EXTREMAL EIGENVALUE CORRELATIONS IN THE GUE MINOR PROCESS AND A LAW OF FRACTIONAL LOGARITHM

• 成果类型: Article
• 署名作者: Paquette, Elliot; Zeitouni, Ofer
• 署名单位: University System of Ohio; Ohio State University; New York University; Weizmann Institute of Science
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1161
• 发表日期: 2017
• 页码: 4112-4166
• 关键词: bulk universality conjecture; airy(2) processes; wigner; asymptotics

摘要

Let lambda((N)) be the largest eigenvalue of the N x N GUE matrix which is the Nth element of the GUE minor process, rescaled to converge to the standard Tracy-Widom distribution. We consider the sequence {lambda((N))}(N >= 1) and prove a law of fractional logarithm for the lim sup: lim sup(N ->infinity) lambda((N)) / (logN)(2/3) = (1/4)(2/3) almost surely. For the lim inf, we prove the weaker result that there are constants c(1), c(2) > 0 so that -c(1) <= lim inf(N ->infinity) lambda((N)) / (logN)(1/3) <= -c(2) almost surely. We conjecture that in fact, c(1) = c(2) = 4(1/3).