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Kerov, Olshanski and Vershik introduced the so-called virtual permutations,defined as families of permutations $(\sigma_N)_{N \geq 1}$, $\sigma_N$ in thesymmetric group of order $N$, such that the cycle structure of $\sigma_N$ canbe deduced from the structure of $\sigma_{N+1}$ simply by removing the element$N+1$. The virtual permutations, and in particular the probability measures onthe corresponding space which are invariant by conjugation, have been studiedin a details by Tsilevich. In the present article, we prove that for a largeclass of such invariant measures (containing in particular the Ewens measure ofany parameter $\theta \geq 0$), it is possible to associate a flow$(T^{\alpha})_{\alpha \in \mathbb{R}}$ of random operators on a suitablefunctional space. Moreover, if $(\sigma_N)_{N \geq 1}$ is a random virtualpermutation 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 madeprecise, of the permutation $\sigma_N^{\alpha_N}$, where $N$ goes to infinityand $\alpha_N$ is equivalent to $\alpha N$. In relation with thisinterpretation, we prove that the eigenvalues of the infinitesimal generator of$(T^{\alpha})_{\alpha \in \mathbb{R}}$ are equal to the limit of the rescaledeigenangles of the permutation matrix associated to $\sigma_N$.

On a flow of operators associated to virtual permutations