10 research outputs found
Moment determinants as isomonodromic tau functions
We consider a wide class of determinants whose entries are moments of the
so-called semiclassical functionals and we show that they are tau functions for
an appropriate isomonodromic family which depends on the parameters of the
symbols for the functionals. This shows that the vanishing of the tau-function
for those systems is the obstruction to the solvability of a Riemann-Hilbert
problem associated to certain classes of (multiple) orthogonal polynomials. The
determinants include Haenkel, Toeplitz and shifted-Toeplitz determinants as
well as determinants of bimoment functionals and the determinants arising in
the study of multiple orthogonality. Some of these determinants appear also as
partition functions of random matrix models, including an instance of a
two-matrix model.Comment: 24 page
Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two--matrix model
We apply the nonlinear steepest descent method to a class of 3x3
Riemann-Hilbert problems introduced in connection with the Cauchy two-matrix
random model. The general case of two equilibrium measures supported on an
arbitrary number of intervals is considered. In this case, we solve the
Riemann-Hilbert problem for the outer parametrix in terms of sections of a
spinorial line bundle on a three-sheeted Riemann surface of arbitrary genus and
establish strong asymptotic results for the Cauchy biorthogonal polynomials.Comment: 31 pages, 12 figures. V2; typos corrected, added reference
Cauchy Biorthogonal Polynomials
The paper investigates the properties of certain biorthogonal polynomials
appearing in a specific simultaneous Hermite-Pade' approximation scheme.
Associated to any totally positive kernel and a pair of positive measures on
the positive axis we define biorthogonal polynomials and prove that their
zeroes are simple and positive. We then specialize the kernel to the Cauchy
kernel 1/{x+y} and show that the ensuing biorthogonal polynomials solve a
four-term recurrence relation, have relevant Christoffel-Darboux generalized
formulae, and their zeroes are interlaced. In addition, these polynomial solve
a combination of Hermite-Pade' approximation problems to a Nikishin system of
order 2. The motivation arises from two distant areas; on one side, in the
study of the inverse spectral problem for the peakon solution of the
Degasperis-Procesi equation; on the other side, from a random matrix model
involving two positive definite random Hermitian matrices. Finally, we show how
to characterize these polynomials in term of a Riemann-Hilbert problem.Comment: 38 pages, partially replaces arXiv:0711.408
The Cauchy two-matrix model
We introduce a new class of two(multi)-matrix models of positive Hermitean
matrices coupled in a chain; the coupling is related to the Cauchy kernel and
differs from the exponential coupling more commonly used in similar models. The
correlation functions are expressed entirely in terms of certain biorthogonal
polynomials and solutions of appropriate Riemann-Hilbert problems, thus paving
the way to a steepest descent analysis and universality results. The
interpretation of the formal expansion of the partition function in terms of
multicolored ribbon-graphs is provided and a connection to the O(1) model. A
steepest descent analysis of the partition function reveals that the model is
related to a trigonal curve (three-sheeted covering of the plane) much in the
same way as the Hermitean matrix model is related to a hyperelliptic curve.Comment: 34 pages, 2 figures. V2: changes only to metadat
Cubic String Boundary Value Problems and Cauchy Biorthogonal Polynomials
Cauchy Biorthogonal Polynomials appear in the study of special solutions to the dispersive nonlinear partial differential equation called the Degasperis-Procesi (DP) equation, as well as in certain two-matrix random matrix models. Another context in which such biorthogonal polynomials play a role is the cubic string; a third order ODE boundary value problem −f ′′ ′ = zgf which is a generalization of the inhomogeneous string problem studied by M.G. Krein. A general class of such boundary value problems going beyond the original cubic string problem associated with the DP equation is discussed under the assumption that the source of inhomogeneity g is a discrete measure. It is shown that by a suitable choice of a generalized Fourier transform associated to these boundary value problems one can establish a Parseval type identity which aligns Cauchy biorthogonal polynomials with certain natural orthogonal systems on L 2 g
Cauchy\u2013Laguerre two-matrix model and the Meijer-G random point field
We apply the general theory of Cauchy biorthogonal polynomials developed in Bertola et al. (Commun Math Phys 287(3):983-1014, 2009) and Bertola et al. (J Approx Th 162(4):832-867, 2010) to the case associated with Laguerre measures. In particular, we obtain explicit formulae in terms of Meijer-G functions for all key objects relevant to the study of the corresponding biorthogonal polynomials and the Cauchy two-matrix model associated with them. The central theorem we prove is that a scaling limit of the correlation functions for eigenvalues near the origin exists, and is given by a new determinantal two-level random point field, the Meijer-G random field. We conjecture that this random point field leads to a novel universality class of random fields parametrized by exponents of Laguerre weights. We express the joint distributions of the smallest eigenvalues in terms of suitable Fredholm determinants and evaluate them numerically. We also show that in a suitable limit, the Meijer-G random field converges to the Bessel random field and hence the behavior of the eigenvalues of one of the two matrices converges to the one of the Laguerre ensemble. \ua9 2013 Springer-Verlag Berlin Heidelberg