A Simple Approach to Global Regime of the Random Matrix Theory

We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of this unique approach we obtain functional equations for the Stieltjes transforms of the limiting normalized eigenvalue counting measure and the bounds for the rate of convergence for the majority known random matrix ensembles.

[1]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[2]  V. Vasilchuk,et al.  On the Law of Multiplication of Random Matrices , 2001 .

[3]  The interband light absorption coefficient in the weak disorder regime: an asymptotically exactly solvable model , 1994 .

[4]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[5]  Rigorous mean-field model for coherent-potential approximation: Anderson model with free random variables , 1994, cond-mat/9410064.

[6]  W. Hackbusch.  Singular Integral Equations , 1995 .

[7]  F. Smithies,et al.  Singular Integral Equations , 1955, The Mathematical Gazette.

[8]  P. Di Francesco,et al.  2D gravity and random matrices , 1993 .

[9]  R. Speicher.  Free Probability Theory , 1996, Oberwolfach Reports.

[10]  L. Pastur.  On the spectrum of random matrices , 1972 .

[11]  L. Pastur,et al.  Limits of infinite interaction radius, dimensionality and the number of components for random operators with off-diagonal randomness , 1993 .

[12]  Random Transfer Matrix Theory and Conductance Fluctuations , 1991 .

[13]  L. Pastur.  On the universality of the level spacing distribution for some ensembles of random matrices , 1992 .

[14]  A. Zee.  LAW OF ADDITION IN RANDOM MATRIX THEORY , 1996, cond-mat/9602146.

[15]  I. M. Glazman,et al.  Theory of linear operators in Hilbert space , 1961 .

[16]  A Random Matrix Model From Two Dimensional Yang-Mills Theory , 1997 .

[17]  L. Pastur,et al.  On the Law of Addition of Random Matrices , 2000 .

[18]  C. Itzykson,et al.  Quantum field theory techniques in graphical enumeration , 1980 .

[19]  L. Pastur,et al.  On the statistical mechanics approach in the random matrix theory: Integrated density of states , 1995 .

[20]  E. M.,et al.  Statistical Mechanics , 2021, Manual for Theoretical Chemistry.

[21]  R. Speicher.  Free Convolution and the Random Sum of Matrices , 1993 .