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THE APPLICATION OF random matrices IN MATHEMATICAL PHYSICS

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Annals of Applied Mathematics 2017年第3期33卷 221-238页

作者： Boling Guo Fangfang Li Institute of Applied Physics and Computational Math. Beijing 100088PR China School of Math.and Statistics Zhoukou Normal UniversityZhoukou 466001HenanPR China

In this paper, we introduce the application of random matrices in mathe- matical physics including Riemann-Hilbert problem, nuclear physics, big data, image processing, compressed sensing and so on. We start with the Riemann- Hilbert problem and state the relation between the probability distribution of nontrivial zeros and the eigenvalues of the random matrices. Through the random matrices theory, we derive the distribution of Neutron width and prob- ability density between energy levels. In addition, the application of random matrices in quantum chromo dynamics and two dimensional Einstein gravity equations is also present in this paper.

关键词： random matrices Riemann Hypothesis Riemann-Hilbert prob-lem nuclear physics

On the Spectral Distribution of Gaussian random matrices

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Acta Mathematicae Applicatae Sinica 2006年第2期22卷 297-312页

作者： B.Delyon J.Yao IRMAR Universit de Rennes I Campus de Beaulieu F-35042 Rennes Cedex France

We consider the empirical spectral distribution （ESD） of a random matrix from the Gaussian Unitary Ensemble. Based on the Plancherel-Rotaeh approximation formula for Hermite polynomials, we prove that the expected empirical spectral distribution converges at the rate of O（n^-1） to the Wigner distribution function uniformly on every compact intervals [u,v] within the limiting support （-1, 1）. Furthermore, the variance of the ESD for such an interval is proved to be （πn）^-2 logn asymptotically which surprisingly enough, does not depend on the details （e.g. length or location） of the interval, This property allows us to determine completely the covariance function between the values of the ESD on two intervals.

关键词： Convergence rate random matrices empirical spectral distribution Wigner distribution

Phase Transitions for Products of Characteristic Polynomials under Dyson Brownian Motion

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Acta Mathematica Sinica,English Series 2021年第3期37卷 509-524页

作者： Peter J.FORRESTER Dang-Zheng LIU ARC Centre of Excellence for Mathematical and Statistical Frontiers School of Mathematics and StatisticsThe University of MelbourneVictoria 3010Australia CAS Key Laboratory of Wu Wen-Tsun Mathematics School of Mathematical SciencesUniversity of Science and Technology of ChinaHefei 230026P.R.China

We study the averaged products of characteristic polynomials for the Gaussian and Laguerre β-ensembles with external source, and prove Pearcey-type phase transitions for particular full rank perturbations of source. The phases are characterised by determining the explicit functional forms of the scaled limits of the averaged products of characteristic polynomials, which are given as certain multidimensional integrals, with dimension equal to the number of products.

关键词： random matrices Dyson Brownian motion β-ensembles phase transitions

Some Comments on Zonal Polynomials and Their Expected Values with respect to Elliptical.Distributions

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Acta Mathematicae Applicatae Sinica 2017年第3期33卷 567-574页

作者： Jose A.Diaz-Garcia Ramon Gutierrez-Jaimez Department of Statistics and Computation Universidad Autonoma Agraria Antonio Narro25350 BuenavistaSaltilloCoahuilaMexico Department of Statistics and O.R University of GranadaGranada 18071Spain

In this paper, we give alternative proofs of some results in [15] （Li R.,1997） about the expected value of zonal polynomials.

关键词： elliptical matrix distributions random matrices zonal polynomials generalised Wishart distribution

Logarithmic Sobolev Inequalities, Matrix Models and Free Entropy

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Acta Mathematica Sinica,English Series 2003年第3期19卷 497-506页

作者： PhilippeBIANE CNRS DepartementdeMathematiquesetApplicationsEcoleNormaleSupericure45rued'Ulm75005ParisFrance

We give two applications of logarithmic Sobolev inequalities to matrix models and free probability. We also provide a new characterization of semi-circular systems through a Poincaré-type inequality.

关键词： Free probability random matrices Logarithmic Sobolev inequalities

Spectrum of SYK Model

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Peking Mathematical Journal 2019年第1期2卷 41-70页

作者： Renjie Feng Gang Tian Dongyi Wei Beijing International Center for Mathematical Research Peking UniversityBeijing 100871China

This is the first part of a series of papers on the spectrum of the SYK model,which is a simple model of the black hole in physics literature.In this paper,we will give a rigorous proof of the almost sure convergence ... 详细信息

关键词： SYK model Global density random matrices