For each N≥1, let GN be a simple random graph on the set of vertices
[N]={1,2,...,N}, which is invariant by relabeling of the vertices. The
asymptotic behavior as N goes to infinity of correlation functions:
CN(T)=E[(i,j)∈T∏(1({i,j}∈GN)−P({i,j}∈GN))],T⊂[N]2finite furnishes informations on the asymptotic
spectral properties of the adjacency matrix AN of GN. Denote by dN=N×P({i,j}∈GN) and assume dN,N−dNN→∞⟶∞. If CN(T)=(NdN)∣T∣×O(dN−2∣T∣) for any
T, the standardized empirical eigenvalue distribution of AN 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 dN-regular graphs GN,dN, under the additional
assumption that ∣2N−dN−ηdN∣N→∞⟶∞ for some η>0. Our method applies also for
simple graphs whose edges are labelled by i.i.d. random variables.Comment: 21 pages, 7 figure