The Annals of Probability

The hyperbolic geometry of random transpositions

Nathanaël Berestycki

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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 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 Gn. 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.

Article information

Ann. Probab., Volume 34, Number 2 (2006), 429-467.

First available in Project Euclid: 9 May 2006

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Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60C05: Combinatorial probability

Random walks Gromov hyperbolic spaces phase transition random transpositions random graphs Cayley graphs


Berestycki, Nathanaël. The hyperbolic geometry of random transpositions. Ann. Probab. 34 (2006), no. 2, 429--467. doi:10.1214/009117906000000043.

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