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.     

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.     

Localization for Random Unitary Operators

We consider unitary analogs of one-dimensional Anderson models on defined by the product U _ω=D _ω S where S is a deterministic unitary and D _ω is a diagonal matrix of i.i.d. random phases. The operator S is an absolutely continuous band matrix which depends on a parameter controlling the size of its off-diagonal elements. We prove that the spectrum of U _ω is pure point almost surely for all values of the parameter of S. We provide similar results for unitary operators defined on together with an application to orthogonal polynomials on the unit circle. We get almost sure localization for polynomials characterized by Verblunsky coefficients of constant modulus and correlated random phases Mathematics Subject Classification. 82B44, 42C05, 81Q05     

Symmetry Transitions in Random Matrix Theory &; <Emphasis Type="Italic">L</Emphasis>-functions

We compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices, defined by averages with respect to Haar measure on SO(2N) and USp(2N), to leading order as N → ∞, on the unit circle as functions of the angle θ measured from one of the two symmetry points in the eigenvalue spectrum . Our results extend previous formulae that relate just to the symmetry points, i.e. to θ = 0. Local spectral statistics are expected to converge to those of random unitary matrices in the limit as N → ∞ when θ is fixed, and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing: if θ = π y/N they become functions of y in the limit when N → ∞. We verify that this is true for the spectral two-point correlation function, but show that it is not true for the moments of the characteristic polynomials, for which the leading order asymptotic approximation is a function of θ rather than y. Symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing. We also investigate the moments of the logarithms of the characteristic polynomials in the same context. The moments of the characteristic polynomials of random matrices are conjectured to be related to the moments of families of L-functions. Previously, moments at the symmetry point θ = 0 have been related to the moments of families of L-functions evaluated at the centre of the critical strip. Our results motivate general conjectures for the moments of orthogonal and symplectic families of L-functions evaluated at a fixed height t up the critical line. These conjectures suggest that the symmetry of the non-trivial zeros of the L-functions influences the moments asymptotically far, on the scale of the mean zero spacing, from the centre of the critical strip. We verify that the second moments of real quadratic Dirichlet L-functions and a family of automorphic L-functions are consistent with our conjectures. JPK is supported by an EPSRC Senior Research Fellowship. BEO was supported by an Overseas Research Scholarship and a University of Bristol Research Scholarship.     

Density of States for Random Band Matrices

 By applying the supersymmetric approach we rigorously prove smoothness of the averaged density of states for a three dimensional random band matrix ensemble, in the limit of infinite volume and fixed band width. We also prove that the resulting expression for the density of states coincides with the Wigner semicircle with a precision 1/W ² , for W large but fixed. Received: 6 February 2002 / Accepted: 17 July 2002 Published online: 7 November 2002 RID="*" ID="*" Supported by NSF grant DMS 9729992     

Exact Wigner Surmise Type Evaluation of the Spacing Distribution in the Bulk of the Scaled Random Matrix Ensembles

Random matrix ensembles with orthogonal and unitary symmetry correspond to the cases of real symmetric and Hermitian random matrices, respectively. We show that the probability density function for the corresponding spacings between consecutive eigenvalues can be written exactly in the Wigner surmise type form a(s)e^–b(s) for a simply related to a Painlevé transcendent and b its anti-derivative. A formula consisting of the sum of two such terms is given for the symplectic case (Hermitian matrices with real quaternion elements).     

Universality for Orthogonal and Symplectic Laguerre-Type Ensembles

We give a proof of the Universality Conjecture for orthogonal (β=1) and symplectic (β=4) random matrix ensembles of Laguerre-type in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5, 1.7). They concern the appropriately rescaled kernels K _{n, β}, correlation and cluster functions, gap probabilities and the distributions of the largest and smallest eigenvalues. Corresponding results for unitary (β=2) Laguerre-type ensembles have been proved by the fourth author in Ref. 23. The varying weight case at the hard spectral edge was analyzed in Ref. 13 for β=2: In this paper we do not consider varying weights. Our proof follows closely the work of the first two authors who showed in Refs. 7, 8 analogous results for Hermite-type ensembles. As in Refs. 7, 8 we use the version of the orthogonal polynomial method presented in Refs. 22, 25, to analyze the local eigenvalue statistics. The necessary asymptotic information on the Laguerre-type orthogonal polynomials is taken from Ref. 23.     

Modified perturbation series for the anharmonic oscillator using linearization technique

The linearization technique of random phase approximation is applied to the anharmonic oscillator to find a modified perturbation series. It is shown that for the anharmonic termλx ⁴, the ground state energyE ₀ upto the second order of perturbation is given byE ₀=(35/48) (3/4)^1/3 λ ^1/3 asλ→∞.     

