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Eigenvalues of random lifts and polynomials of random permutation matrices
Annals of Mathematics ( IF 4.9 ) Pub Date : 2019-01-01 , DOI: 10.4007/annals.2019.190.3.3
Charles Bordenave 1 , Benoît Collins 2
Affiliation  

Consider 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 goes to infinity, these permutations viewed as operators on the (n-1) dimensional vector space orthogonal to the vector with all coordinates equal to 1, 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. As a byproduct, we show that the non-trivial eigenvalues of random n-lifts of a fixed based graphs approximately achieve the Alon-Boppana bound with high probability in the large n limit. This result generalizes Friedman's Theorem stating that with high probability, the Schreier graph generated by a finite number of independent random permutations is close to Ramanujan. Finally, we extend our results to tensor products of random permutation matrices. This extension is especially relevant in the context of quantum expanders.

中文翻译:

随机提升的特征值和随机置换矩阵的多项式

考虑一个有限的独立随机排列序列,在所有排列中或在 n 个点的所有匹配中统一选择。我们证明,在概率中,随着 n 趋于无穷大,这些被视为 (n-1) 维向量空间上的运算符的排列与所有坐标等于 1 的向量正交,是渐近强自由的。我们的证明依赖于非回溯算子理论的矩阵版本和精细跟踪方法的发展。作为副产品,我们展示了基于固定图的随机 n 提升的非平凡特征值在大 n 限制中以高概率近似达到 Alon-Boppana 界限。这个结果概括了弗里德曼定理,指出很有可能,由有限数量的独立随机排列生成的 Schreier 图接近于 Ramanujan。最后,我们将结果扩展到随机置换矩阵的张量积。这种扩展在量子扩展器的上下文中尤其重要。
更新日期:2019-01-01
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