Random matrices with row constraints and eigenvalue distributions of graph Laplacians

Abstract

Symmetric matrices with zero row sums occur in many theoretical settings and in real-life applications. When the offdiagonal elements of such matrices are i.i.d. random variables and the matrices are large, the eigenvalue distributions converge to a peculiar universal curve p_zrs(λ) that looks like a cross between the Wigner semicircle and a Gaussian distribution. An analytic theory for this curve, originally due to Fyodorov, can be developed using supersymmetry-based techniques.

We extend these derivations to the case of sparse matrices, including the important case of graph Laplacians for large random graphs with N vertices of mean degree c. In the regime 1<<c<<N , the eigenvalue distribution of the ordinary graph Laplacian (diffusion with a fixed transition rate per edge) tends to a shifted and scaled version of p_zrs (λ), centered at c with width ∼sqrt(c). At smaller c, this curve receives corrections in powers of 1/sqrt(c) accurately captured by our theory. For the normalized graph Laplacian (diffusion with a fixed transition rate per vertex), the large c limit is a shifted and scaled Wigner semicircle, again with corrections captured by our analysis.