The Pfaff lattice and skew-orthogonal polynomials (original) (raw)
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The Pfaff lattice on symplectic matrices
Journal of Physics A: Mathematical and Theoretical, 2010
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 recent paper [Intern. Math. Res. Notices, (2007) rnm120], we studied the Pfaff lattice hierarchy for the case where the Lax matrix is defined to be a lower Hessenberg matrix. In this paper we deal with the case of a symplectic lower Hessenberg Lax matrix, this forces the Lax matrix to take a tridiagonal shape. We then show that the odd members of the Pfaff lattice hierarchy are trivial, while the even members are equivalent to the indefinite Toda lattice hierarchy defined in [Y. Kodama and J. Ye, Physica D, 91 (1996) 321-339]. This is analogous to the case of the Toda lattice hierarchy in the relation to the Kac-van Moerbeke system. In the case with initial matrix having only real or imaginary eigenvalues, the fixed points of the even flows are given by 2 × 2 block diagonal matrices with zero diagonals. We also consider a family of skew-orthogonal polynomials with symplectic recursion relation related to the Pfaff lattice, and find that they are succinctly expressed in terms of orthogonal polynomials appearing in the indefinite Toda lattice.
Geometry of the Pfaff Lattices
International Mathematics Research Notices, 2010
The (semi-infinite) Pfaff lattice was introduced by Adler and van Moerbeke [2] to describe the partition functions for the random matrix models of GOE and GSE type. The partition functions of those matrix models are given by the Pfaffians of certain skew-symmetric matrices called the moment matrices, and they are the τ -functions of the Pfaff lattice. In this paper, we study a finite version of the Pfaff lattice equation as a Hamiltonian system. In particular, we prove the complete integrability in the sense of Arnold-Liouville, and using a moment map, we describe the real isospectral varieties of the Pfaff lattice. The image of the moment map is a convex polytope whose vertices are identified as the fixed points of the flow generated by the Pfaff lattice.
Partition functions for matrix models and isomonodromic tau functions
Journal of Physics A: Mathematical and General, 2003
We derive the explicit relationship between the partition function for (generalized) one-matrix models with polynomial potentials and the isomonodromic tau function for the 2 × 2 polynomial differential system satisfied by the associated orthogonal polynomials.
On tau functions for orthogonal polynomials and matrix models
Journal of Physics A: Mathematical and Theoretical, 2011
Let v be a real polynomial of even degree, and let ρ be the equilibrium probability measure for v with support S; so that, v(x) ≥ 2 log |x−y| ρ(dy)+C v for some constant C v with equality on S. Then S is the union of finitely many bounded intervals with endpoints δ j , and ρ is given by an algebraic weight w(x) on S. The system of orthogonal polynomials for w gives rise to the Magnus-Schlesinger differential equations. This paper identifies the τ function of this system with the Hankel determinant det[ x j+k ρ(dx)] n−1 j,k=0 of ρ. The solutions of the Magnus-Schlesinger equations are realised by a linear system, which is used to compute the tau function in terms of a Gelfand-Levitan equation. The tau function is associated with a potential q and a scattering problem for the Schrödinger operator with potential q. For some algebro-geometric q, the paper solves the scattering problem in terms of linear systems. The theory extends naturally to elliptic curves and resolves the case where S has exactly two intervals. MSC (2000) classification: 60B20 (37K15)
Skew Young Diagram Method in Spectral Decomposition of Integrable Lattice Models
Communications in Mathematical Physics, 1997
The spectral decomposition of the path space of the vertex model associated to the vector representation of the quantized affine algebra U q (sl n) is studied. We give a one-to-one correspondence between the spin configurations and the semistandard tableaux of skew Young diagrams. As a result we obtain a formula of the characters for the degeneracy of the spectrum in terms of skew Schur functions. We conjecture that our result describes the sl n-module contents of the Yangian Y (sl n)module structures of the level 1 integrable modules of the affine Lie algebra sl n. An analogous result is obtained also for a vertex model associated to the quantized twisted affine algebra U q (A (2) 2n), where Y (B n) characters appear for the degeneracy of the spectrum. The relation to the spectrum of the Haldane-Shastry and the Polychronakos models are also discussed.
