Large Deviations for Permutations Avoiding Monotone Patterns

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

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