Séminaire de Probabilités XLVI pp 481-512

Part of the Lecture Notes in Mathematics book series (LNM, volume 2123)

# On a Flow of Operators Associated to Virtual Permutations

Chapter

## 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 $$(\sigma _{N})_{N\geq 1}$$, σN in the symmetric group of order N, such that the cycle structure of σN can be deduced from the structure of σN+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 $$(\sigma _{N})_{N\geq 1}$$ is a random virtual permutation following a distribution in the class described above, the operator Tα 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 αN 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 σN.

### Keywords

Central measure Flow of operators Infinitesimal generator Random operator Virtual permutation

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