Large deviations for the and processes

• 成果类型: Article
• 署名作者: Holcomb, Diane; Valko, Benedek
• 署名单位: University of Arizona; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0594-4
• 发表日期: 2015
• 页码: 339-378
• 关键词: gap probability; asymptotics; constant

摘要

The process is the bulk point process limit of the Gaussian -ensemble. For and 4 this process gives the limit of the GOE, GUE and GSE random matrix models. The process is obtained similarly as the bulk scaling limit of the spectrum of certain discrete one-dimensional random Schrodinger operators. Both point processes have asymptotically constant average density, in our chosen normalization one expects close to points in a large interval of length . We prove large deviation principles for the average densities of the processes, identifying the rate function in both cases. Our approach is based on the representation of the counting functions using coupled systems of stochastic differential equations. Our techniques work for the full range of parameter values. The results are novel even in the classical and 4 cases for the process. They are consistent with the existing rigorous results on large gap probabilities and confirm the physical predictions made using log-gas arguments.