Properties from random matrix theory allow us to uncover naturally embedded signals from different data sets. While there are many parameters that can be changed, including the probability distribution of the entries,...Properties from random matrix theory allow us to uncover naturally embedded signals from different data sets. While there are many parameters that can be changed, including the probability distribution of the entries, the introduction of noise, and the size of the matrix, the resulting eigenvalue and eigenvector distributions remain relatively unchanged. However, when there are certain anomalous eigenvalues and their corresponding eigenvectors that do not follow the predicted distributions, it could indicate that there’s an underlying non-random signal inside the data. As data and matrices become more important in the sciences and computing, so too will the importance of processing them with the principles of random matrix theory.展开更多
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...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.展开更多
文摘Properties from random matrix theory allow us to uncover naturally embedded signals from different data sets. While there are many parameters that can be changed, including the probability distribution of the entries, the introduction of noise, and the size of the matrix, the resulting eigenvalue and eigenvector distributions remain relatively unchanged. However, when there are certain anomalous eigenvalues and their corresponding eigenvectors that do not follow the predicted distributions, it could indicate that there’s an underlying non-random signal inside the data. As data and matrices become more important in the sciences and computing, so too will the importance of processing them with the principles of random matrix theory.
基金Supported by NSFC(Grant Nos.12171038,11871008)985 Projects。
文摘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.
