Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs

2009/11/30 by M. Shcherbina, B. Tirozzi · 1 citation
Physics and Astronomy · Mathematics · #math-ph #math.MP #msc:15A52 #msc:15A57

paper · pdf · doi:10.1063/1.3299297

22 pages

arxiv created 2009/11/30 · arxiv updated 2015/05/14

Abstract

We consider the adjacency matrix A of a large random graph and study fluctuations of the function f_n(z,u)=(1)/(n)∑_k=1ⁿexp\-uG_kk(z)\ with G(z)=(z-iA)^-1. We prove that the moments of fluctuations normalized by n^-1/2 in the limit n→∞ satisfy the Wick relations for the Gaussian random variables. This allows us to prove central limit theorem for \hboxTrG(z) and then extend the result on the linear eigenvalue statistics \hboxTrϕ(A) of any function ϕ:ℝ→ℝ which increases, together with its first two derivatives, at infinity not faster than an exponential.

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• Eigenvalue distribution of bipartite large weighted random graphs. Resolvent approach

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