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Eigenvalues of random lifts and polynomials of random permutation matrices. (English) Zbl 1446.60004
The authors prove, along with a number of other results, the following: suppose that $$(\sigma_1,\dots,\sigma_d)$$ is a finite sequence of independent random permutations, chosen uniformly among all permutations or among all matchings on $$N$$ points. Then it is shown that, in probability, as $$N\to \infty$$, these permutations viewed as operators on the $$(N-1)$$-dimensional vector space $$\left\{(x_1,\dots,x_N)\in \mathbb{C}^N:\sum_{i=1}^Nx_i=0 \right\}$$ are asymptotically strongly free. Moreover, as a byproduct, the authors obtain that the non-trivial eigenvalues of random $$N$$-lifts of a fixed base graph achieve, up to a vanishing term, the so-called Alon-Boppana lower bound with high probability as $$N\to \infty$$. This settles a conjecture of J. Friedman [Duke Math. J. 118, No. 1, 19–35 (2003; Zbl 1035.05058)].
The main contribution of this paper can in fact be understood as a novel use of non-backtracking operators, since the authors reduce the results above to establishing analogous ones for non-backtracking operators. Then, the strategy of proof of these results on non-backtracking operators is similar to the one followed by the first author to give a new proof of Friedman’s theorem in [“A new proof of Friedman’s second eigenvalue theorem and its extension to random lifts”, Preprint, arXiv:1502.04482], but with a number of significant refinements which require a very delicate analysis.
Finally, extensions of the results mentioned above to tensor products of random permutation matrices are also discussed.

##### MSC:
 60B20 Random matrices (probabilistic aspects) 05C80 Random graphs (graph-theoretic aspects) 46L54 Free probability and free operator algebras
Zbl 1035.05058
Full Text:
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