Application of the τ-Function Theory¶of Painlevé Equations to Random Matrices:¶PIV, PII and the GUE

Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We generalise these results to the evaluation of , where for and otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of . Of particular interest are and F _N (λ;2), and their scaled limits, which give the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto τ-function theory of PIV and PII, for which we give a self contained presentation based on the recent work of Noumi and Yamada. We point out that the same approach can be used to study the quantities and F _N (λ;a) for the other classical matrix ensembles. Received: 27 June 2000 / Accepted: 8 December 2000     

Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State

A recent conjecture regarding the average of the minimum eigenvalue of the reduced density matrix of a random complex state is proved. In fact, the full distribution of the minimum eigenvalue is derived exactly for both the cases of a random real and a random complex state. Our results are relevant to the entanglement properties of eigenvectors of the orthogonal and unitary ensembles of random matrix theory and quantum chaotic systems. They also provide a rare exactly solvable case for the distribution of the minimum of a set of N strongly correlated random variables for all values of N (and not just for large N).     

Random matrix theory and equilibrium statistics for the quasienergies of classically chaotic systems

We calculate thek-level correlation function for the eigenphases of theN-dimensional Floquet operator of kicked dynamics whose classical counterpart is chaotic in the limitN by applying standard equilibrium statistical mechanics to a fictitiousN-particle system which is constructed from the eigenvalue equation of the Floquet operator and show that they become equal to those of random matrices from the circular ensemble associated with the appropriate universality class.     

Correlations between eigenvalues of a random matrix

Exact analytical expressions are found for the joint probability distribution functions ofn eigenvalues belonging to a random Hermitian matrix of orderN, wheren is any integer andN. The distribution functions, like those obtained earlier forn=2, involve only trigonometrical functions of the eigenvalue differences.     

Spherically symmetric free fall collapse

The general dynamical equations for spherical gravitational collapse are derived by introducing the eigenvalue of the conformal Weyl tensor in the 2-2 component of the Einstein tensor and assuming the material content of the models to be a perfect fluid. Since this eigenvalue is coupled always with the material energy density, it has been interpreted as theenergy density of the free gravitational field whose presence is related with anisotropy and inhomogeneity. As a particular case, the collapse of a spherically symmetric dust (zero pressure) with vanishing radial acceleration (free fall collapse) is discussed. It is shown that the model is inhomogeneous with non-vanishing shear of the congruence of world lines of the dust particles. The model contains gravitational radiation by Szekere’s criterion since both shear invariant and the spatial gradient of density are non-vanishing. This is in contrast to the Oppenheimer-Synder model for which both the above mentioned characteristics are absent. A particular solution which is anisotropic and inhomogeneous has been given to prove the emission of gravitational radiation by the freely falling dust and in this case the energy density of the free gravitational field contains a typeN term superposed on the coulombian field.     

New Symmetries for the Uq(slN) 6-j Symbols from the Eigenvalue Conjecture1

In the present paper, we discuss the eigenvalue conjecture, suggested in 2012, in the particular case of U_q(sl_N) 6-j The eigenvalue conjecture provides a certain symmetry for Racah coefficients and we prove that the eigenvalue conjecture is provided by the Regge symmetry for U_q(sl_N) 6-j, when three representations coincide. This in perspective provides us a kind of generalization of the Regge symmetry to arbitrary U_q(sl_N) 6-j.

     

A Random Matrix Decimation Procedure Relating <Emphasis Type="Italic">β</Emphasis> = 2/(<Emphasis Type="Italic">r</Emphasis> + 1) to <Emphasis Type="Italic">β</Emphasis> = 2(<Emphasis Type="Italic">r</Emphasis> + 1)

Classical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry. These results are shown to be the case r = 1 of a family of inter-relations between eigenvalue probability density functions for generalizations of the classical random matrix ensembles referred to as β-ensembles. The inter-relations give that the joint distribution of every (r + 1)^st eigenvalue in certain β-ensembles with β  =  2/(r + 1) is equal to that of another β-ensemble with β  =  2(r + 1). The proof requires generalizing a conditional probability density function due to Dixon and Anderson.     

