## Strong convergence for random permutations

#### Benoit CollinsKyoto University

We show that given $s_1,…,s_k$ $k$ iid random permutations of the permutation $S_n$, they are asymptotically strongly free with high probability in the large $n$ limit when seen as endomorphisms of the orthogonal of the Perron Frobenius vector in $C^n$. Time allowing, we will explain the interest of this result from the point of view of random graph theory, and give an outline of the proof, which relies on developing a non-commutative version of non-backtracking operators. This is joint work with Charles Bordenave.

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