Support of Dyson Brownian Motion

Abstract

We consider beta-Dyson Brownian motion, with $\beta >=1$, started from a deterministic configuration with uniformly bounded support. Let ${\mu }_t$ be the semicircular free-convolution flow issued from the initial empirical measure, and set $S_t=supp({\mu }_t)$. For every fixed $T,\epsilon >0$, with probability at least $1-C\exp (-(logn{)}^2)$, 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 $o((n\eta {)}^{(}-1))$ 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.

17 pages