Random sorting networks: edge limit

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 $n\to \mathrm{\infty }$. 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.