Uniform regular weighted graphs with large degree: Wigner's law, asymptotic freeness and graphons limit

For each $N\ge 1$, let $G_N$ be a simple random graph on the set of vertices $[N]=\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},N\}$, which is invariant by relabeling of the vertices. The asymptotic behavior as $N$ goes to infinity of correlation functions: $${\mathfrak{C}}_N(T)=\mathbb{E}\left[\prod _{(i,j)\in T}\right({\mathbf{1}}_{(\{i,j\}\in G_N)}-\mathbb{P}(\{i,j\}\in G_N)\left)\right],T\subset [N{]}^2\text{finite}$$ furnishes informations on the asymptotic spectral properties of the adjacency matrix $A_N$ of $G_N$. Denote by $d_N = N\times \mathbb P({i,j} \in G_N) $andassume$d_N, N-d_N\underset{N \rightarrow \infty}{\longrightarrow} \infty$\mathrm{.}If$\mathfrak C_N(T) =\big(\frac{d_N}N\big)^{|T|} \times O\big(d_N^{-\frac {|T|}2}\big)$forany$T$,thestandardizedempiricaleigenvaluedistributionof$A_N$ converges in expectation to the semicircular law and the matrix satisfies asymptotic freeness properties in the sense of free probability theory. We provide such estimates for uniform $d_N$-regular graphs $G_{N,d_N}$, under the additional assumption that $|\frac N 2 - d_N- \eta \sqrt{d_N}| \underset{N \rightarrow \infty}{\longrightarrow} \infty$forsome$\eta>0$. Our method applies also for simple graphs whose edges are labelled by i.i.d. random variables.