Asymptotic freeness of large graphs with large degree

Let $A_1,\dots,A_L$ be adjacency matrices of independent random graphs $G_1,\dots,G_L$ on the vertex set $\{1,...,N\}$. Assume for each $\ell=1,\dots,L$ that the expected degree of a vertex of $G_\ell$ uniformly chosen at random goes to infinity, and that each graph is invariant in law by relabelling of its vertices. We state a quantitative estimate of decorrelation on the edges of the graphs that implies the asymptotic freeness of well-normalized versions of $A_1,\dots,A_L$, as well as their asymptotic freeness with arbitrary matrices. We prove that this estimate holds for the uniform simple $d_N$-regular graph with $|d_N-\frac N 2|$ going to infinity fast enough. The proof is based on asymptotic traffic independence and combinatorial manipulations.Non UBCUnreviewedAuthor affiliation: Université Bordeaux & CNRSFacult