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20
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 quantization ..."
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Cited by 86 (19 self)
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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.
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 ..."
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Cited by 32 (10 self)
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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 ..."
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Cited by 29 (7 self)
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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.
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 ..."
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Cited by 24 (1 self)
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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
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 ..."
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Cited by 20 (0 self)
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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.
Universality and scaling of zeros on symplectic manifolds
"... Abstract. This article is concerned with random holomorphic polynomials and their generalizations to algebraic and symplectic geometry. A natural algebrogeometric generalization studied in our prior work involves random holomorphic sections H 0 (M, L N) of the powers of any positive line bundle L → ..."
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Cited by 17 (12 self)
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Abstract. This article is concerned with random holomorphic polynomials and their generalizations to algebraic and symplectic geometry. A natural algebrogeometric generalization studied in our prior work involves random holomorphic sections H 0 (M, L N) of the powers of any positive line bundle L → M over any complex manifold. Our main interest is in the statistics of zeros of k independent sections (generalized polynomials) of degree N as N → ∞. We fix a point P and focus on the ball of radius 1 / √ N about P. Under a microscope magnifying the ball by the factor √ N, the statistics of the configurations of simultaneous zeros of random ktuples of sections tends to a universal limit independent of P, M, L. We review this result and generalize it further to the case of prequantum line bundles over almostcomplex symplectic manifolds (M, J, ω). Following [SZ2], we replace H 0 (M, L N) in the complex case with the ‘asymptotically holomorphic ’ sections defined by Boutet de MonvelGuillemin and (from another point of view) by Donaldson and Auroux. Using a generalization to an mdimensional setting of the KacRice formula for zero correlations together with the results of [SZ2], we prove that the scaling limits of the correlation functions for zeros of random ktuples of asymptotically holomorphic sections belong to the same universality class as in the complex case. 1.
A growth model in a random environment
, 2000
"... We consider a model of interface growth in two dimensions, given by a height function on the sites of the one–dimensional integer lattice. According to the discrete time update rule, the height above the site x increases to the height above x − 1, if the latter height is larger; otherwise the heig ..."
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Cited by 14 (5 self)
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We consider a model of interface growth in two dimensions, given by a height function on the sites of the one–dimensional integer lattice. According to the discrete time update rule, the height above the site x increases to the height above x − 1, if the latter height is larger; otherwise the height above x increases by 1 with probability px. We assume that px are chosen independently at random with a common distribution F, and that the initial state is such that the origin is far above the other sites. We explicitly identify the asymptotic shape and prove that, in the pure regime, the fluctuations about that shape, normalized by the square root of time, are asymptotically normal. This contrasts with the quenched version: conditioned on the environment, and normalized by the cube root of time, the fluctuations almost surely approach a distribution known from random matrix theory.
Large deviations and stochastic calculus for large random matrices
, 2004
"... Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of math ..."
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Cited by 13 (0 self)
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Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of mathematical breakthroughs allowing for instance a better understanding of local properties of their spectrum, answering universality questions, connecting these issues with growth processes etc. In this survey, we shall discuss the problem of the large deviations of the empirical measure of Gaussian random matrices, and more generally of the trace of words of independent Gaussian random matrices. We shall describe how such issues are motivated either in physics/combinatorics by the study of the socalled matrix models or in free probability by the definition of a noncommutative entropy. We shall show how classical large deviations techniques can be used in this context. These lecture notes are supposed to be accessible to non probabilists and non freeprobabilists.