Random matrices with row constraints and eigenvalue distributions of graph Laplacians

TL;DR

This paper extends the analytic understanding of eigenvalue distributions of symmetric matrices with zero row sums, especially graph Laplacians, revealing their universal behavior and corrections in sparse regimes.

Contribution

It develops supersymmetry-based analytic derivations for eigenvalue distributions of sparse matrices and graph Laplacians, including corrections for small mean degrees.

Findings

Eigenvalue distribution of large graph Laplacians approaches a shifted, scaled version of a universal curve.

For normalized Laplacians, the distribution tends to a shifted Wigner semicircle at large degrees.

The theory accurately captures corrections at small mean degrees.

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 $p_{\mathrm{zrs}}(\lambda)$ 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 $1\ll c\ll N$, the eigenvalue distribution of the ordinary graph Laplacian (diffusion with a fixed transition rate per edge) tends to a shifted and scaled version of $p_{\mathrm{zrs}}(\lambda)$, centered at $c$ with width $\sim\sqrt{c}$. 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.