2014/10/29 by Camille Male, Sandrine Péché, Male, Camille +1 · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Operator Algebras (math.OA) #Probability (math.PR) #Random Matrices and Applications #math.CO #math.OA #math.PR
paper · pdf · doi:10.48550/arxiv.1410.8126
21 pages, 7 figures
arxiv created 2014/10/29 · openalex publication_date 2014/10/29 · arxiv updated 2014/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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: \mathfrak CN(T)= \mathbb E[ ∏(i,j) ∈ T (\mathbf 1_(\i,j\ ∈ GN ) - \mathbb P(\i,j\ ∈ GN) )], T ⊂ [N]2 \textrmfinite furnishes informations on the asymptotic spectral properties of the adjacency matrix AN of GN. Denote by dN = N× \mathbb P(\i,j\ ∈ GN) and assume dN, N-dN\undersetN → ∞\longrightarrow ∞. If \mathfrak CN(T) =(\fracdNN)|T| × O(dN^-\frac |T|2) 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 |\frac N 2 - dN- η√(dN)| \undersetN → ∞\longrightarrow ∞ for some η>0. Our method applies also for simple graphs whose edges are labelled by i.i.d. random variables.