Empirical spectral distributions of arbitrary-degree polynomials in Ginibre matrices converge to their Brown measures as matrix size tends to infinity.
On Gaussian Brunn-Minkowski inequalities
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abstract
In this paper, we are interested in Gaussian versions of the classical Brunn-Minkowski inequality. We prove in a streamlined way a semigroup version of the Ehrard inequality for $m$ Borel or convex sets based on a previous work by Borell. Our method also allows us to have semigroup proofs of the geometric Brascamp-Lieb inequality and of the reverse one which follow exactly the same lines.
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Brown measure convergence for the spectrum of polynomials in Ginibre matrices
Empirical spectral distributions of arbitrary-degree polynomials in Ginibre matrices converge to their Brown measures as matrix size tends to infinity.