On a flow of operators associated to virtual permutations

TL;DR

This paper constructs a flow of random operators linked to virtual permutations, showing their eigenvalues relate to the asymptotic behavior of permutation matrices under invariant measures like Ewens measure.

Contribution

It introduces a novel flow of operators associated with invariant measures on virtual permutations, connecting operator limits to permutation eigenangles.

Findings

Flow of operators exists for a broad class of invariant measures.

Eigenvalues of the operator's generator match limits of permutation eigenangles.

Operator limits relate to the asymptotic spectral properties of permutation matrices.

Abstract

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 $\sigma_N$.