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- W3110606956 abstract "In (Comptes Rend Acad Sci Paris 316:773–778, 1993), Kerov, Olshanski and Vershik introduce the so-called virtual permutations, defined as families of permutations ((sigma _{N})_{Ngeq 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 more detailed way by Tsilevich in (J Math Sci 87(6):4072–4081, 1997) and (Theory Probab Appl 44(1):60–74, 1999). 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^{alpha })_{alpha in mathbb{R}}) of random operators on a suitable function space. Moreover, if ((sigma _{N})_{Ngeq 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 (sigma _{N}^{alpha _{N}}), where N goes to infinity and α N is equivalent to α N. In relation with this interpretation, we prove that the eigenvalues of the infinitesimal generator of ((T^{alpha })_{alpha in mathbb{R}}) are equal to the limit of the rescaled eigenangles of the permutation matrix associated to σ N ." @default.
- W3110606956 created "2020-12-07" @default.
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- W3110606956 date "2014-01-01" @default.
- W3110606956 modified "2023-10-01" @default.
- W3110606956 title "On a Flow of Operators Associated to Virtual Permutations" @default.
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- W3110606956 doi "https://doi.org/10.1007/978-3-319-11970-0_21" @default.
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