Random sorting networks

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 → ∞ the spacetime 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 Hölder-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. τ s 1 τ s 2 · · · τ s N = ρ

A combinatorial version of Sylvester's four-point problem

“…Most notably, Angel et al [2] consider several convergence questions. V. Reiner [11] computes the expected number of possible Yang-Baxter moves for such a reduced expression in the symmetric group while B. Tenner [14] performs the analogous calculation for the hyperoctahedral group.…”

“…Fix a closed, bounded region R in the plane along with a directed line . The uniform distribution on R induces a probability distribution f n on φ(P n ) for each n. We can think of picking four points at random from R as picking one of the 16 elements of R( [4,3,2,1]) according to f 4 . The distribution f 4 is displayed for several different shapes in Fig.…”

“…Running over all unordered pairs of points, these θ define a sequence 0 θ 1 < θ 2 < · · · < θ ( n 2 ) < π. (2) We now label the points according to their projections onto . (In the probability-zero case that θ 1 = 0, we instead project onto (− ) for some sufficiently small .)…”

Geodesics and almost geodesic cycles in random regular graphs

“…We do not define “shape”, but one might take the minimum metric distortion of an embedding of one graph in another to be a measure of the difference of their shapes. In connection with the influence of geodesics upon the shape of a graph, Angel et al 2 study random geodesics in a graph defined from permutations, and conjecture that the geodesics lie close to great circles in a particular Euclidean embedding of the graph.…”

Limiting behavior for the distance of a random walk

“…Recently, Angel et al [1] have also used the simple exclusion process to analyze a process on the Cayley graph of the symmetric group generated by adjacent transpositions, but this time in the context of sorting networks.…”