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# Random sorting networks: edge limit

Jul 2022

A sorting network is a shortest path from $12\dots n$ to $n\dots 21$ in theCayley graph of the symmetric group $\mathfrak S_n$ spanned by adjacenttranspositions. The paper computes the edge local limit of the uniformly randomsorting networks as $n\to\infty$. We find the asymptotic distribution of thefirst occurrence of a given swap $(k,k+1)$ and identify it with the law of thesmallest positive eigenvalue of a $2k\times 2k$ aGUE (an aGUE matrix has purelyimaginary Gaussian entries that are independently distributed subject toskew-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 auniformly random sorting network. Two definitions lead to two differentexpressions for the asymptotic laws expressed in terms of derivatives ofFredholm determinants.

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