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J. Harnad

Publications and source records attributed to J. Harnad.

At least 55 records · Page 3Linked to original sources
arXiv2006

Integrals of rational symmetric functions, two matrix models and biorthogonal polynomials

We give a new method for the evaluation of a class of integrals of rational symmetric functions in N pairs of variables {x_a, y_a}_{a=1,... N} arising in coupled matrix models, valid for a broad class of two-variable measures. The result is expressed as the determinant of a matrix whose entries consist of the associated biorthogonal polynomials, their Hilbert transforms, evaluated at the zeros and poles of the integrand, and bilinear expressions in these. The method is elementary and direct, using only standard determinantal identities, partial fraction expansions and the property of biorthogonality. The corresponding result for one-matrix models and integrals of rational symmetric functions in N variables {x_a}_{a=1,... N} is also rederived in a simple way using this method

math-ph↗
arXiv2006

Superintegrability, Lax matrices and separation of variables

We show how the superintegrability of certain systems can be deduced from the presence of multiple parameters in the rational Lax matrix representation. This is also related to the fact that such systems admit a separation of variables in parametric families of coordinate systems.

nlin.SI↗
arXiv2005

Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions

A new representation of the 2N fold integrals appearing in various two-matrix models that admit reductions to integrals over their eigenvalues is given in terms of vacuum state expectation values of operator products formed from two-component free fermions. This is used to derive the perturbation series for these integrals under deformations induced by exponential weight factors in the measure, expressed as double and quadruple Schur function expansions, generalizing results obtained earlier for certain two-matrix models. Links with the coupled two-component KP hierarchy and the two-component Toda lattice hierarchy are also derived.

math-ph↗
arXiv2005

Constrained Reductions of 2D dispersionless Toda Hierarchy, Hamiltonian Structure and Interface Dynamics

Finite-dimensional reductions of the 2D dispersionless Toda hierarchy, constrained by the ``string equation'' are studied. These include solutions determined by polynomial, rational or logarithmic functions, which are of interest in relation to the ``Laplacian growth'' problem governing interface dynamics. The consistency of such reductions is proved, and the Hamiltonian structure of the reduced dynamics is derived. The Poisson structure of the rationally reduced dispersionless Toda hierarchies is also derived

math-ph↗
arXiv2004

Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions

The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such measures. These are shown to preserve the generalized monodromy of the associated rank-2 rational covariant derivative operators. The corresponding matrix models, consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed, and it is shown that all coefficients of the associated spectral curves are given by logarithmic derivatives of the partition function or, more generally, the gap probablities. The associated isomonodromic tau functions are shown to coincide, within an explicitly computed factor, with these partition functions.

nlin.SI↗
arXiv2004

Janossy densities, multimatrix spacing distributions and Fredholm resolvents

A simple proof is given for a generalized form of a theorem of Soshnikov. The latter states that the Janossy densities for multilevel determinantal ensembles supported on measurable subspaces of a set of measure spaces are constructed by dualization of bases on dual pairs of N-dimensional function spaces with respect to a pairing given by integration on the complements of the given measurable subspaces. The generalization extends this to dualization with respect to measures modified by arbitrary sets of weight functions.

math-ph↗
arXiv2003

On the bilinear equations for Fredholm determinants appearing in random matrices

It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space variables introduced by Tracy and Widom. The essential step is to recast the latter as isomonodromic deformation equations for families of rational covariant derivative operators on the Riemann sphere and interpret the Fredholm determinants as isomonodromic $τ$-functions.

solv-int↗
arXiv2002

Scalar products of symmetric functions and matrix integrals

We present relations between Hirota-type bilinear operators, scalar products on spaces of symmetric functions and integrals defining matrix model partition functions. Using the fermionic Fock space representation, a proof of the expansion of an associated class of KP and 2-Toda tau functions $τ_{r,n}$ in a series of Schur functions generalizing the hypergeometric series is given and related to the scalar product formulae. It is shown how special cases of such $τ$-functions may be identified as formal series expansions of partition functions. A closed form exapnsion of $\log τ_{r,n}$ in terms of Schur functions is derived.

