Random Sorting Networks

2006/09/19 by Omer Angel, Alexander E. Holroyd, Dan Romik +1 · 1 citation
Mathematics · #math.PR #math.CO #msc:60C05 #msc:05E10 #msc:68P10

paper · pdf · doi:10.1016/j.aim.2007.05.019

published as Adv. Math. 215 (2007), no. 2, 839-868 · 38 pages, 12 figures

arxiv created 2006/09/19 · arxiv updated 2011/11/10

Abstract

A sorting network is a shortest path from 12...n to n...21 in the Cayley graph of S_n generated by nearest-neighbour swaps. We prove that for a uniform random sorting network, as n->infinity the space-time process of swaps converges to the product of semicircle law and Lebesgue measure. We conjecture that the trajectories of individual particles converge to random sine curves, while the permutation matrix at half-time converges to the projected surface measure of the 2-sphere. We prove that, in the limit, the trajectories are Holder-1/2 continuous, while the support of the permutation matrix lies within a certain octagon. A key tool is a connection with random Young tableaux.

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