Circular support in random sorting networks

Dauvergne, Duncan and Virág, Bálint (2020) Circular support in random sorting networks. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 373 (3). pp. 1529-1553. ISSN 0002-9947

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Abstract

A sorting network is a shortest path from 12 center dot center dot center dot n to n center dot center dot center dot 21 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 of the edge are within root 2c of a sine curve in uniform norm.[GRAPHICS]

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