On a flow of operators associated to virtual permutations

2010/08/29 by Joseph Najnudel, Najnudel, Joseph, Ashkan Nikeghbali +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1008.4972

arxiv created 2010/08/29 · arxiv updated 2010/08/31

Abstract

Kerov, Olshanski and Vershik introduced the so-called virtual permutations, defined as families of permutations (σ_N)_{N ≥ 1}, σ_N in the symmetric group of order N, such that the cycle structure of σ_N can be deduced from the structure of σ_N+1 simply by removing the element N+1. The virtual permutations, and in particular the probability measures on the corresponding space which are invariant by conjugation, have been studied in a details by Tsilevich. In the present article, we prove that for a large class of such invariant measures (containing in particular the Ewens measure of any parameter θ≥ 0), it is possible to associate a flow (T^α)_{α∈ ℝ} of random operators on a suitable functional space. Moreover, if (σ_N)_{N ≥ 1} is a random virtual permutation following a distribution in the class described above, the operator T^α can be interpreted as the limit, in a sense which has to be made precise, of the permutation σ_N^α_N, where N goes to infinity and α_N is equivalent to αN. In relation with this interpretation, we prove that the eigenvalues of the infinitesimal generator of (T^α)_{α∈ ℝ} are equal to the limit of the rescaled eigenangles of the permutation matrix associated to σ_N.

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