Spectrum of L\'evy-Khintchine Random Laplacian Matrices

We consider the spectrum of random Laplacian matrices of the form $L_n=A_n-D_n$ where $A_n$ is a real symmetric random matrix and $D_n$ is a diagonal matrix whose entries are equal to the corresponding row sums of $A_n$. If $A_n$ is a Wigner matrix with entries in the domain of attraction of a Gaussian distribution the empirical spectral measure of $L_n$ is known to converge to the free convolution of a semicircle distribution and a standard real Gaussian distribution. We consider real symmetric random matrices $A_n$ with independent entries (up to symmetry) whose row sums converge to a purely non-Gaussian infinitely divisible distribution, which fall into the class of L\'evy-Khintchine random matrices first introduced by Jung [Trans Am Math Soc, \textbf{370}, (2018)]. Our main result shows that the empirical spectral measure of $L_n$ converges almost surely to a deterministic limit. A key step in the proof is to use the purely non-Gaussian nature of the row sums to build a random operator to which $L_n$ converges in an appropriate sense. This operator leads to a recursive distributional equation uniquely describing the Stieltjes transform of the limiting empirical spectral measure.