Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

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

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

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

0. Introduction. The notion of the q-analogue of universal enveloping algebras is introduced independently by V. G. Drinfeld and M. Jimbo in 1985 in their study of exactly solvable models in the statistical mechanics. This algebra Uq(g) contains a parameter q, and, when q 1, this coincides with the universal enveloping algebra. In the context of exactly solvable models, the parameter q is that of temperature, and q 0 corresponds to the absolute temperature zero. For that reason, we can expect that the q-analogue has a simple structure at q 0. In [K1] we named crystallization the study at q 0, and we introduced the notion of crystal bases. Roughly speaking, crystal bases are bases of Uq(9)-modules at q 0 that satisfy certain axioms. There, we proved the existence and the uniqueness of crystal bases of finite-dimensional representations of U(g) when g is one of the classical Lie algebras A,, B,, C, and D,. K. Misra and T. Miwa ([M]) proved the existence of a crystal base of the basic representation of U,(A1)) and gave its combinatorial description. The aim of this article is to give the proof of the existence and uniqueness theorem of crystal bases for an arbitrary symmetrizable Kac-Moody Lie algebra I. Moreover, we globalize this notion. Namely, with the aid of a crystal base we construct a base named the global crystal base of any highest weight irreducible integrable

On crystal bases of the $Q$-analogue of universal enveloping algebras

Introduction §1. Fock Representation of gf(°°) §2. T Functions and the KP Hierarchy §3. Reduction to A[\" §4. Fermions with 2 Components §5. Algebras B^ and Co §6. Spin Representation of J&TO §7. Algebras D^ and £C §8. Reduction to Kac-Moody Lie Algebras §9. Time Evolutions with Singularities Other than k = 00 —The 2 Dimensional Toda Lattice— §10. Difference Equations —The Principal Chiral Field— Appendix 1. Bilinear Equations for the (Modified) KP Hierarchies Appendix 2. Bilinear Equations for the 2 Component Reduced KP Hierarchy Appendix 3. Bilinear Equations Related to the Spin Representations of Bca Appendix 4. Bilinear Equations Related to the Spin Representations of D^

https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=19&iss=3&rank=4

Solitons and Infinite Dimensional Lie Algebras

Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II

Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III

Transformation groups for soliton equations

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Tetsuji Miwa

Papers

Solitons and Infinite Dimensional Lie Algebras

Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II

Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III

Transformation groups for soliton equations

Algebraic Analysis of Solvable Lattice Models.