nlin.SI↗
arXiv2002

Multi-Hamiltonian structures for r-matrix systems

For the rational, elliptic and trigonometric r-matrices, we exhibit the links between three "levels" of Poisson spaces: (a) Some finite-dimensional spaces of matrix-valued holomorphic functions on the complex line; (b) Spaces of spectral curves and sheaves supported on them; (c) Symmetric products of a surface. We have, at each level, a linear space of compatible Poisson structures, and the maps relating the levels are Poisson. This leads in a natural way to Nijenhuis coordinates for these spaces. At level (b), there are Hamiltonian systems on these spaces which are integrable for each Poisson structure in the family, and which are such that the Lagrangian leaves are the intersections of the symplective leaves over the Poisson structures in the family. Specific examples include many of the well-known integrable systems.

math-ph↗
arXiv2002

Matrix integrals as Borel sums of Schur function expansions

The partition function for unitary two matrix models is known to be a double KP tau-function, as well as providing solutions to the two dimensional Toda hierarchy. It is shown how it may also be viewed as a Borel sum regularization of divergent sums over products of Schur functions in the two sequences of associated KP flow variables.

nlin.SI↗
arXiv2002

Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem

We consider biorthogonal polynomials that arise in the study of a generalization of two--matrix Hermitian models with two polynomial potentials V_1(x), V_2(y) of any degree, with arbitrary complex coefficients. Finite consecutive subsequences of biorthogonal polynomials (`windows'), of lengths equal to the degrees of the potentials, satisfy systems of ODE's with polynomial coefficients as well as PDE's (deformation equations) with respect to the coefficients of the potentials and recursion relations connecting consecutive windows. A compatible sequence of fundamental systems of solutions is constructed for these equations. The (Stokes) sectorial asymptotics of these fundamental systems are derived through saddle-point integration and the Riemann-Hilbert problem characterizing the differential equations is deduced.

nlin.SI↗
arXiv2001

Duality of spectral curves arising in two-matrix models

The two matrix model is considered, with measure given by the exponential of a sum of polynomials in two different variables. It is shown how to derive a sequence of pairs of ``dual'' finite size systems of ODEs for the corresponding biorthonormal polynomials. An inverse theorem is proved showing how to reconstruct such measures from pairs of semi-infinite finite band matrices defining the recursion relations and satisfying the string equation. A proof is given in the $N\to \infty$ limit that the dual systems obtained share the same spectral curve.

nlin.SI↗
arXiv2001

Duality, Biorthogonal Polynomials and Multi-Matrix Models

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 V_1, V_2 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 V_1, V_2. 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 V_1 or V_2 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 V_1 and V_2. 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.

nlin.SI↗
arXiv1999

Picard-Fuchs Equations, Hauptmoduls and Integrable Systems

The Schwarzian equations satisfied by certain Hauptmoduls (i.e., uniformizing functions for Riemann surfaces of genus zero) are derived from the Picard-Fuchs equations for families of elliptic curves and associated surfaces. The inhomogeneous Picard-Fuchs equations associated to elliptic integrals with varying endpoints are derived and used to determine solutions of equations that are algebraically related to a class of Painlevé VI equations.

solv-int↗
arXiv1999

Modular Invariants and Generalized Halphen Systems

Generalized Halphen systems are solved in terms of functions that uniformize genus zero Riemann surfaces, with automorphism groups that are commensurable with the modular group. Rational maps relating these functions imply subgroup relations between their automorphism groups and symmetrization relations between the associated differential systems.

solv-int↗
arXiv1998

Modular Solutions to Equations of Generalized Halphen Type

Solutions to a class of differential systems that generalize the Halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group. These functions all uniformize Riemann surfaces of genus zero and have $q$--series with integral coefficients. Rational maps relating these functions are derived, implying subgroup relations between their automorphism groups, as well as symmetrization maps relating the associated differential systems.

solv-int↗
arXiv1997

Bispectral Operators, Dual Isomonodromic Deformations and the Riemann-Hilbert Dressing Method

A comparison is made between bispectral systems and dual isomonodromic deformation equations. A number of examples are given, showing how bispectral systems may be embedded into isomonodromic ones. Sufficiency conditions are given for the construction of rational solutions of isomonodromic deformation equations through the Riemann-Hilbert problem dressing method, and these are shown, in certain cases, to reduce to bispectral systems.

solv-int↗