Large deviation principles for pattern-avoiding permutations, and limit shapes for constrained Mallows permutations

TL;DR

This paper analyzes pattern-avoiding Mallows permutations, establishing their convergence to explicit limit shapes and deriving large deviation principles, with applications to $q$-Catalan number asymptotics.

Contribution

It introduces explicit limit permutons for pattern-avoiding Mallows permutations and proves a large deviation principle in this setting.

Findings

Permutations converge to explicit deterministic permutons depending on pattern and parameter.

Large deviation principle established for uniform pattern-avoiding permutations.

Asymptotic estimates for $q$-Catalan numbers in the regime $q=e^{eta/n}$.

Abstract

We study Mallows random permutations conditioned to avoid a given pattern $\alpha$ of length~$3$. When the bias parameter is of the form $e^{\beta/n}$, we prove that these permutations converge to a non-trivial explicit deterministic permuton that depends on the pattern $\alpha$ and on the parameter $\beta$. Along the way, we provide parametrizations for $\alpha$-avoiding permutons, and establish a large deviation principle for uniform $\alpha$-avoiding permutations. As a byproduct of the proof, we also obtain asymptotic estimates of two versions of $q$-Catalan numbers in the regime $q=e^{\beta/n}$.