Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models ofN×N hermitian matrices and then going to the limitN→∞ leads to the Fredholm determinant of thesine kernel sinπ(x−y)/π(x−y). Similarly a scaling limit at the “edge of the spectrum” leads to theAiry kernel [Ai(x)Ai(y)−Ai′(x)Ai(y)]/(x−y). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, Mori, and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painleve transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for generaln, of the probability that an interval contains preciselyn eigenvalues.

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Level spacing distributions and the Airy kernel

One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

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An Introduction to Random Matrices

We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

Shape Fluctuations and Random Matrices

Past, present and future of nonlinear system identification in structural dynamics

The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next is a book about the history of physics from Copernicus forward. It is also a book that discusses the current state of physics research, particularly the dominion that string theory holds over the field. The author covers many diverse topics, and, while the title singles out string theory, this is a book about much more. Smolin starts with the five great unanswered problems in physics today. I replicate them here as they appear in Chapter 1:

The trouble with Physics: The rise of string theory, the fall of a Science, and what comes next

Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part
 
$$\tilde{\mathfrak{g}}^+$$

 of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouvile-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. As illustrative examples, the caseg =sl(2), together with its real forms, is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, with the Liouville generating function expressed in hyperellipsoidal coordinates. Forg =sl(3), the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schrodinger equation, a case which requires further symplectic constraints in order to deal with singularities in the spectral data at ∞.

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Darboux coordinates and Liouville-Arnold integration in loop algebras

A moment map Jr\JiA^>(gl(r) + )* is constructed from the Poisson manifold JίA of rank-r perturbations of a fixed N xN matrix A to the dual (&'(r)+)* °f Λe positive part of the formal loop algebra ^I(r) = g/(r)®C((Λ.,Λ,~1)). The Adler-Kostant-Symes theorem is used to give hamiltonians which generate commutative isospectral flows on (g/(r)+)*. The pull-back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in JίA. The latter may be identified with flows on finite dimensional coadjoint orbits in (g/(>)+)* and linearized on the Jacobi variety of an invariant spectral curve Xr which, generically, is an r-sheeted Riemann surface. Reductions of JίA are derived, corresponding to subalgebras of gl(r, <C) and sl(r, C), determined as the fixed point set of automorphism groupes generated by involutions (i.e.,jdl_the classical algebras), as well as reductions to twisted subalgebras of s/(r, (C). The theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of PDE's.

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Isospectral Hamiltonian Flows in Finite and Infinite Dimensions I. Generalized Moser Systems and Moment Maps into Loop Algebras

The Hamiltonian structure of the monodromy preserving deformation equations of Jimboet al [JMMS] is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to “dual” pairs of matrix differential operators whose monodromy is preserved under the same family of deformations. As illustrative examples, involving discrete and continuous reductions, a higher rank generalization of the Hamiltonian equations governing the correlation functions for an impenetrable Bose gas is obtained, as well as dual pairs of isomonodromy representations for the equations of the Painleve transcendentsPV andVI.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1104271613

Dual isomonodromic deformations and moment maps to loop algebras

The statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized Christoffel–Darboux form constructed from sequences of biorthogonal polynomials. For measures involving exponentials of a pair of polynomials V1, V2 in two different variables, these kernels may be expressed in terms of finite dimensional “windows” spanned by finite subsequences having length equal to the degree of one or the other of the polynomials V1, V2. The vectors formed by such subsequences satisfy “dual pairs” of first order systems of linear differential equations with polynomial coefficients, having rank equal to one of the degrees of V1 or V2 and degree equal to the other. They also satisfy recursion relations connecting the consecutive windows, and deformation equations, determining how they change under variations in the coefficients of the polynomials V1 and V2. Viewed as overdetermined systems of linear difference-differential-deformation equations, these are shown to be compatible, and hence to admit simultaneous fundamental systems of solutions. The main result is the demonstration of a spectral duality property; namely, that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters.

John Harnad

Papers

The trouble with Physics: The rise of string theory, the fall of a Science, and what comes next

Darboux coordinates and Liouville-Arnold integration in loop algebras

Isospectral Hamiltonian Flows in Finite and Infinite Dimensions I. Generalized Moser Systems and Moment Maps into Loop Algebras

Dual isomonodromic deformations and moment maps to loop algebras

Duality, Biorthogonal Polynomials¶and Multi-Matrix Models