Support of Dyson Brownian Motion
Abstract
We consider beta-Dyson Brownian motion, with β>= 1, started from a deterministic configuration with uniformly bounded support. Let μt be the semicircular free-convolution flow issued from the initial empirical measure, and set St = supp(μt). For every fixed T, ε> 0, with probability at least 1 − C exp(−(log n)2), every particle remains within an epsilon-neighborhood of St for all 0 <= t <= T. The result holds from time zero, requires no regularity assumption at the initial spectral edges, and applies to multi-cut supports with macroscopic interior gaps. A key ingredient is a deterministic local resolvent exclusion principle: an o((n η)(−1)) comparison of Stieltjes transforms on a complex disc above a real point separated from the reference support excludes eigenvalues from the corresponding real interval. This gives a model-independent mechanism for converting local resolvent estimates into spectral confinement.
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BibTeX
@misc{huang2026support,
title = {{Support of Dyson Brownian Motion}},
author = {Jiaoyang Huang and Shengjing Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01770},
eprint = {2609.01770},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01770},
}
RIS
TY - UNPB
TI - Support of Dyson Brownian Motion
AU - Jiaoyang Huang
AU - Shengjing Xu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01770
UR - https://arxiv.org/abs/2609.01770
AB - We consider beta-Dyson Brownian motion, with β>= 1, started from a deterministic configuration with uniformly bounded support. Let μ_t be the semicircular free-convolution flow issued from the initial empirical measure, and set S_t = supp(μ_t). For every fixed T, ε> 0, with probability at least 1 − C exp(−(log n)^2), every particle remains within an epsilon-neighborhood of S_t for all 0 <= t <= T. The result holds from time zero, requires no regularity assumption at the initial spectral edges, and applies to multi-cut supports with macroscopic interior gaps. A key ingredient is a deterministic local resolvent exclusion principle: an o((n η)^(−1)) comparison of Stieltjes transforms on a complex disc above a real point separated from the reference support excludes eigenvalues from the corresponding real interval. This gives a model-independent mechanism for converting local resolvent estimates into spectral confinement.
ER -