Random matrices with row constraints and eigenvalue distributions of graph Laplacians

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

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