Random matrices with row constraints and eigenvalue distributions of graph Laplacians

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.