RIGIDITY OF EIGENVALUES FOR β ENSEMBLE IN MULTI-CUT REGIME

• 成果类型: Article
• 署名作者: Li, Yiting
• 署名单位: Korea Advanced Institute of Science & Technology (KAIST)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP1943
• 发表日期: 2023
• 页码: 5111-5144
• 关键词: CENTRAL-LIMIT-THEOREM; bulk universality; matrix models; general beta; edge universality; semicircle law; statistics; fluctuations; delocalization; CONVERGENCE

摘要

For a beta ensemble on Sigma((N)) = {(x(1),... , x(N)) is an element of R-N |x(1) <= center dot center dot center dot <= x(N)} with real analytic potential and general beta > 0, under the assumption that its equilibrium measure is supported on q intervals where q > 1, we prove the following rigidity property for its particles. 1. In the bulk of the spectrum, with overwhelming probability, the distance between a particle and its classical position is of order O (N-1+epsilon). 2. If k is close to 1 or close to N, that is, near the extreme edges of the spectrum, then with overwhelming probability, the distance between the kth largest particle and its classical position is of order O(N-2/3+epsilon min(k, N + 1 - k)(-1/3)). Here epsilon > 0 is an arbitrarily small constant. Our main idea is to decompose the multi-cut beta ensemble as a product of probability measures on spaces with lower dimensions and show that each of these measures is very close to a beta ensemble in one-cut regime for which the rigidity of particles is known.