PreprintCC BY 4.0arXiv
On the heat flow conjecture for random matrices
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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BibTeX
@misc{assiotis2026heat,
title = {{On the heat flow conjecture for random matrices}},
author = {Theodoros Assiotis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.05185},
eprint = {2608.05185},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.05185},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the heat flow conjecture for random matrices
AU - Theodoros Assiotis
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2608.05185
UR - https://arxiv.org/abs/2608.05185
AB - 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.
ER -