Increasing subsequences and the hard-to-soft edge transition in matrix ensembles

TL;DR

This paper explores the probabilities of maximum increasing subsequences in various random permutation ensembles, linking them to matrix ensemble gap probabilities and analyzing their transition from hard to soft edge regimes.

Contribution

It establishes the equivalence of these probabilities to matrix ensemble gap probabilities and proves their limiting forms, confirming previous theorems by Baik-Deift-Johansson and Baik-Rains.

Findings

Probabilities equal to matrix ensemble gap probabilities.

Limiting forms match soft edge gap probabilities.

Reconfirmation of key theorems in random matrix theory.

Abstract

Our interest is in the cumulative probabilities Pr(L(t) \le l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively. The gap probabilities can be written as a sum over correlations for certain determinantal point processes. From these expressions a proof can be given that the limiting form of Pr(L(t) \le l) in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively, thereby reclaiming theorems due to Baik-Deift-Johansson and Baik-Rains.