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 $μ_t$ 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.