AI 中文总结
该研究针对β≥1的β-戴森布朗运动，通过确定性局部预解式排除原理，证明了粒子在指定时间内保持在半圆自由卷积流支撑集的ε邻域内的高概率结论，适用于多分支支撑集且无需初始谱边界正则性假设。
AI 中文摘要
我们考虑β≥1的β-戴森布朗运动，其初始构型为具有一致有界支撑集的确定性构型。设μ_t为从初始经验测度出发的半圆自由卷积流，并令S_t = supp(μ_t)。对任意固定的T, ε>0，存在至少1 - C exp(-(log n)^2)的概率，使得所有粒子在0≤t≤T的整个区间内都保持在S_t的ε邻域内。该结果从时间零点起成立，无需初始谱边界处的正则性假设，且适用于具有宏观内部间隙的多分支支撑集。一个关键要素是确定性局部预解式排除原理：在与参考支撑集分离的实点上方的复圆盘上，对Stieltjes变换进行o((nη)⁻¹)级别的比较，可将特征值排除在对应的实区间之外。这提供了一种与模型无关的机制，用于将局部预解式估计转化为谱限制。
英文摘要
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.
Comments17 pages
