# Log-Gases and Random Matrices (LMS-34)

P.J. Forrester
Pages: 808
https://www.jstor.org/stable/j.ctt7t5vq

1. Front Matter
(pp. i-iv)
2. Preface
(pp. v-x)
Peter Forrester
(pp. xi-xiv)
4. Chapter One Gaussian matrix ensembles
(pp. 1-52)

The Gaussian ensembles are introduced as Hermitian matrices 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...

5. Chapter Two Circular ensembles
(pp. 53-84)

Invariance of the probability measure on the space of matrices under conjugation by the appropriate unitary matrices does not uniquely determine the Gaussian ensembles. This fact prompted Dyson to develop a theory of random unitary matrices with the same invariances as the Hermitian matrices used to model quantum Hamiltonians. In quantum mechanics, scattering matrices and Floquet operators are quantities which can be modeled by random unitary matrices. In the case of no time reversal symmetry, the unitary matrices have no further constraints and form the group U(N). It is well known that the Haar measure is the unique uniform measure...

6. Chapter Three Laguerre and Jacobi ensembles
(pp. 85-132)

A Hermitian random matrix X can be formed out of a rectangular Gaussian matrix in the top right block, and its Hermitian conjugate in the bottom left block, with zeros elsewhere. This structure, which defines the chiral ensembles, can be motivated by the consideration of Dirac operators in the context of quantum chromodynamics, and time reversal symmetry distinguishes the cases of real, complex and real quaternion elements. The positive eigenvalues of matrices from the chiral ensembles are the singular values of X, or equivalently the nonzero eigenvalues of${{\bf{X}}^\dag }{\bf{X}}$. The ratio of the largest to smallest singular value is precisely...

7. Chapter Four The Selberg integral
(pp. 133-185)

The normalization of the Jacobi$\beta$-ensemble, known as the Selberg integral, is studied in its own right. Four different derivations of the evaluation of the Selberg integral, including the original one of Selberg, are presented. Two of these derivations give extensions of the Selberg integral which are of use in the calculation of correlation functions considered in subsequent chapters. Furthermore, the derivation due to Anderson has an interpretation in terms of the eigenvalues of a random corank 1 projection of a fixed matrix and leads to a further random three-term recurrence for the characteristic polynomial of the Jacobi$\beta$-ensemble. Alternative...

8. Chapter Five Correlation functions at $\beta = 2$
(pp. 186-235)

At the special coupling$\beta = 2$, the generaln-particle correlation function for the one-component log-gas can be expressed as an$n\; \times \;n$determinant involving orthogonal polynomials. A number of different viewpoints on this result are presented. These include integration formulas, functional differentiation and a formulation in terms of an integral equation. Yet another is to first deduce a determinant formula for the canonical average of what in the random matrix interpretation corresponds to ratios of characteristic polynomials. A general element of the$n\; \times \;n$determinant, referred to as the correlation kernel, is independent ofn. It is written as a sum of orthogonal...

9. Chapter Six Correlation functions at $\beta = 1$ and 4
(pp. 236-282)

At$\beta = 1$and 4, generalizations of the orthogonal polynomial technique used to calculate the correlation functions of the log-gas at$\beta = 2$are possible. These generalizations involve introducing skew orthogonal polynomials and quaternion determinants, or equivalently Pfaffians. For classical one-body potentials, the required skew orthogonal polynomials can be written in terms of the corresponding orthogonal polynomials, and the correlation kernel can be expressed as the Christoffel-Darboux summation plus a correction involving these polynomials. To obtain the quaternion determinant form both integration formulas and the method of functional differentiation, known from the studies of the previous chapter, are used. For$\beta = 1$the...

10. Chapter Seven Scaled limits at $\beta = 1,\;2$ and 4
(pp. 283-327)

In the previous two chapters the generaln-point correlations for log-gas systems at$\beta = 1,\;2$and 4 have been evaluated as determinants or quaternion determinants with entries given in terms of certain correlation kernels. In the classical cases the explicit form of the latter can readily be analyzed in certain scaling limits. Our main technique is to make use of known asymptotic expansions of the classical polynomials. The bulk, soft edge and hard edge scaling limits all refer to moving the origin to the respective portions of the support of the density, choosing the length scale to be of the order...

11. Chapter Eight Eigenvalue probabilities — Painlevé systems approach
(pp. 328-379)

The generating function for the probability that there are exactlyneigenvalues in an intervalJof a classical matrix ensemble with unitary symmetry can, for certainJ, be identified with the$\tau$-function of a Painlevé system. In particular, this is possible wheneverJis a single interval containing an endpoint of the support of the density. This allows the distribution of certain eigenvalue probabilities relating to the largest and smallest eigenvalue, and the bulk spacing, to be characterized in terms of the solution of nonlinear equations of Painlevé type. A practical consequence is the rapid computation of the power...

