Spectrum of Lévy–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 Lévy–Khintchine random matrices first introduced by Jung [Trans Am Math Soc, 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.