Dedicated to the memory of Gert Kjærg˚ Pedersen ard Abstract In the process of developing the theory of free probability and free entropy, Voiculescu introduced in 1991 a random matrix model for a free semicircular system. Since then, random matrices have played a key role in von Neumann algebra theory (cf. [V8], [V9]). The main result of this paper is the follow(n) (n) ing extension of Voiculescu’s random matrix result: Let (X1 , . . . , Xr ) be a system of r stochastically independent n × n Gaussian self-adjoint random matrices as in Voiculescu’s random matrix paper...
Let A be an n × n matrix, whose entries are independent copies of a centered random variable satisfying the subgaussian tail estimate. We prove that the operator norm of A−1 does not exceed Cn3/2 with probability close to 1. 1. Introduction Let A be an n × n matrix, whose entries are independent, identically distributed random variables. The spectral properties of such matrices, in particular invertibility, have been extensively studied (see, e.g. [M] and the survey [DS]).
Random matrices are widely and successfully used in physics for almost
60-70 years, beginning with the works of Wigner and Dyson. Initially proposed
to describe statistics of excited levels in complex nuclei, the Random
Matrix Theory has grown far beyond nuclear physics, and also far beyond just
level statistics. It is constantly developing into new areas of physics and mathematics,
and now constitutes a part of the general culture and curriculum of a
The present book collects most of the courses and seminars delivered at the
meeting entitled “ Frontiers in Number Theory, Physics and Geometry”, which
took place at the Centre de Physique des Houches in the French Alps, March 9-
21, 2003. It is divided into two volumes. Volume I contains the contributions on
three broad topics: Random matrices, Zeta functions and Dynamical systems.
The present volume contains sixteen contributions on three themes: Conformal
field theories for strings and branes, Discrete groups and automorphic forms
and finally, Hopf algebras and renormalization....
CHAPTER 9 Random Matrices. The step from random vectors to random matrices (and higher order random arrays) is not as big as the step from individual random variables to random vectors. We will ﬁrst give a few quite trivial veriﬁcations that the expected value operator is indeed a linear operator
This lecture is an overview of the fundamental tools and techniques for numeric computation. For some, numerics are everything. For many, numerics are occasionally essential. Here, we present the basic problems of size, precision, truncation, and error handling in standard mathematical functions. We present multidimensional matrices and the standard library complex numbers.
CHAPTER 3 Random Variables. 3.1. Notation Throughout these class notes, lower case bold letters will be used for vectors and upper case bold letters for matrices, and letters that are not bold for scalars. The (i, j) element of the matrix A is aij , and the ith element of a vector b is bi