Results 1  10
of
26
Matrix models for betaensembles
 J. Math. Phys
, 2002
"... This paper constructs tridiagonal random matrix models for general (β> 0) βHermite (Gaussian) and βLaguerre (Wishart) ensembles. These generalize the wellknown Gaussian and Wishart models for β = 1,2,4. Furthermore, in the cases of the βLaguerre ensembles, we eliminate the exponent quantizati ..."
Abstract

Cited by 172 (22 self)
 Add to MetaCart
(Show Context)
This paper constructs tridiagonal random matrix models for general (β> 0) βHermite (Gaussian) and βLaguerre (Wishart) ensembles. These generalize the wellknown Gaussian and Wishart models for β = 1,2,4. Furthermore, in the cases of the βLaguerre ensembles, we eliminate the exponent quantization present in the previously known models. We further discuss applications for the new matrix models, and present some open problems.
Random matrix theory
, 2005
"... Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We includ ..."
Abstract

Cited by 80 (4 self)
 Add to MetaCart
Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We include the important mathematics which is a very modern development, as well as the computational software that is transforming the theory into useful practice.
Orthogonal polynomial ensembles in probability theory
 Prob. Surv
, 2005
"... Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary ..."
Abstract

Cited by 63 (1 self)
 Add to MetaCart
Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other wellknown ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, noncolliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther
The problem of harmonic analysis on the infinitedimensional unitary group
 J. Funct. Anal. 205, no
"... Abstract. The infinite–dimensional unitary group U(∞) is the inductive limit of growing compact unitary groups U(N). In this paper we solve a problem of harmonic analysis on U(∞) stated in [Ol3]. The problem consists in computing spectral decomposition for a remarkable 4–parameter family of characte ..."
Abstract

Cited by 58 (20 self)
 Add to MetaCart
(Show Context)
Abstract. The infinite–dimensional unitary group U(∞) is the inductive limit of growing compact unitary groups U(N). In this paper we solve a problem of harmonic analysis on U(∞) stated in [Ol3]. The problem consists in computing spectral decomposition for a remarkable 4–parameter family of characters of U(∞). These characters generate representations which should be viewed as analogs of nonexisting regular representation of U(∞). The spectral decomposition of a character of U(∞) is described by the spectral measure which lives on an infinite–dimensional space Ω of indecomposable characters. The key idea which allows us to solve the problem is to embed Ω into the space of point configurations on the real line without 2 points. This turns the spectral measure into a stochastic point process on the real line. The main result of the paper is a complete description of the processes corresponding to our concrete family of characters. We prove that each of the processes is a determinantal point process. That is, its correlation functions have determinantal form with a certain kernel. Our kernels have a special ‘integrable ’ form and are expressed through the Gauss
Fluctuations of the onedimensional polynuclear growth model in half space
 J. STAT. PHYS
, 2004
"... We consider the multipoint equal time height fluctuations of a onedimensional polynuclear growth model in a half space. For special values of the nucleation rate at the origin, the multilayer version of the model is reduced to a determinantal process, for which the asymptotics can be analyzed. In ..."
Abstract

Cited by 54 (9 self)
 Add to MetaCart
We consider the multipoint equal time height fluctuations of a onedimensional polynuclear growth model in a half space. For special values of the nucleation rate at the origin, the multilayer version of the model is reduced to a determinantal process, for which the asymptotics can be analyzed. In the scaling limit, the fluctuations near the origin are shown to be equivalent to those of the largest eigenvalue of the orthogonal/symplectic to unitary transition ensemble at soft edge in random matrix theory.
On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review
, 2010
"... Abstract. In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) βensembles and t ..."
Abstract

Cited by 36 (5 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) βensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the kth largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplectic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.
Universality for mathematical and physical systems
, 2006
"... Abstract. All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all ..."
Abstract

Cited by 33 (0 self)
 Add to MetaCart
(Show Context)
Abstract. All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. In this paper we describe some recent history of universality ideas in physics starting with Wigner’s model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting. 1.
Sparse regular random graphs: Spectral density and eigenvectors
"... Abstract. We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to ..."
Abstract

Cited by 26 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to the semicircle law. Moreover, we prove concentration estimates on the number of eigenvalues over progressively smaller intervals. We also show that, with high probability, all the eigenvectors are delocalized. 1.