Über die Autorin bzw. den Autor

Peter J. Forrester is professor of mathematics at the University of Melbourne.

Von der hinteren Coverseite

"Encyclopedic in scope, this book achieves an excellent balance between the theoretical and physical approaches to the subject. It coherently leads the reader from first-principle definitions, through a combination of physical and mathematical arguments, to the full derivation of many fundamental results. The vast amount of material and impeccable choice of topics make it an invaluable reference."--Eduardo Dueñez, University of Texas, San Antonio

"This self-contained treatment starts from the basics and leads to the 'high end' of the subject. Forrester often gives new derivations of old results that beginners will find helpful, and the coverage of comprehensive topics will be useful to practitioners in the field."--Boris Khoruzhenko, Queen Mary, University of London

Aus dem Klappentext

"Encyclopedic in scope, this book achieves an excellent balance between the theoretical and physical approaches to the subject. It coherently leads the reader from first-principle definitions, through a combination of physical and mathematical arguments, to the full derivation of many fundamental results. The vast amount of material and impeccable choice of topics make it an invaluable reference."--Eduardo Dueñez, University of Texas, San Antonio

"This self-contained treatment starts from the basics and leads to the 'high end' of the subject. Forrester often gives new derivations of old results that beginners will find helpful, and the coverage of comprehensive topics will be useful to practitioners in the field."--Boris Khoruzhenko, Queen Mary, University of London

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Log-Gases and Random Matrices

PRINCETON UNIVERSITY PRESS

Contents

Preface.....................................................................................vChapter 1. Gaussian matrix ensembles........................................................1Chapter 2. Circular ensembles...............................................................53Chapter 3. Laguerre and Jacobi ensembles....................................................85Chapter 4. The Selberg integral.............................................................133Chapter 5. Correlation functions at = 2...................................................186Chapter 6. Correlation functions at = 1 and 4.............................................236Chapter 7. Scaled limits at = 1, 2 and 4..................................................283Chapter 8. Eigenvalue probabilities-Painlev systems approach...............................328Chapter 9. Eigenvalue probabilities-Fredholm determinant approach...........................380Chapter 10. Lattice paths and growth models.................................................440Chapter 11. The Calogero-Sutherland model...................................................505Chapter 12. Jack polynomials................................................................543Chapter 13. Correlations for general ......................................................592Chapter 14. Fluctuation formulas and universal behavior of correlations.....................658Chapter 15. The two-dimensional one-component plasma........................................701Bibliography................................................................................765Index.......................................................................................785

Chapter One

The Gaussian ensembles are introduced as Hermitianmatrices with independent elements distributed as Gaussians, and joint distribution of all independent elements invariant under conjugation by appropriate unitary matrices. The Hermitian matrices are divided into classes according to the elements being real, complex or real quaternion, and their invariance under conjugation by orthogonal, unitary, and unitary symplectic matrices, respectively. These invariances are intimately related to time reversal symmetry in quantum physics, and this in turn leads to the eigenvalues of the Gaussian ensembles being good models of the highly excited spectra of certain quantum systems. Calculation of the eigenvalue p.d.f.'s is essentially an exercise in change of variables, and to calculate the corresponding Jacobians both wedge products and metric forms are used. The p.d.f.'s coincide with the Boltzmann factor for a log-gas system at three special values of the inverse temperature = 1, 2 and 4. Thus the eigenvalues behave as charged particles, all of like sign, which are in equilibrium. The Coulomb gas analogy, through the study of various integral equations, allows for the prediction of the leading asymptotic form of the eigenvalue density. After scaling, this leading asymptotic form is referred to as the Wigner semicircle law. The Wigner semicircle law is applied to the study of the statistics of critical points for a model of high-dimensional energy landscapes, and to relating matrix integrals to some combinatorial problems on the enumeration of maps. Conversely, the latter considerations also lead to the proof of the Wigner semicircle law in the case of the GUE. The shifted mean Gaussian ensembles are introduced, and it is shown how the Wigner semicircle law can be used to predict the condition for the separation of the largest eigenvalue. In the last section a family of random tridiagonal matrices, referred to as the Gaussian -ensemble, are presented. These interpolate continuously between the eigenvalue p.d.f.'s of the Gaussian ensembles studied previously.

1.1 RANDOM REAL SYMMETRIC MATRICES

Quantum mechanics singles out three classes of random Hermitian matrices. We will begin our study by specifying one of these-Hermitian matrices with all entries real, or equivalently real symmetric matrices. The independent elements are taken to be distributed as independent Gaussians, but with the variance different for the diagonal and off-diagonal entries.

Definition 1.1.1 A random real symmetric N x N matrix X is said to belong to the Gaussian orthogonal ensemble (GOE) if the diagonal and upper triangular elements are independently chosen with p.d.f.'s

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

respectively.

The p.d.f.'s of Definition 1.1.1 are examples of the normal (or Gaussian) distribution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

denoted N[, [sigma]]. With this notation, note that an equivalent construction of GOE matrices is to let Y be an N x N random matrix of independent standard Gaussians N[0, 1] and to form X = 1/2 (Y + [Y.sup.T]).

The joint p.d.f. of all the independent elements is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where [A.sub.N] is the normalization and Tr denotes the trace. This structure is behind the choices of the independent Gaussians in Definition 1.1.1. It provides the starting point to identify features of the GOE which make it relevant to quantum physics [447].

Proposition 1.1.2 Let X be a member of the GOE and let R be an N ?N real orthogonal matrix. One has P ([R.sup.T] XR) = P(X). Furthermore, the most general p.d.f. satisfying this equation which has the factorization property [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for [f.sub.jk] differentiable is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. See Exercises 1.1 q.1.

Proposition 1.1.3 Define the entropy S of the joint p.d.f. P of the independent elements of X by S[P] := - [??] P log P (d X) =: -P>P where]TIL (dX) := [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then P as given by (1.1) maximizes S subject to the constraint [.sub.P] = [N.sub.2].

Proof. Because of the constraint on the second moment, and the normalization constraint, we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [lambda] and -(log A + 1) are Lagrange multipliers. The condition for a maximum is dS = 0, where the variation is made with respect to P. This gives

-log P - [lambda]Tr[X.sup.2] + log A = 0

and thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The value of 1/2 is determined to be 1 2 from the given constraint.

From these properties an understanding of the applicability of the GOE in the study of quantum energy spectra can be obtained. However as a further prerequisite some theory from quantum mechanics is required [401], [284].

1.1.1 Time reversal in quantum systems

First it is necessary to understand the relevance of an N x N matrix to quantum energy spectra. A basic axiom of quantum mechanics says the energy spectrum of a quantum system is given by the eigenvalues of its (Hermitian) Hamiltonian operator H, the latter being in general infinite dimensional. Now, to model the discrete portion of the spectrum of a complicated quantum system, a reasonable approximation is to replace H by a finite-dimensional N x N Hermitian matrix, which has a discrete...