The Cauchy Two-Matrix Model

We introduce a new class of two(multi)-matrix models of positive Hermitian 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 Hermitian matrix model is related to a hyperelliptic curve. Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). Work supported in part by NSF Grant DMD-0400484. Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), Grant. No. 138591-04.     

Cauchy–Laguerre 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.     

Poisson Structures Compatible with the Cluster Algebra Structure in Grassmannians

The present paper is a first step toward establishing connections between solutions of the classical Yang–Baxter equations and cluster algebras. We describe all Poisson brackets compatible with the natural cluster algebra structure in the open Schubert cell of the Grassmannian G _k (n) and show that any such bracket endows G _k (n) with a structure of a Poisson homogeneous space with respect to the natural action of SL _n equipped with an R-matrix Poisson–Lie structure. The corresponding R-matrices belong to the simplest class in the Belavin–Drinfeld classification. Moreover, every compatible Poisson structure can be obtained this way.     

On the orthogonal polynomials in several variables

We study the connection between orthogonal polynomials in several variables and families of commuting symmetric operators of a special form.     

Asymptotics of the eigenvalues of the dirichlet and neumann problems for the abstract sturm-liouville operator

Let A>0 be an unbounded self-adjoint operator in a Hilbert space H. In the Hilbert space H₁=L₂ (0, π; H) we study the spectrum of the differential equations−y″(x)+Ay=λy, y (0)=y(π)=0,−y″(x)+Ay=λy, y′(0) =y′(π)=0. We find the principal terms of the asymptotics of the functions N(λ) for these problems and we ascertain the conditions under which they are asymptotically not equivalent. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 209–212, February, 1977.     

Self-adjoint abstract differential operators

Let H be an abstract separable Hilbert space. We will consider the Hilbert space H₁ whose elements are functionsf(x) with domain H and we will also consider the set of self-adjoint operators Q(x) in H of the form Q(x)=A+B(x). In this formula AE, B(x)0, and the operator B(x) is bounded for all x. An operator L₀ is defined on the set of finite, infinitely differentiable (in the strong sense) functions y(x) H₁ according to the formula: L₀y=–y + Q(x)y (–0 is a self-adjoint operator in H₁ under the given assumptions.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 65–72, July, 1969.