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

Abstract

We study Mallows random permutations conditioned to avoid a given pattern $\hat{I}\pm$ of length~$3$. When the bias parameter is of the form $e^{{\hat{I}}^2/n}$, we prove that these permutations converge to a non-trivial explicit deterministic permuton that depends on the pattern $\hat{I}\pm$ and on the parameter ${\hat{I}}^2$. Along the way, we provide parametrizations for $\hat{I}\pm$-avoiding permutons, and establish a large deviation principle for uniform $\hat{I}\pm$-avoiding permutations. As a byproduct of the proof, we also obtain asymptotic estimates of two versions of $q$-Catalan numbers in the regime $q=e^{{\hat{I}}^2/n}$.

38 pages