Monotone Subsequences in Locally Uniform Random Permutations

TL;DR

This paper studies the asymptotic behavior of decreasing subsequences in locally uniform random permutations, revealing a limit surface described by a variational problem and connecting to classical limit shape results.

Contribution

It introduces a new geometric framework for understanding decreasing subsequences in random permutations via limit surfaces and variational problems, extending classical results.

Findings

Largest union of decreasing subsequences converges to a limit surface

Proves existence of a limit shape for the associated Young diagram

Recovers the Vershik-Kerov and Logan-Shepp limit shape in a special case

Abstract

A locally uniform random permutation is generated by sampling $n$ points independently from some absolutely continuous distribution $\rho$ on the plane and interpreting them as a permutation by the rule that $i$ maps to $j$ if the $i$th point from the left is the $j$th 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 $r\ge0$, the largest union of $\lfloor r\sqrt{n}\rfloor$ 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 $\rho$ is the uniform distribution on the diamond $|x|+|y|<1$ 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.