The hyperbolic geometry of random transpositions

Abstract

Turn the set of permutations of n objects into a graph G_n by connecting two permutations that differ by one transposition, and let σ_t be the simple random walk on this graph. In a previous paper, Berestycki and Durrett [In Discrete Random Walks (2005) 17–26] showed that the limiting behavior of the distance from the identity at time cn/2 has a phase transition at c=1. Here we investigate some consequences of this result for the geometry of G_n. Our first result can be interpreted as a breakdown for the Gromov hyperbolicity of the graph as seen by the random walk, which occurs at a critical radius equal to n/4. Let T be a triangle formed by the origin and two points sampled independently from the hitting distribution on the sphere of radius an for a constant 0<a<1. Then when a<1/4, if the geodesics are suitably chosen, with high probability T is δ-thin for some δ>0, whereas it is always O(n)-thick when a>1/4. We also show that the hitting distribution of the sphere of radius an is asymptotically singular with respect to the uniform distribution. Finally, we prove that the critical behavior of this Gromov-like hyperbolicity constant persists if the two endpoints are sampled from the uniform measure on the sphere of radius an. However, in this case, the critical radius is a=1−log2.