Combinatorics of Dispersionless Integrable Systems and Universality in Random Matrix Theory

It is well-known that the partition function of the unitary ensembles of random matrices is given by a τ-function of the Toda lattice hierarchy and those of the orthogonal and symplectic ensembles are τ-functions of the Pfaff lattice hierarchy. In these cases the asymptotic expansions of the free energies given by the logarithm of the partition functions lead to the dispersionless (i.e. continuous) limits for the Toda and Pfaff lattice hierarchies. There is a universality between all three ensembles of random matrices, one consequence of which is that the leading orders of the free energy for large matrices agree. In this paper, this universality, in the case of Gaussian ensembles, is explicitly demonstrated by computing the leading orders of the free energies in the expansions. We also show that the free energy as the solution of the dispersionless Toda lattice hierarchy gives a solution of the dispersionless Pfaff lattice hierarchy, which implies that this universality holds in general for the leading orders of the unitary, orthogonal, and symplectic ensembles. We also find an explicit formula for the two point function F _nm which represents the number of connected ribbon graphs with two vertices of degrees n and m on a sphere. The derivation is based on the Faber polynomials defined on the spectral curve of the dispersionless Toda lattice hierarchy, and \frac1nmF_nm{\frac{1}{nm}F_{nm}} are the Grunsky coefficients of the Faber polynomials.     

On the Relation Between Orthogonal,Symplectic and Unitary Matrix Ensembles

For the unitary ensembles of N×N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are 2×2 matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever w/w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w/w. General formulas are obtained for these extra terms. We do not use skew-orthogonal polynomials in the derivations     

Transition from Poisson to circular unitary ensemble

Transitions to universality classes of random matrix ensembles have been useful in the study of weakly-broken symmetries in quantum chaotic systems. Transitions involving Poisson as the initial ensemble have been particularly interesting. The exact two-point correlation function was derived by one of the present authors for the Poisson to circular unitary ensemble (CUE) transition with uniform initial density. This is given in terms of a rescaled symmetry breaking parameter Λ. The same result was obtained for Poisson to Gaussian unitary ensemble (GUE) transition by Kunz and Shapiro, using the contour-integral method of Brezin and Hikami. We show that their method is applicable to Poisson to CUE transition with arbitrary initial density. Their method is also applicable to the more general ℓCUE to CUE transition where ℓCUE refers to the superposition of ℓ independent CUE spectra in arbitrary ratio.     

Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

We consider unitary random matrix ensembles on the space of Hermitian n × n matrices M, where the confining potential V _s,t is such that the limiting mean density of eigenvalues (as n→∞ and s,t→ 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the P _I ² equation, which is a fourth order analogue of the Painlevé I equation. In order to prove our result, we use the well-known connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the P _I ² equation. In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights on . The special solution of the P _I ² equation pops up in the n ^−2/7-term of the asymptotics.     

Generalized Fokker-Planck equation for a class of stochastic dynamical systems driven by additive Gaussian and Poissonian fractional white noises of order <Emphasis Type="Italic">α</Emphasis>

In a first stage, the paper deals with the derivation and the solution of the equation of the probability density function of a stochastic system driven simultaneously by a fractional Gaussian white noise and a fractional Poissonian white noise both of the same order. The key is the Taylor’s series of fractional order f(x + h) = E _α(h^αD _x ^α)f(x) where E _α() denotes the Mittag-Leffler function, and D _x ^α is the so-called modified Riemann-Liouville fractional derivative which removes the effects of the non-zero initial value of the function under consideration. The corresponding fractional linear partial differential equation is solved by using a suitable extension of the Lagrange’s technique involving an auxiliary set of fractional differential equations. As an example, one considers a half-oscillator of fractional order driven by a fractional Poissonian noise.      

Characteristic Polynomials of Random Matrices

Number theorists have studied extensively the connections between the distribution of zeros of the Riemann ζ-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices, which are the expectation values of the characteristic polynomials of the matrix. It turns out that these expectation values are quite interesting. For instance, the moments of order 2K scale, for unitary invariant ensembles, as the density of eigenvalues raised to the power K ²; the prefactor turns out to be a universal number, i.e. it is independent of the specific probability distribution. An equivalent behaviour and prefactor had been found, as a conjecture, within number theory. The moments of the characteristic determinants of random matrices are computed here as limits, at coinciding points, of multi-point correlators of determinants. These correlators are in fact universal in Dyson's scaling limit in which the difference between the points goes to zero, the size of the matrix goes to infinity, and their product remains finite. Received: 1 October 1999 / Accepted: 18 May 2000     

Classical Skew Orthogonal Polynomials and Random Matrices

Skew orthogonal polynomials arise in the calculation of the n-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the case that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed-form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre, and Jacobi cases.     

