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Let $(\sigma _{1}, \ldots , \sigma _d)$ be a finite sequence of independent random permutations, chosen uniformly either among all permutations or among all matchings on $n$ points. We show that, in probability, as $n\to \infty $, these permutations viewed as operators on the $n-1$ dimensional vector space $\{(x_1,\ldots , x_n) \in \mathbb {C}^n, \sum x_i=0\}$, are asymptotically strongly free. Our proof relies on the development of a matrix version of the non-backtracking operator theory and a refined trace method.
@article{10_4007_annals_2019_190_3_3,
author = {Charles Bordenave and Beno{\^\i}t Collins},
title = {Eigenvalues of random lifts and polynomials of random permutation matrices},
journal = {Annals of mathematics},
pages = {811--875},
publisher = {mathdoc},
volume = {190},
number = {3},
year = {2019},
doi = {10.4007/annals.2019.190.3.3},
mrnumber = {4024563},
zbl = {07128141},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.190.3.3/}
}
TY - JOUR AU - Charles Bordenave AU - Benoît Collins TI - Eigenvalues of random lifts and polynomials of random permutation matrices JO - Annals of mathematics PY - 2019 SP - 811 EP - 875 VL - 190 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.190.3.3/ DO - 10.4007/annals.2019.190.3.3 LA - en ID - 10_4007_annals_2019_190_3_3 ER -
%0 Journal Article %A Charles Bordenave %A Benoît Collins %T Eigenvalues of random lifts and polynomials of random permutation matrices %J Annals of mathematics %D 2019 %P 811-875 %V 190 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.190.3.3/ %R 10.4007/annals.2019.190.3.3 %G en %F 10_4007_annals_2019_190_3_3
Charles Bordenave; Benoît Collins. Eigenvalues of random lifts and polynomials of random permutation matrices. Annals of mathematics, Tome 190 (2019) no. 3, pp. 811-875. doi: 10.4007/annals.2019.190.3.3
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