the partition function limit.

Connected the limit to the Fredholm determinant of an integral operator.

Extended understanding of the asymptotic behavior of Gibbs measures on permutations.

Abstract

Let $\Sym{n}$ denote the set of all permutations on $n$ labels. Let $c:[0, 1]^2\to [0, \infty)$ be a twice continuously differentiable function. A subfamily of the Mallows model is the Gibbs probability measures on $\Sym{n}$ such that $\mathbb{P}(X=\sigma)=L_n^{-1} \prod_{i=1}^{n}\exp(-c(i/n, \sigma(i)/n))$. Mukherjee [Ann. Stat., Vol. 44(2), pp 853--875 (2016)] computed the limit of the log partition function and showed that $\lim_{n\to \infty}\frac{1}{n}\log L_n=-\Gamma_0$ where $\Gamma_0$ is the optimal cost associated with an entropy regularized optimal transport problem. In the KRP Memorial Volume of the Indian Journal of Pure and Applied Math, Pal conjectured an exact value for the limit $\lim_{n\to \infty} e^{-n\Gamma_0}L_n$ in terms of the Fredholm determinant of an integral operator and provided a partial proof. We give a complete proof of Pal's conjecture.