On a Flow of Operators Associated to Virtual Permutations

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 \((\sigma _{N})_{N\geq 1}\) 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 .