Mathematical Physics Laboratory
I'll attempt to cover some recent papers by several groups, mainly following Dubrovin's description of the subject. The general aim (not attained as yet) is to establish exact relations between (generalized) matrix model integrals (starting with just the usual Hermitian one-matrix model) in the 1/N-expansion pattern (not restricting to just the planar contribution) and the related hierarchies of integrable equations (especially, in the case of multicut solutions).
This is a continuation of the topic begun in the previous week's seminar: moduli spaces of punctured Riemann spheres, relations to tau functions and matrix integrals. The results of Kontsevich and Witten.
An introduction to the topic of moduli spaces and Strebel differentials,
We continue the introduction to the topic of moduli spaces and Strebel
differentials, with examples.
We review in some detail the coordinatization of Teichmuller spaces of marked algebraic curves of higher genus.
We show that the general (multicut, domain-walls) solutions of the
Hermitian 1-matrix model are Whitham tau-functions in the leading order of
1/N-expansion . We develop the technique for solving the loop equation
(master equation governing the model) in the 1/N-expansion pattern; this
technique is a (very) special form of (Feynmann) diagrammatic technique
well-suited for calculating the one- and higher-point resolvents. We
demonstrate how to get the free energy from the answer for the 1-point
resolvent in full generality. We discuss the momentum technique and why it
may be helpful when finding the integrable system tau-functions.
Continuation of Part I.
Continuation of Part II.
Many (perhaps all) known identities involving
determinants and integrals can be interpreted as special
cases of Wick's theorem for free Fermions.
In this talk, I will show how a number of such results follow, and how the expression of correlation functions involving products and quotients of characteristic polynomials of random matrices may be deduced either using "direct methods", which are based on the Cauchy-Binet identity and partial fraction decompositions of symmetric rational functions (both special instances of the above), or by realizing the corresponding matrix integrals as vacuum state expectation values of products of opertors constructed from free fermion creation and annihilation operrators.
This approach makes clear why these integrals may all be interpreted as KP or 2-Toda tau functions.
In this informal (hopefully informative) talk I will try to present a result by
Krichever and Novikov about the classification of commuting difference
operators, in a particular case (for sake of simplicity) relevant to
orthogonal polynomials. The problem can be easily formulated as: classify
all pairs of infinite matrices P,Q where Q is symmetric and tridiagonal
(a Jacobi matrix) and P is antisymmetric, under the condition that they commute [P,Q]=0 (plus some technical requirements).
The reconstruction can be achieved explicitly in terms of Theta functions on a
This "exercise" has interesting and potentially far-reaching applications in the study of the (formal) asymptotics of orthogonal polynomials (in particular
also semiclassical ones).
This last is part of an ongoing research project and may not be completely
detailed: in particular it applies to a much wider setting, including but not
limited to biorthogonal polynomials for the two-matrix model.
I plan to give the essentials of the tools
As a follow-up to last week's talk, I will show how spinors and Szego kernels
appear quite naturally in the (formal) asymptotics of (bi)orthogonal
polynomials, reproducing formulae obtained by Deift et al for the (strong)
asymptotics of orthogonal polynomials.
This is a continuation of the talk of Dec. 1
(see abstract above).
All the relevant quantities necessary to demonstrate such determinantal identities will be expressed as fermionic VOV's; including: Vandermonde determinants, Schur functions, partition functions; determinantal correlation functions; biothogonal polynomials, Christoffel Darboux kernels, and second type (Hilbert transform) solutions.