The local limit of random sorting networks

A sorting network is a geodesic path from 12 · · · n to n · · · 21 in the Cayley graph of S_n generated by adjacent transpositions. For a uniformly random sorting network, we establish the existence of a local limit of the process of space–time locations of transpositions in a neighbourhood of an for a ∈ [0, 1] as n → ∞. Here time is scaled by a factor of 1/n and space is not scaled. The limit is a swap process U on Z. We show that U is stationary and mixing with respect to the spatial shift and has time-stationary increments. Moreover, the only dependence on a is through time scaling by a factor of a(1 − a). To establish the existence of U, we find a local limit for staircase-shaped Young tableaux. These Young tableaux are related to sorting networks through a bijection of Edelman and Greene.