On the partition function of a class of Mallows model

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 $\mathbb{P}(X=σ)=L_n^{-1} \prod_{i=1}^{n}\exp(-c(i/n, σ(i)/n))$. Mukherjee [Ann. Stat., Vol. 44(2), pp 853--875 (2016)] computed the limit of the log partition function and showed that $\lim_{n\to \infty}\frac{1}{n}\log L_n=-Γ_0$ where $Γ_0$ 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 $\lim_{n\to \infty} e^{-nΓ_0}L_n$ in terms of the Fredholm determinant of an integral operator and provided a partial proof. We give a complete proof of Pal's conjecture.