Abstract
We prove a conjecture raised by the work of Diaconis and Shahshahani (Z Wahrscheinlichkeitstheorie Verwandte Geb 57(2):159–179, 1981) about the mixing time of random walks on the permutation group induced by a given conjugacy class. To do this we exploit a connection with coalescence and fragmentation processes and control the Kantorovich distance by using a variant of a coupling due to Oded Schramm as well as contractivity of the distance. Recasting our proof in the language of Ricci curvature, our proof establishes the occurrence of a phase transition, which takes the following form in the case of random transpositions: at time cn / 2, the curvature is asymptotically zero for cle 1 and is strictly positive for c>1.
Highlights
1.1 Main resultsLet Sn denote the multiplicative group of permutations of {1, . . . , n}
For our results it will turn out to be convenient to view the symmetric group as a metric space equipped with the metric d which is the word metric induced by the set T of transpositions
The two appendices contain respectively a proof of the lower bound on the mixing time; and an adaptation of Schramm’s argument [24] for the Poisson–Dirichlet structure of cycles inside the giant component, which is needed in the proof
Summary
Let Sn denote the multiplicative group of permutations of {1, . . . , n}. Let ⊂ Sn be a fixed conjugacy class in Sn, i.e., = {gγ g−1 : g ∈ Sn} for some fixed permutation γ ∈ Sn. It has long been conjectured that for a general conjugacy class such that | | = o(n) (where here and in the rest of the paper, | | denotes the number of non fixed points of any permutation γ ∈ ), a similar result should hold at a time (1/| |)n log n. This has been verified for k-cycles with a fixed k ≥ 2 by Berestycki et al [6]. The conjecture is already established for c ≤ 1 and so is only interesting for c > 1
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