Monotone subsequences in locally uniform random permutations

Abstract

A locally uniform random permutation is generated by sampling n points independently from some absolutely continuous distribution ρ on the plane and interpreting them as a permutation by the rule that i maps to j if the ith point from the left is the jth point from below. As n tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences give rise to a surface. We show that, under the correct scaling, for any $\mathit{r}\ge 0$, the largest union of $\lfloor \mathit{r}\sqrt{\mathit{n}}\rfloor$ decreasing subsequences approaches a limit surface as n tends to infinity, and the limit surface is a solution to a specific variational problem. As a corollary, we prove the existence of a limit shape for the Young diagram associated to the random permutation under the Robinson–Schensted correspondence. In the special case where ρ is the uniform distribution on the diamond $|\mathit{x}|+|\mathit{y}|<1$, we conjecture that the limit shape is triangular, and assuming the conjecture is true, we find an explicit formula for the limit surfaces of a uniformly random permutation and recover the famous limit shape of Vershik, Kerov and Logan, Shepp.