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The oriented swap process

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The oriented swap process is a random walk on the symmetric group of order N. Starting from the identity permutation, at each step an adjacent swap is chosen uniformly and applied to the current permutation, but only if it increases the number of inversions. Eventually the walk terminates when it reaches the permutation with maximal number of inversions. In recent work with Omer Angel and Alexander Holroyd, we analyzed the asymptotic behavior of the oriented swap process when N tends to infinity using the theory of totally asymmetric exclusion processes, deriving formulas for the limiting trajectories of individual numbers (“particles”) in the permutation and for the flow of particles en masse. An interesting connection to random matrix theory also makes an appearance. I will explain these results and show computer simulations.

This talk is part of the Probability series.

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