Skew-Orthogonal Polynomials in the Complex Plane and Their Bergman-Like Kernels
Communications in Mathematical Physics, 2021
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 skew-orthogonal polynomials. We develop their theory in providing an explicit construction of skew-orthogonal polynomials in terms of orthogonal polynomials that satisfy a three-term recurrence relation, for general weight functions in the complex plane. New examples for symplectic ensembles are provided, based on recent developments in orthogonal polynomials on planar domains or curves in the complex plane. Furthermore, Bergman-like kernels of skew-orthogonal Hermite and Laguerre polynomials are derived, from which the conjectured universality of the elliptic symplectic Ginibre ensemble and its chiral partner follow in the limit of strong non-Hermiticity at the origin. A Christoffel perturbation of skew-orthogonal polynomials as it appears in applications to quantum field theory is provided.
Kloosterman integrals for skew symmetric matrices
Pacific Journal of Mathematics, 1992
If G is a reductive group quasi-split over a number field F and K the kernel of the trace formula, one can integrate K in the two variables against a generic character of a maximal unipotent subgroup N to obtain the Kuznietsov trace formula. If H is the fixator of an involution of G, one can also integrate K in one variable over H and in the other variable against a generic character of N: one obtains then a "relative" version of the Kuznietsov trace formula. We propose as a conjecture that the relative Kuznietsov trace formula can be "matched" with the Kuznietsov trace formula for another group G'. A consequence of this formula would be the characterization of the automorphic representations of G which admit an element whose integral over H is non-zero: they should be functorial image of representations of G'. In this article, we study the case where H is the symplectic group inside the linear group; we prove the "fundamental lemma" for the situation at hand and outline the identity of the trace formulas. This case is elementary and should serve as a model for the general case.
Semiclassical Orthogonal Polynomials, Matrix Models and Isomonodromic Tau Functions
Communications in Mathematical Physics, 2006
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 probabilities. The associated isomonodromic tau functions are shown to coincide, within an explicitly computed factor, with these partition functions.
Unitary Matrix Integrals in the Framework of the Generalized Kontsevich Model
International Journal of Modern Physics A, 1996
We advocate a new approach to the study of unitary matrix models in external fields which emphasizes their relationship to generalized Kontsevich models (GKM's) with nonpolynomial potentials. For example, we show that the partition function of the Brezin–Gross–Witten model (BGWM), which is defined as an integral over unitary N × N matrices, [Formula: see text], can also be considered as a GKM with potential [Formula: see text]. Moreover, it can be interpreted as the generating functional for correlators in the Penner model. The strong and weak coupling phases of the BGWM are identified with the "character" (weak coupling) and "Kontsevich" (strong coupling) phases of the GKM, respectively. This type of GKM deserves classification as a p = −2 model (i.e. c = 28 or c = −2) when in the Kontsevich phase. This approach allows us to further identify the Harish-Chandra–Itzykson–Zuber integral with a peculiar GKM, which arises in the study of c = 1, theory, and, furth...
In this article, we define and study a geometry on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an integer parameter NNN. Then we emulate the theory of random matrices in a combinatorial framework: for any parameter NNN, we introduce a family of linear forms on the partition algebras which allows us to define a notion of weak convergence similar to the convergence in moments in random matrices theory. A renormalization of the partition algebras allows us to consider the weak convergence as a simple convergence in a fixed space. This leads us to the definition of a deformed partition algebra for any integer parameter NNN and to the definition of two transforms: the cumulants transform and the exclusive moments transform. Using an improved triangular inequality for the distance defined on partitions, we prove that the deformed partition algebras, endowed with a deformation of the linear forms converge as NNN go to infinity. This result allows us to prove combinatorial properties about geodesics and a convergence theorem for semi-groups of functions on partitions. At the end we study a sub-algebra of functions on infinite partitions with finite support : a new addition operation and a notion of mathcalR\mathcal{R}mathcalR-transform are defined. We introduce the set of multiplicative functions which becomes a Lie group for the new addition and multiplication operations. For each of them, the Lie algebra is studied. The appropriate tools are developed in order to understand the algebraic fluctuations of the moments and cumulants for converging sequences. This allows us to extend all the results we got for the zero order of fluctuations to any order.