The Pfaff lattice and skew-orthogonal polynomials

@article{Adler1999ThePL,
  title={The Pfaff lattice and skew-orthogonal polynomials},
  author={Mark Adler and Emil Horozov and Pierre van Moerbeke},
  journal={International Mathematics Research Notices},
  year={1999},
  volume={1999},
  pages={569-588}
}
Consider a semi-inflnite skew-symmetric moment matrix, m1 evolving according to the vector flelds @m=@tk = ⁄ k m + m⁄ >k ; where ⁄ is the shift matrix. Then The skew-Borel decomposition m1 := Q i1 JQ >i1 leads to the so-called Pfafi Lattice, which is integrable, by virtue of the AKS theorem, for a splitting involving the a‐ne symplectic algebra. The tau-functions for the system are shown to be pfa‐ans and the wave vectors skew-orthogonal polynomials; we give their explicit form in terms of… 

Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions

Skew-orthogonal polynomials (SOPs) arise in the study of the n-point distribution function for orthogonal and symplectic random matrix ensembles. Motivated by the average of characteristic

Rational solutions to the Pfaff lattice and Jack polynomials

The finite Pfaff lattice is given by a commuting Lax pair involving a finite matrix L (zero above the first subdiagonal) and a projection onto sp(N). The lattice admits solutions such that the

Toda versus Pfaff lattice and related polynomials

The Pfaff lattice was introduced by us in the context of a Lie algebra splitting of gl (infinity) into sp (infinity) and an algebra of lower-triangular matrices. The Pfaff lattice is equivalent to a

Geometry of the Pfaff lattices

Pfaff lattice was introduced by Adler and van Moerbeke to describe the partition functions for the random matrix models of GOE and GSE type. The partition functions of those matrix models are given

Skew-Orthogonal Polynomials in the Complex Plane and Their Bergman-Like Kernels

Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of

The Pfaff lattice, Matrix integrals and a map from Toda to Pfaff

We study the Pfaff lattice, introduced by us in the context of a Lie algebra splitting of gl(infinity) into sp(infinity) and lower-triangular matrices. We establish a set of bilinear identities,

Pfaff τ -functions

Consider the two-dimensional Toda lattice, with certain skew-symmetric initial condition, which is preserved along the locus s =− t of the space of time variables. Restricting the solution to s =− t,

The Pfaff lattice on symplectic matrices

The Pfaff lattice is an integrable system arising from the SR-group factorization in an analogous way to how the Toda lattice arises from the QR-group factorization. In our earlier paper (Kodama and

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,

Classical discrete symplectic ensembles on the linear and exponential lattice: skew orthogonal polynomials and correlation functions

The eigenvalue probability density function for symplectic invariant random matrix ensembles can be generalized to discrete settings involving either a linear or an exponential lattice. The
...

References

SHOWING 1-10 OF 19 REFERENCES

The Pfaff lattice, Matrix integrals and a map from Toda to Pfaff

We study the Pfaff lattice, introduced by us in the context of a Lie algebra splitting of gl(infinity) into sp(infinity) and lower-triangular matrices. We establish a set of bilinear identities,

Symmetric random matrices and the Pfaff lattice

Consider a symmetric (finite) matrix ensemble, with a certain probability distribution. What is the probability that the spectrum belongs to a certain interval or union of intervals on the real line?

Infinite dimensional Lie algebras: Frontmatter

Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials

into the algebras of skew-symmetric As and lower triangular (including the diagonal) matrices Ab (Borel matrices). We show that this splitting plays a prominent role also in the construction of the

The Spectrum of coupled random matrices

The study of the spectrum of coupled random matrices has received rather little attention. To the best of our knowledge, coupled random matrices have been studied, to some extent, by Mehta. In this

Group factorization, moment matrices, and Toda lattices

∗appears in: International Mathematics Research notices, 12 (1997), received 4 september 1996. †Department of Mathematics, Brandeis University, Waltham, Mass 02254, USA (adler@math.brandeis.edu). The

The geometry of spinors and the multicomponent BKP and DKP hierarchies

We develop a formalism of multicomponent BKP hierarchies using elementary geometry of spinors. The multicomponent KP and the modified KP hierarchy (hence all their reductions like KdV, NLS, AKNS or

Random matrices, 2nd ed

  • 1991