The longest increasing subsequence in a random permutation and a unitary random matrix model

Abstract

If LN is the expected length of the longest increasing subsequence in a randomperm utation, then LN ∼ 2 √ N as N →∞ . We give a new proof of this result using a connection with a certain unitary random matrix model. The asymptotic formula is directly related to a third order phase transition in this model found by Gross and Witten.