The hyperbolic geometry of random transpositions

Turn the set of permutations of $n$ objects into a graph $G_n$ by connecting two permutations that differ by one transposition, and let $\sigma_t$ 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 $\delta$-thin for some $\delta>0$, 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 $a=1-\log2$.