Marco Bertola

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 V1, V2 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 V1, V2. 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 V1 or V2 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 V1 and V2. 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.

Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann–Hilbert Problem

AbstractWe consider biorthogonal polynomials that arise in the study of a generalization of two–matrix Hermitian models with two polynomial potentials V1(x), V2(y) of any degree, with arbitrary complex coefficients. Finite consecutive subsequences of biorthogonal polynomials (‘‘windows’’), of lengths equal to the degrees of the potentials V1 and V2, 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.

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 probabilities. The associated isomonodromic tau functions are shown to coincide, within an explicitly computed factor, with these partition functions.

Partition functions for matrix models and isomonodromic tau functions

For one-matrix models with polynomial potentials, the explicit relationship between the partition function and the isomonodromic tau function for the 2 × 2 polynomial differential systems satisfied by the associated orthogonal polynomials is derived.

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.

Free energy of the two-matrix model/dToda tau-function

Abstract We provide an integral formula for the free energy of the two-matrix model with polynomial potentials of arbitrary degree (or formal power series). This is known to coincide with the τ -function of the dispersionless two-dimensional Toda hierarchy. The formula generalizes the case of conformal maps of Jordan curves studied by Kostov, Krichever, Mineev-Weinstein, Wiegmann, Zabrodin and separately Takhtajan. Finally we generalize the formula found in genus zero to the case of spectral curves of arbitrary genus with certain fixed data.

Bilinear semiclassical moment functionals and their integral representation

We introduce the notion of bilinear moment functional and study their general properties. The analogue of Favards theorem for moment functionals is proven. The notion of semiclassical bilinear functionals is introduced as a generalization of the corresponding notion for moment functionals and motivated by the applications to multi-matrix random models. Integral representations of such functionals are derived and shown to be linearly independent.

The Dependence on the Monodromy Data of the Isomonodromic Tau Function

The isomonodromic tau function defined by Jimbo-Miwa-Ueno vanishes on the Malgrange’s divisor of generalized monodromy data for which a vector bundle is nontrivial, or, which is the same, a certain Riemann–Hilbert problem has no solution. In their original work, Jimbo, Miwa, Ueno provided an algebraic construction of its derivatives with respect to isomonodromic times. However the dependence on the (generalized) monodromy data (i.e. monodromy representation and Stokes’ parameters) was not derived. We fill the gap by providing a (simpler and more general) description in which all the parameters of the problem (monodromy-changing and monodromy-preserving) are dealt with at the same level. We thus provide variational formulæ for the isomonodromic tau function with respect to the (generalized) monodromy data. The construction applies more generally: given any (sufficiently well-behaved) family of Riemann–Hilbert problems (RHP) where the jump matrices depend arbitrarily on deformation parameters, we can construct a one-form Ω (not necessarily closed) on the deformation space (Malgrange’s differential), defined off Malgrange’s divisor. We then introduce the notion of discrete Schlesinger transformation: it means that we allow the solution of the RHP to have poles (or zeros) at prescribed point(s). Even if Ω is not closed, its difference evaluated along the original solution and the transformed one, is shown to be the logarithmic differential (on the deformation space) of a function. As a function of the position of the points of the Schlesinger transformation, it yields a natural generalization of the Sato formula for the Baker–Akhiezer vector even in the absence of a tau function, and it realizes the solution of the RHP as such BA vector. Some exemplifications in the setting of the Painlevé II equation and finite Töplitz/Hankel determinants are provided.

The Transition between the Gap Probabilities from the Pearcey to the Airy Process—a Riemann–Hilbert Approach

We consider the gap probability for the Pearcey and Airy processes; we set up a Riemann–Hilbert approach (different from the standard one) whereby the asymptotic analysis for large gap/large time of the Pearcey process is shown to factorize into two independent Airy processes using the Deift–Zhou steepest descent analysis. Additionally, we relate the theory of Fredholm determinants of integrable kernels and the theory of isomonodromic tau function. Using the Riemann–Hilbert problem mentioned above, we construct a suitable Lax pair formalism for the Pearcey gap probability and re-derive the two nonlinear PDEs recently found and additionally find a third one not reducible to those.

A general construction of conformal field theories from scalar anti-de Sitter quantum field theories

We provide a new general setting for scalar interacting fields on the covering of a ( d+1 )-dimensional AdS spacetime. The formalism is used at first to construct a one-parameter family of field theories, each living on a corresponding spacetime submanifold of AdS, which is a cylinder R×Sd−1 . We then introduce a limiting procedure which directly produces Luscher–Mack CFTs on the covering of the AdS asymptotic cone. Our generalized AdS → CFT construction is nonperturbative, and is illustrated by a complete treatment of two-point functions, the case of Klein–Gordon fields appearing as particularly simple in our context. We also show how the Minkowskian representation of these boundary CFTs can be directly generated by an alternative limiting procedure involving Minkowskian theories in horocyclic sections (nowadays called ( d− 1)-branes, 3-branes for AdS5 ). These theories are restrictions to the brane of the ambient AdS field theory considered. This provides a more general correspondence between the AdS field theory and a Poincare invariant QFT on the brane, satisfying all the Wightman axioms. The case of two-point functions is again studied in detail from this viewpoint as well as the CFT limit on the boundary.