Probability and Mathematical Physics Seminar

What is the behavior of the longest increasing subsequence of a uniformly random permutation? Its length is of order 2n^{1/2} plus Tracy–Widom fluctuations of order n^{1/6}. Its scaling limit is the directed geodesic of the directed landscape.

This talk discusses how this behavior changes dramatically when one looks at universal Brownian–type permutations, i.e., permutations σₙ sampled from the Brownian separable permutons μₚ for p∈(0,1). We show that LIS(σₙ)/nᵅ converges almost surely to X, where $α=α (p)∈(1/2,1)  is the unique solution to an equation involving a number-theoretic function and X=X(p) is a non-deterministic and a.s. positive and finite random variable, which is a measurable function of the Brownian separable permuton μₚ.

Based on joint work with Arka Adhikari, Thomas Budzinski, William Da Silva and Delphin Sénizergues.