On the 1/n Expansion for Some Unitary Invariant Ensembles of Random Matrices

We present a version of the 1/n-expansion for random matrix ensembles known as matrix models. The case where the support of the density of states of an ensemble consists of one interval and the case where the density of states is even and its support consists of two symmetric intervals is treated. In these cases we construct the expansion scheme for the Jacobi matrix determining a large class of expectations of symmetric functions of eigenvalues of random matrices, prove the asymptotic character of the scheme and give an explicit form of the first two terms. This allows us, in particular, to clarify certain theoretical physics results on the variance of the normalized traces of the resolvent of random matrices. We also find the asymptotic form of several related objects, such as smoothed squares of certain orthogonal polynomials, the normalized trace and the matrix elements of the resolvent of the Jacobi matrices, etc. Received: 9 November 2000 / Accepted: 26 July 2001     

Central Limit Theorem¶for Stochastic Hamilton–Jacobi Equations

We study the asymptotic behavior of , where u solves the Hamilton–Jacobi equation u _t +H(x,u _x ) ≡ 0 with H a stationary ergodic process in the x-variable. It was shown in Rezakhanlou–Tarver [RT] that u ^ɛ converges to a deterministic function provided H(x,p) is convex in p and the convex conjugate of H in the p-variable satisfies certain growth conditions. In this article we establish a central limit theorem for the convergence by showing that for a class of examples, u ^ɛ(x,t) can be (stochastically) represented as , where Z(x,t) is a suitable random field. In particular we establish a central limit theorem when the dimension is one and , where ω is a random function that enjoys some mild regularity. Received: 15 February 1999 / Accepted: 14 December 1999     

Generalized Orthogonal Polynomials, Discrete KP and Riemann–Hilbert Problems

Classically, a single weight on an interval of the real line leads to moments, orthogonal polynomials and tridiagonal matrices. Appropriately deforming this weight with times t= (t ₁, t ₂, …), leads to the standard Toda lattice and τ-functions, expressed as hermitian matrix integrals. This paper is concerned with a sequence of t-perturbed weights, rather than one single weight. This sequence leads to moments, polynomials and a (fuller) matrix evolving according to the discrete KP-hierarchy. The associated τ-functions have integral, as well as vertex operator representations. Among the examples considered, we mention: nested Calogero–Moser systems, concatenated solitons and m-periodic sequences of weights. The latter lead to 2m+ 1-band matrices and generalized orthogonal polynomials, also arising in the context of a Riemann–Hilbert problem. We show the Riemann–Hilbert factorization is tantamount to the factorization of the moment matrix into the product of a lower–times upper–triangular matrix. Received: 8 September 1998 / Accepted: 27 April 1999     

Density of Complex Zeros of a System of Real Random Polynomials

We study the density of complex zeros of a system of real random SO(m+1) polynomials in m variables. We show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case. We also show that the behavior the scaled density of complex zeros near ℝ ^m of the system of real random polynomials is different in the m≥2 case than in the m=1 case: the density approaches infinity instead of tending linearly to zero.     

Moment Matrices and Multi-Component KP,with Applications to Random Matrix Theory

Questions on random matrices and non-intersecting Brownian motions have led to the study of moment matrices with regard to several weights. The main result of this paper is to show that the determinants of such moment matrices satisfy, upon adding one set of “time” deformations for each weight, the multi-component KP-hierarchy: these determinants are thus “tau-functions” for these integrable hierarchies. The tau-functions, so obtained, with appropriate shifts of the time-parameters (forward and backwards) will be expressed in terms of multiple orthogonal polynomials for these weights and their Cauchy transforms. The main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials: it concerns an identity between the orthogonality of polynomials on the real line, the bilinear identity in KP theory and a generating functional for the full KP theory. An additional fact not discussed in this paper is that these τ-functions satisfy Virasoro constraints with respect to these time parameters. As one of the many examples worked out in this paper, we consider N non-intersecting Brownian motions in leaving from the origin, with n _i particles forced to reach p distinct target points b _i at time t  =  1; of course, . We give a PDE, in terms of the boundary points of the interval E, for the probability that the Brownian particles all pass through an interval E at time 0  <  t  <  1. It is given by the determinant of a (p + 1)  ×  (p + 1) matrix, which is nearly a wronskian. This theory is also applied to biorthogonal polynomials and orthogonal polynomials on the circle. The support of a National Science Foundation grant # DMS-07-04271 is gratefully acknowledged. The support of a National Science Foundation grant # DMS-07-04271, a European Science Foundation grant (MISGAM), a Marie Curie Grant (ENIGMA), a FNRS grant and a “Interuniversity Attraction Pole” grant is gratefully acknowledged. The support of a European Science Foundation grant (MISGAM), a Marie Curie Grant (ENIGMA) and a ANR grant (GIMP) is gratefully acknowledged.