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arXiv · 2608.05185

On the heat flow conjecture for random matrices

Theodoros Assiotis
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Abstract

We prove two families of distinguished cases of the heat flow conjecture for random matrices of Hall-Ho: elliptic Gaussian sources with planar targets, and deterministic Hermitian sources with targets on the boundary of the covariance disk. In particular, the empirical zero measure of the heat-evolved characteristic polynomial of a complex Ginibre matrix converges almost surely to the semicircle law. The strategy of the proof is to transfer estimates for the corresponding well-understood Gaussian matrix ensembles, through exact heat identities, into control of the zeros of the heat-evolved characteristic polynomials.

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Theodoros Assiotis. 2026-09-17. On the heat flow conjecture for random matrices. https://arxiv.org/abs/2608.05185

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