# On a flow of operators associated to virtual permutations - Zurich Open Repository and Archive

Najnudel, Joseph; Nikeghbali, Ashkan (2014). On a flow of operators associated to virtual permutations. In: Donati-Martin, Catherine; Lejay, Antoine; Rouault, Alain. Séminaire de Probabilités XLVI. Cham, Switzerland: Springer, 481-512.

## 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$(\sigma _{N})_{N\geq 1}$, 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 $(\sigma _{N})_{N\geq 1}1$ 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 .

## 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$(\sigma _{N})_{N\geq 1}$, 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 $(\sigma _{N})_{N\geq 1}1$ 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 .

## Altmetrics

Item Type: Book Section, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 30 October 2014 14 Jan 2016 11:10 05 Apr 2016 19:16 Springer Lecture Notes in Mathematics 2123 0075-8434 978-3-319-11969-4 (P), 978-3-319-11969-4 (E) https://doi.org/10.1007/978-3-319-11970-0_21 http://www.recherche-portal.ch/ZAD:default_scope:ebi01_prod010391461 (Library Catalogue)