N-fold joint photocounting distribution for modulated laser radiation: Transmission through the turbulent atmosphere

In this paper a generalised result for theN-fold joint photoelectron counting distribution for independently modulated radiation is given. We extend the recent results of Diament and Teich, for the one-fold photoelectron counting distribution for light propagated through an atmosphere characterised by log-normal irradiance fluctuations, to theN-fold joint photoelectron counting distribution. An approximate solution for thisN-fold distribution is obtained, for detection intervals {T_i} « _a where _a is the characteristic time of the atmospheric turbulence. We present specifically the two-fold joint photocounting distribution for amplitude-stabilised laser radiation passing through such an atmosphere for several levels of turbulence and degrees of correlation. Cases including additive, independent, non-interfering Poisson noise are considered. Computer generated plots of the photocounting distribution are presented. For noise-free detection, the otherwise narrow-peaked photocounting distribution is seen to broaden markedly and shift its peak to lower counts as the turbulence level increases. Furthermore, a non-singular counting distribution is obtained for fully correlated detection. In the presence of additive noise and varying only the signal-to-noise ratio, the probability surface is intermediate between that of the Poisson and that of the noise-free log-normal fading counting distribution. The peak, however, is observed to decrease and then again increase in magnitude as , for correlated detection only. These results are expected to be of use in the study of atmospheric turbulence, as well as in the evaluation of certain stochastic functionals that occur in optical communication theory for the turbulent atmospheric channel.This work was supported in part by the US National Science Foundation under Grant Number NSF-GK-16649.     

On Eigenvalues of the Sum of Two Random Projections

We study the behavior of eigenvalues of matrix P _N +Q _N where P _N and Q _N are two N-by-N random orthogonal projections. We relate the joint eigenvalue distribution of this matrix to the Jacobi matrix ensemble and establish the universal behavior of eigenvalues for large N. The limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the Bessel or the Airy kernel at the edge depending on parameters. We also study an exceptional case when the local behavior of eigenvalues of P _N +Q _N is not universal in the usual sense.     

Some comments on the solvable chiral potts model

We present a simple proof of the conjecture produced by Baxter, Perk and Au-Yang on the structure of the normalization factorR(p, q, r) corresponding to their new solution of the star-triangle equation related with the generalized Fermat curve. Some important properties of the underlying curvex ^N y ^N+x ^N+y ^N+1/k ²=0 for theN=3 state case are also established. Particularly, we calculate exactly its matrix of theb-periods for some normalized basis of holomorphic differentials. We also show that associated four-dimensional theta function may be decomposed into a sum containing 12 terms, each term being the product of four one-dimensional theta functions. We also derive Picard-Fuchs equations for the periods of holomorphic differentials of the same curve. The remarkable appearance of the hypergeometric functions in our answers seems to be closely related with an expression for the groundstate energy per site, obtained for the superintegrable case by Albertini, Perk, and McCoy and independently by Baxter, although for a moment the connection is not clear.     

The Largest Eigenvalue of Real Symmetric,Hermitian and Hermitian Self-dual Random Matrix Models with Rank One External Source,Part I

We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric (β=1), Hermitian (β=2), and Hermitian self-dual (β=4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the “one-band” condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value. When the nonzero eigenvalue of the external source matrix is less than or equal to the critical value, the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper. In this paper we also give a definition of the external source model for all β>0.     

The bi-Hamiltonian structure of fully supersymmetric Korteweg-de Vries systems

The bi-Hamiltonian structure of integrable supersymmetric extensions of the Korteweg-de Vries (KdV) equation related to theN=1 and theN=2 superconformal algebras is found. It turns out that some of these extensions admit inverse Hamiltonian formulations in terms of presymplectic operators rather than in terms of Poisson tensors. For one extension related to theN=2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical KdV. They can be explained by a factorization of the corresponding Lax operator. All the bi-Hamiltonian formulations are derived in a systematic way from the Lax operators.     

Nonintersecting Brownian Motions on the Half-Line and Discrete Gaussian Orthogonal Polynomials

We study the distribution of the maximal height of the outermost path in the model of N nonintersecting Brownian motions on the half-line as N→∞, showing that it converges in the proper scaling to the Tracy-Widom distribution for the largest eigenvalue of the Gaussian orthogonal ensemble. This is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the Airy₂ process minus a parabola. Our proof is based on Riemann-Hilbert analysis of a system of discrete orthogonal polynomials with a Gaussian weight in the double scaling limit as this system approaches saturation. We consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation. Both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point.     

Fermionic strings on the compactified moduli space

On the compactified moduli space we consider theN=2,N=4 local supersymmetric string theories. It would be proven that theN=2,N=4 fermionic string theories might not develop any tachyon pole, which might imply theg-loop partition functions forN=2,N=4 fermionic string would be finite.     

