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On the mixing time of random conjugacy walks

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Let G be a finite graph and consider a random walk on this graph. How long does it take for this walk to be well mixed, i.e., to be close to its equilibrium distribution? A striking phenomenon, discovered in the early 80’s by Aldous and Diaconis independently, is that convergence to equilibrium often occurs abruptly: this is known as the cutoff phenomenon. In this talk we shall consider the classical example of random transpositions over the symmetric group. In this case, Diaconis and Shahshahani used representation theory to prove that such a cutoff occurs at time (1/2) n log n. We present a new, probabilistic proof of this result, which extends readily to other walks where the step distribution is uniform over a given conjugacy class. This proves a conjecture of Roichman (1996) that the mixing time of this process is (1/C) n log n, where C is the size of the conjugacy class. This is joint work with Oded Schramm and Ofer Zeitouni

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