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. [ABSTRACT FROM AUTHOR]