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Circular support in random sorting networks

By Duncan Dauvergne and Bálint Virág

Abstract

A sorting network is a shortest path from $12 \cdots n$ to $n \cdots 2 1$ in the Cayley graph of the symmetric group generated by adjacent transpositions. For a uniform random sorting network, we prove that in the global limit, particle trajectories are supported on $\pi$-Lipschitz paths. We show that the weak limit of the permutation matrix of a random sorting network at any fixed time is supported within a particular ellipse. This is conjectured to be an optimal bound on the support. We also show that in the global limit, trajectories of particles that start within distance $\epsilon$ of the edge are within $\sqrt{2\epsilon}$ of a sine curve in uniform norm.Comment: 28 pages, 5 figure

Topics: Mathematics - Probability, Mathematics - Combinatorics, 60C05, 05E10, 68P10
Publisher: 'American Mathematical Society (AMS)'
Year: 2018
DOI identifier: 10.1090/tran/7819
OAI identifier: oai:arXiv.org:1802.08933

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