Coherent motion of a hole in the copper-oxygen planes of high-temperature superconductors

The dispersion relation for the coherent propagation of a hole moving in a two-dimensional (CuO₂) _N system is discussed. The (CuO_2)N planes constitute the most important structural element in the high-T _c superconducting materials. The system is described by the Kondo-Heisenberg Hamiltonian, which is a simplified version of the extended Hubbard or Emery model. The calculations are based on the introduction of a trial wave function in the unitary space of the Cu spins and the O degrees of freedom. They generalize an approach recently proposed for the coherent motion of a hole in thet-J model. The propagation is mainly determined by the spin-fluctuation part of the superexchange between the copper spins. Minor contributions to the coherent hole motion are due to an effective tunneling of the hole to second and third nearest neighbors along spiral paths in the (CuO₂) _N plane. This mechanism can be considered as the analogue of a mechanism for coherent hole motion in thet-J model first discussed by Trugman. For the dispersion relation a cosine-band-like form is found similar to that for thet-J model. The band width, however, is somewhat increased. Except for this difference, our results seem to support the point of view of Zhang and Rice, who have claimed that there exists a one-to-one mapping between the low-lying states of the two-band model and the effectivet-J model.     

Eigenvalues of the one-dimensional Smoluchowski equation

The eigenvalues and eigenfunctions of the Smoluchowski equation are investigated for the case of potentials withN deep wells. The small parameter =kT/V, which measures the ratio of the thermal energy to a typical well depth, is used in connection with the method of matched asymptotic expansion to obtained asymptotic approximations to all the eigenvalues and eigenfunctions. It is found that the eigensolutions fall into two classes, namely (i) the top-of-the-well and (ii) the bottom-of-the-well eigensolutions. The eigenvalues for both classes of solutions are integer multiples of the squqres of the frequencies at the top or bottom of the various wells. The eigenfunctions are, in general, localized to the top or bottom of the corresponding well. The very small eigenvalues require special consideration because the asymptotic analysis is incapable of distinguishing them from the zero eigenvalue with multiplicityN. Another approximation reveals that, in addition to the true zero eigenvalue, there areN-1 eigenvalues of order exp(–). The case of other possible multiple eigenvalues is also examined.     

On the problem of spacetime symmetries in the theory of supergravity. part II:N=2 supergravity and spinorial Lie derivatives

The definition of a spacetime symmetry, developed in a previous paper in the framework of simple (N=1) supergravity, is extended to theN=2 theory. As an application, the properties of theN=2 plane wave are studied. The mathematically related question of defining the Lie derivative of a spinor is also considered.     

The distribution of Lyapunov exponents: Exact results for random matrices

Simple exact expressions are derived for all the Lyapunov exponents of certainN-dimensional stochastic linear dynamical systems. In the case of the product of independent random matrices, each of which has independent Gaussian entries with mean zero and variance 1/N, the exponents have an exponential distribution asN. In the case of the time-ordered product integral of exp[N ^–1/2 dW], where the entries of theN×N matrixW(t) are independent standard Wiener processes, the exponents are equally spaced for fixedN and thus have a uniform distribution as N.John S. Guggenheim Memorial Fellow. Research supported in part by NSF Grant MCS 80-19384     

Effective schrödinger equation for a classical massless dislocation plasma

Summary The statistical behaviour of classical massless excitations finds an increasing importance in the physics of low-dimensional condensed matter. Dislocation and disclination dipole-gases and plasmas play such a relevant role in the theory of 2D melting. Here the equilibrium statistical mechanics of a system of strongly interactingparticles of this type is faced searching for the approximate stationary solution of the multivariate associated Fokker-Planck equation corresponding to zero eigenvalue. The problems, encountered in a preceding paper, involved by the nonhermiticity of Fokker-Planck operator, are evaded, following Risken, building anequivalent many-body Schr?dinger equation. This last is solved self-consistently in a Hartree-like way starting with a free-particleproduct wavefunction in the case of a uniform background whosecharge is of sign opposite to that of theparticles. Unlike thetrue quantum case, here the integral part of the equivalent Hamiltonian operator is not simply Coulomb-like and defines a more difficult novel integrodifferential problem which is solved using a convergence in mean procedure. The author of this paper has agreed to not receive the proofs for correction.     

A multidimensional radiation-filled universe

We consider a radiation-filled universe which possesses the product symmetry: (N-dimensional space of constant curvature) × (n sphere). The solutions of all the types, within this class, to the classical field equations are given. In the case of theN-dimensional space of zero or negative curvature constant, the solutions exhibit a tendency to approach asymptotically the Kasner-like state in which theN-dimensional subspace expands while then sphere shrinks to the final singularity. Our conclusions based on the phase-diagram method are in agreement with the results concerning the ^N × S ⁿ universe calculated by Sahdev with the help of numerical methods.