Limit Theorems and Large Deviations forβ-Jacobi Ensembles at Scaling Temperatures

查看全文 作　　者：Yu Tao [1]MA 机构地区：[1]School of Mathematical Sciences&Laboratory of Mathematics and Complex Systems of Ministry of Education,Beijing Normal University,Beijing 100875,P.R.China 出　　处：《Acta Mathematica Sinica,English Series》索引2023年第39卷第10期,共21页 基　　金：Supported by NSFC(Grant Nos.12171038,11871008);985 Projects。 摘　　要：Letλ=(λ_(1),...,λ_(n))beβ-Jacobi ensembles with parameters p_(1),p_(2),n andβwhileβvarying with n.Setγ=lim_(n→∞)n/p_(1)andσ=lim_(n→∞)p_(1)/p_(2).In this paper,supposing lim_(n→∞)log_(n)/β_(n)=0,we prove that the empirical measures of different scaledλconverge weakly to a Wachter distribution,a Marchenko–Pastur law and a semicircle law corresponding toσγ>0,σ=0 orγ=0,respectively.We also offer a full large deviation principle with speedβn^(2)and a good rate function to precise the speed of these convergences.As an application,the strong law of large numbers for the extremal eigenvalues ofβ-Jacobi ensembles is obtained. 关 键 词：β-Jacobi ensemble large deviation principle semi-circle law Marchenko-Pastur law Wachter law