Limit theorems for linear eigenvalue statistics of overlapping matrices

The paper proves several limit theorems for linear eigenvalue statistics of overlapping Wigner and sample covariance matrices. It is shown that the covariance of the limiting multivariate Gaussian distribution is diagonalized by choosing the Chebyshev polynomials of the first kind as the basis for the test function space. The covariance of linear statistics for the Chebyshev polynomials of sufficiently high degree depends only on the first two moments of the matrix entries. Proofs are based on a graph-theoretic interpretation of the Chebyshev linear statistics as sums over non-backtracking cyclic pathsComment: 44 pages, 4 figures, some typos are corrected and proofs clarified. Accepted to the Electronic Journal of Probabilit

Recent results of quantum ergodicity on graphs and further investigation

We outline some recent proofs of quantum ergodicity on large graphs and give new applications in the context of irregular graphs. We also discuss some remaining questions.Comment: To appear in "Annales de la facult\'e des Sciences de Toulouse

Finite reflection groups and graph norms

Given a graph H on vertex set {1, 2, · · · , n} and a function f : [0, 1]2 → R, define kfkH := Z Y ij∈E(H) f(xi , xj )dµ|V (H)| 1/|E(H)| , where µ is the Lebesgue measure on [0, 1]. We say that H is norming if k·kH is a semi-norm. A similar notion k·kr(H) is defined by kfkr(H) := k|f|kH and H is said to be weakly norming if k·kr(H) is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes are all weakly norming. We demonstrate that any graph whose edges percolate in an appropriate way under the action of a certain natural family of automorphisms is weakly norming. This result includes all previously known examples of weakly norming graphs, but also allows us to identify a much broader class arising from finite reflection groups. We include several applications of our results. In particular, we define and compare a number of generalisations of Gowers’ octahedral norms and we prove some new instances of Sidorenko’s conjectur