12. Chapter Nine Eigenvalue probabilities — Fredholm determinant approach
(pp. 380-439)

The theme of characterizing eigenvalue probabilities as solutions of nonlinear equations is continued, this time using various function theoretic and integrable systems aspects of Fredholm determinants, an approach complimentary to that of Painlevé systems. The starting point is a Fredholm determinant formula for the generating function of the gap probability in the case of matrix ensembles with unitary symmetry. Fredholm determinants with analytic kernels are well suited to numerical evaluation, and this allows tables of statistical properties of various distributions obtained in the previous chapter using power series to be extended. Function theoretic properties of the Fredholm determinants are used...

13. Chapter Ten Lattice paths and growth models
(pp. 440-504)

Nonintersecting lattice paths are the space-time trajectories of random walkers on a one-dimensional lattice. In some circumstances the probability distribution for the final position of the paths can be written in the form of the Boltzmann factor of a$\beta = 1$or 2 log-gas in which the particles are confined to lattice sites, or as an average over the unitary or symplectic groups. An important role is played by the Schur polynomials, which can be defined combinatorially in terms of weighted lattice paths. Tiling of a hexagon by three species of rhombi is equivalent to some nonintersecting lattice path configurations. By...

14. Chapter Eleven The Calogero—Sutherland model
(pp. 505-542)

Consideration of shifted mean parameter-dependent Gaussian random matrices, or equivalently Hermitian matrices with entries undergoing Brownian motion, leads to the Dyson Brownian motion model of the onecomponent log-gas. In the classical cases, a similarity transformation of the corresponding Fokker-Planck operator gives the Schrödinger operator for the Calogero-Sutherland model, which is the name given to the quantum many-body system for particles interacting on a line or a circle via the$1\,/\,{r^2}$pair potential. By generalizing these Schrödinger operators to include exchange terms, decompositions into more elementary operators can be exhibited, and these operators can be used to establish integrability. For the...

15. Chapter Twelve Jack polynomials
(pp. 543-591)

We have seen that the calculation of dynamical correlation functions for the Brownian evolution of the loggas requires knowledge of the Green function solution of the Fokker-Planck equation. It has also been noted that the Green function can be expressed in terms of the eigenvalues and eigenfunctions of the corresponding Schrödinger operator. In the cases of interest, these eigenfunctions factorize into a product of the ground state wave function times a multivariable polynomial. The most fundamental case is the Schrödinger operator${H^{({\text{C,}}\,{\text{Ex)}}}}$. The polynomial part of the eigenfunctions are then termed the nonsymmetric Jack polynomials, and they form the natural...

16. Chapter Thirteen Correlations for general $\beta$
(pp. 592-657)

Our ability to calculate correlations for log-gas systems beyond the special random matrix couplings$\beta = 1,\;2$and 4 relies on Jack polynomial theory. One application of the theory is to particular generalizations of the Selberg integral—the so-called Selberg correlation integrals—which allow the exact calculation of the two particle correlation for the log-gas on a circle at even$\beta$. Another is to the exact calculation of the densitydensity correlation in the case of two different initial conditions: the first corresponding to a perfect gas and the second, which is restricted to$\beta$rational, to the equilibrium state. In the latter...

17. Chapter Fourteen Fluctuation formulas and universal behavior of correlations
(pp. 658-700)

In previous chapters, asymptotic properties of log-gas systems relating to the two-point correlation, density profiles at the edge and spacing distributions were calculated exactly at the random matrix couplings$\beta = 1,\;2$and 4. Here macroscopic physical characterizations of the log-gas will be used to predict extensions of the asymptotic forms to the general β case, and furthermore to study fluctuation formulas for certain statistics. As an example, the log-gas has the physical property that it will perfectly screen an external charge density in the long wavelength limit, and this can be used to predict universal asymptotic forms for the two-point correlation....

18. Chapter Fifteen The two-dimensional one-component plasma
(pp. 701-764)

The two-dimensional one-component plasma (2dOCP) consists of log-potential charges of the same sign in a two-dimensional domain which contains a smeared out neutralizing background, and so is the twodimensional version of the one-component log-gas. Although only one value of the coupling allows an exact solution, there are a number of different two-dimensional geometries and boundary conditions for which this exact solution is possible. Here the exact solutions for disk, sphere and antisphere geometries are considered, as well as the exact solution for metallic and Neumann boundary conditions. The first three of these allow for interpretations as eigenvalue p.d.f.’s, and as...

19. Bibliography
(pp. 765-784)
20. Index
(pp. 785-791)