On a flow of operators associated to virtual permutations

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 $\sigma_N$.