The hyperbolic geometry of random transpositions

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, wh...

Compositions of random transpositions

Let Y = (y1, y2, . . . ), y1 ≥ y2 ≥ · · · , be the list of sizes of the cycles in the composition of c n transpositions on the set {1, 2, . . . , n}. We prove that if c > 1/2 is constant and n → ∞, the distribution of f(c)Y/n converges to PD(1), the Poisson-Dirichlet distribution with paramenter 1, where the function f is known explicitly. A new proof is presented of the theorem by Diaconis, Ma...

Hyperbolic Geometry

Hyperbolic Geometry: Isometry Groups of Hyperbolic Space

The goal of this paper is twofold. First, it consists of an introduction to the basic features of hyperbolic geometry, and the geometry of an important class of functions of the hyperbolic plane, isometries. Second, it identifies a group structure in the set of isometries, specifically those that preserve orientation, and deals with the topological properties of their discrete subgroups. In the...

On the hyperbolic unitary geometry

Hans Cuypers (Preprint) describes a characterisation of the geometry on singular points and hyperbolic lines of a finite unitary space—the hyperbolic unitary geometry—using information about the planes. In the present article we describe an alternative local characterisation based on Cuypers’ work and on a local recognition of the graph of hyperbolic lines with perpendicularity as adjacency. Th...