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arXiv · 2308.16171

Spectrum of Laplacian matrices associated with large random elliptic matrices

Sean O'Rourke·Zhi Yin·Ping Zhong
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Abstract

A Laplacian matrix is a square matrix whose row sums are zero. We study the limiting eigenvalue distribution of a Laplacian matrix formed by taking a random elliptic matrix and subtracting the diagonal matrix containing its row sums. Under some mild assumptions, we show that the empirical spectral distribution of the Laplacian matrix converges to a deterministic probability distribution as the size of the matrix tends to infinity. The limiting measure can be interpreted as the Brown measure of the sum of an elliptic operator and a freely independent normal operator with a Gaussian distribution.

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Sean O'Rourke, Zhi Yin, Ping Zhong. 2023-12-15. Spectrum of Laplacian matrices associated with large random elliptic matrices. https://arxiv.org/abs/2308.16171

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