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 spectrum only.

Next we need to understand the significance of real symmetric matrices in quantum mechanics. This is related to the fact that in general the structure of a matrix modeling H is constrained by the symmetries of H.

Definition 1.1.4 A quantum Hamiltonian H is said to have a symmetry A if

[H,A] = 0,

where [, ] denotes the commutator.

One basic symmetry of most quantum systems is time reversal.

Definition 1.1.5 A general time reversal operator T is any antiunitary operator, which means T = UK where U is unitary and K is the complex conjugation operator.

Hence we say a quantum system has a time reversal symmetry if the Hamiltonian commutes with an antiunitary operator.

Study of time reversal operators in the context of physical systems further restricts their form. For systems with an even number or no spin 1/2 particles, it is required that

[T.sup.2] = 1,

while for a finite-dimensional system with an odd number of spin 1/2 particles

[T.sup.2] = -1 and T = [Z.sub.2N]K,

where [Z.sub.2N] is a 2N x 2N block diagonal matrix with each 2 x 2 diagonal block given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

(a tensor product formula for [Z.sub.2N] is given in Exercises 1.1 q.2) which has the effect of reversing the spins. Real symmetric matrices arise in the former situation.

Proposition 1.1.6 Let H be a quantum Hamiltonian which is invariant with respect to a time reversal symmetry T, where T has the additional property [T.sup.2] = 1. Then H can always be given a T -invariant orthogonal basis, and with respect to this basis the (in general infinite) matrix representation of H is real.

Proof. See Exercises 1.1 q.3.

The above result tells us that a matrix chosen to model the discrete energy spectra of a quantum system with a time reversal symmetry T such that [T.sup.2] = 1 must be real symmetric. A further general property in quantum mechanics is that two operators related by a similarity transformation of unitary operators are equally valid descriptions of the operator, in that all observables are the same for both operators. A requirement of (1.1) is therefore that any two real symmetric matrices related by a similarity transformation of unitary matrices must have the same p.d.f. for the elements. For the two real symmetric matrices to be so related the unitary matrix must be real orthogonal (or i times a real orthogonal matrix; see Exercises 1.1 q.4). Thus this requirement is guaranteed by Proposition 1.1.2.

We are assuming no information on the Hamiltonian other than the time reversal symmetry. Proposition 1.1.3 says that the p.d.f. (1.1) is the most random subject to the given constraint, in that it maximizes the entropy.

These considerations thus show the applicability of the GOE in the study of quantum spectra. Explicitly, it is hypothesized that the statistical properties of the highly excited states of a complex quantum system with a time reversal symmetry [T.sup.2] = 1 coincide with the statistical properties of the bulk eigenvalues from large GOE matrices (see Section 7.1.1 for the notion of bulk eigenvalues). Here it is assumed that both spectra have been scaled (technically referred to as unfolded) so that the mean spacing is unity. The meaning of a complex quantum system requires further explanation. Wigner first made this hypothesis for the spectra of heavy nuclei in the 1950's. In 1984 Bohigas, Giannoni and Schmit made the same hypothesis for a single particle quantum billiard system, provided the underlying classical mechanics is chaotic and the system has a time reversal symmetry [T.sup.2] = 1. It is of interest to note that a GOE hypothesis also applies to eigenmodes of microwave cavities (this is not surprising as the Helmholtz equation is formally equivalent to the stationary Schrodinger equation), and also to the eigenmodes of systems governed by classical wave equations - vibrations of irregular shaped metal plates, electromechanical eigenmodes of aluminium and quartz blocks, among other examples. (For references to the original literature, and an extended discussion of GOE hypotheses, see [276].)

Exercises 1.1 1. The objective of this exercise is to prove Proposition 1.1.2.

(i) Note that the invariance P([R.sup.T]XR) = P(X) with R a permutation matrix requires that the distribution of all elements on the diagonal be equal, [f.sub.jj] = f, and similarly the distribution of all elements on the off diagonal be equal, [f.sub.jk] = g (j < k), for some f and g.

(ii) Choose

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where |[element of]| << 1. Ignoring terms O([[element of].sup.2]), show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the elements * are such that the matrix is symmetric.

(iii) Use the result of (ii) to show that at first order in _ the requirement

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which in turn, by separation of variables, implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some constant ?.

(iv) By a further separation of variables in the last equation conclude

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some constant b. Solve this differential equation.

(v) Note that the invariance P([R.sup.T]XR) = P(X) requires that P be a symmetric function of the eigenvalues, and thus a function of Tr([X.sup.k]) k = 1, 2,.... Now combine the results of (i) and (iv) to deduce the result. 2. Let A = [[a.sub.ij]] be a p x q matrix and B = [b.sub.i,j] be an r x s matrix. The tensor product, denoted A [??] B, is the pr x qs matrix with elements

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

With [Z.sub.2N] defined as above (1.2), show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

3. [401] Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[alpha].sub.1] is a scalar, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are vectors, T is anti-unitary and [T.sup.2] = 1. Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Here Proposition 1.1.6 will be established.

(i) From the antiunitarity property it follows that in general [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where <|> denotes the inner product. Use this to show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (ii) Suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is orthogonal to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Use (i) to show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is orthogonal to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and note how this construction can be used to create an orthogonal basis of vectors with the T-invariance property [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(iii) Consider a Hamiltonian H which has symmetry T. Use the above properties of T to show that with respect to the basis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the matrix elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are real.

4. Let X be an arbitrary real symmetric N x N matrix and suppose X' = [U.sup.-1]XU, where U is unitary and X' is real symmetric. Assume that the only symmetry of X and X' in general (other than some constant times the identity) is the time reversal operator T with [T.sup.2] = 1.

(i) Deduce that TU]T.sup.-1][U.sup.-1] commutes with X.

(ii) Use (i) to show TU = cUT and take the inverse of this equation to conclude c = 1.

(iii) Use (ii) and q.3(i) to show that with respect to the T invariant basis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence conclude that U has either real elements (c = 1) or pure imaginary elements (c = -1) and is thus either a real orthogonal matrix or i times a real orthogonal matrix.

1.2 THE EIGENVALUE P.D.F. FOR THE GOE

The p.d.f. for the elements of the matrices in the GOE is given by (1.1).We want to calculate the corresponding eigenvalue p.d.f. This was first accomplished as long ago as 1939 [299]. We will follow a more recent treatment [410].

(Continues...)

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