Dense kernel-based random graphs on a torus with Pareto weights have a limiting spectral measure, given by a free multiplicative convolution with the semicircle law for the product kernel, and the limit is absolutely continuous.
A simple invariance theorem
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We present a simple extension of Lindeberg's argument for the Central Limit Theorem to get a general invariance result. We apply the technique to prove results from random matrix theory, spin glasses, and maxima of random fields.
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The spectrum of dense kernel-based random graphs
Dense kernel-based random graphs on a torus with Pareto weights have a limiting spectral measure, given by a free multiplicative convolution with the semicircle law for the product kernel, and the limit is absolutely continuous.