Author:
Campbell Andrew,O’Rourke Sean
Abstract
AbstractWe consider the spectrum of random Laplacian matrices of the form $$L_n=A_n-D_n$$
L
n
=
A
n
-
D
n
where $$A_n$$
A
n
is a real symmetric random matrix and $$D_n$$
D
n
is a diagonal matrix whose entries are equal to the corresponding row sums of $$A_n$$
A
n
. If $$A_n$$
A
n
is a Wigner matrix with entries in the domain of attraction of a Gaussian distribution, the empirical spectral measure of $$L_n$$
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 $$A_n$$
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 $$L_n$$
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 $$L_n$$
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.
Funder
Division of Mathematical Sciences
Publisher
Springer Science and Business Media LLC
Subject
Statistics, Probability and Uncertainty,General Mathematics,Statistics and Probability
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