The hyperbolic geometry of random transpositions

Turn the set of permutations of n objects into a graph Gn by connecting two permutations that differ by one transposition, and let σt be the continuous time simple random walk on this graph. In a previous paper, Berestycki and Durrett (2004) showed that the limiting behavior of the distance from the identity at time cn/2 has a phase transition at c = 1. When c < 1, it is asymptotically cn/2, while for c > 1 it is u(c)n with u(c) < c/2. Here we investigate some consequences of this result for the geometry of Gn. Our first result is that when we consider the sphere of radius an centered at the origin, and pick two points independently according to the hitting distribution, then Gromov hyperbolicity breaks down at critical radius a = 1/4. When a < 1/4 the space is hyperbolic but also displays behavior that is much different from manifolds of negative curvature it is shown that there are many geodesics that may travel much different paths to get to a point. 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 qualitative behavior of the Gromov hyperbolicity persists if we pick points independently according to the uniform measure on the sphere of radius an. However, in this case, the critical radius is a = 1− log 2.