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

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Abstract

For each $N, let G N$ 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 certain correlation functions furnishes informations on the asymptotic spectral properties of the adjacency matrix $A N of G N$. Denote by $d N = N × P({i, j} ∈ G N)$ and assume $d N , N − d → N →∞$ . If the correlation functions are small enough, the standardized empirical eigenvalue distribution of 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 $| N 2 − d N − η √ d N | → N →∞$ for some $η > 0$. Our method applies also for simple graphs whose edges are labelled by i.i.d. random variables.
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hal-01458155 , version 1 (06-02-2017)

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Camille Male, Sandrine Péché. Uniform regular weighted graphs with large degree: Wigner's law, asymptotic freeness and graphons limit. 2014. ⟨hal-01458155⟩
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