Random sorting networks: edge limit

TL;DR

This paper analyzes the asymptotic edge behavior of uniformly random sorting networks, revealing the distribution of swap occurrences and connecting them to eigenvalues of a special class of random matrices.

Contribution

It computes the edge local limit of random sorting networks and links swap occurrence distributions to eigenvalues of aGUE matrices, introducing new formal definitions of spacing.

Findings

Distribution of first swap occurrence matches smallest eigenvalue of aGUE matrices

Asymptotic laws expressed via derivatives of Fredholm determinants

Two different definitions of spacing lead to distinct asymptotic expressions

Abstract

A sorting network is a shortest path from $12\dots n$ to $n\dots 21$ in the Cayley graph of the symmetric group $\mathfrak S_n$ spanned by adjacent transpositions. The paper computes the edge local limit of the uniformly random sorting networks as $n\to\infty$. We find the asymptotic distribution of the first occurrence of a given swap $(k,k+1)$ and identify it with the law of the smallest positive eigenvalue of a $2k\times 2k$ aGUE (an aGUE matrix has purely imaginary Gaussian entries that are independently distributed subject to skew-symmetry). Next, we give two different formal definitions of a spacing -- the time distance between the occurrence of a given swap $(k,k+1)$ in a uniformly random sorting network. Two definitions lead to two different expressions for the asymptotic laws expressed in terms of derivatives of Fredholm determinants.