Large Deviations for Permutations Avoiding Monotone Patterns

TL;DR

This paper investigates the large deviations behavior of pattern-avoiding permutations, specifically calculating exponential decay rates of probabilities for fixed positions, and finds similar limits for certain non-monotone patterns.

Contribution

It provides explicit calculations of large deviations limits for pattern-avoiding permutations, extending understanding beyond monotone patterns to specific non-monotone cases.

Findings

Exponential decay rates for probabilities in pattern-avoiding permutations.

Limit of probability ratios matches for certain non-monotone patterns.

Large deviations regime characterized for specific pattern-avoidance scenarios.

Abstract

For a given permutation $\tau$, let $P_N^{\tau}$ be the uniform probability distribution on the set of $N$-element permutations $\sigma$ that avoid the pattern $\tau$. For $\tau=\mu_k:=123\cdots k$, we consider $P_N^{\mu_k}(\sigma_I=J)$ where $I\sim \gamma N$ and $J\sim \delta N$ for $\gamma,\delta \in (0,1)$. If $\gamma+\delta\neq 1$ then we are in the large deviations regime with the probability decaying exponentially, and we calculate the limiting value of $P_N^{\mu_k}(\sigma_I=J)^{1/N}$. We also observe that for $\tau = \lambda_{k,\ell} := 12\ldots\ell k(k-1)\ldots(\ell+1)$ and $\gamma+\delta<1$, the limit of $P_N^{\tau}(\sigma_I=J)^{1/N}$ is the same as for $\tau=\mu_k$.