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- W1546905482 abstract "Kerov, Olshanski and Vershik introduced the so-called virtual permutations, defined as families of permutations $(sigma_N)_{N geq 1}$, $sigma_N$ in the symmetric group of order $N$, such that the cycle structure of $sigma_N$ can be deduced from the structure of $sigma_{N+1}$ 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 $theta geq 0$), it is possible to associate a flow $(T^{alpha})_{alpha in mathbb{R}}$ of random operators on a suitable functional space. Moreover, if $(sigma_N)_{N geq 1}$ is a random virtual permutation following a distribution in the class described above, the operator $T^{alpha}$ 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 $alpha_N$ is equivalent to $alpha N$. 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 $sigma_N$." @default.
- W1546905482 created "2016-06-24" @default.
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- W1546905482 date "2010-08-29" @default.
- W1546905482 modified "2023-10-01" @default.
- W1546905482 title "On a flow of operators associated to virtual permutations" @default.
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