Non-Commutative Painlevé Equations and Hermite-Type Matrix Orthogonal Polynomials

We study double integral representations of Christoffel–Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the Its–Izergin–Korepin–Slavnov (IIKS) theory with a certain Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax pair whose compatibility conditions lead to a non-commutative version of the Painlevé IV differential equation for each family.     

Non-linear PDEs for gap probabilities in random matrices and KP theory

Airy and Pearcey-like kernels and generalizations arising in random matrix theory are expressed as double integrals of ratios of exponentials, possibly multiplied with a rational function. In this work it is shown that such kernels are intimately related to wave functions for polynomial (Gelfand–Dickey) reductions or rational reductions of the KP-hierarchy; their Fredholm determinant also satisfies linear PDEs (Virasoro constraints), yielding, in a systematic way, non-linear PDEs for the Fredholm determinant of such kernels. Examples include Fredholm determinants giving the gap probability of some infinite-dimensional diffusions, like the Airy process, with or without outliers, and the Pearcey process, with or without inliers.     

Riemann–Hilbert approach to multi-time processes: The Airy and the Pearcey cases

We prove that matrix Fredholm determinants related to multi-time processes can be expressed in terms of determinants of integrable kernels à la Its–Izergin–Korepin–Slavnov (IIKS) and hence related to suitable Riemann–Hilbert problems, thus extending the known results for the single-time case. We focus on the Airy and Pearcey processes. As an example of applications we re-deduce a third order PDE, found by Adler and van Moerbeke, for the two-time Airy process.     

Fredholm Determinants and Pole-free Solutions to the Noncommutative Painlevé II Equation

We extend the formalism of integrable operators à la Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a semi–infinite interval and to matrix integral operators with a kernel of the form \fracE₁^T(l) E₂(m)l+m{\frac{E_1^T(\lambda) E_2(\mu)}{\lambda+\mu}}, thus proving that their resolvent operators can be expressed in terms of solutions of some specific Riemann-Hilbert problems. We also describe some applications, mainly to a noncommutative version of Painlevé II (recently introduced by Retakh and Rubtsov) and a related noncommutative equation of Painlevé type. We construct a particular family of solutions of the noncommutative Painlevé II that are pole-free (for real values of the variables) and hence analogous to the Hastings-McLeod solution of (commutative) Painlevé II. Such a solution plays the same role as its commutative counterpart relative to the Tracy–Widom theorem, but for the computation of the Fredholm determinant of a matrix version of the Airy kernel.     

Borodin–Okounkov formula,string equation and topological solutions of Drinfeld–Sokolov hierarchies

Letters in Mathematical Physics - We give a general method to compute the expansion of topological tau functions for Drinfeld–Sokolov hierarchies associated with an arbitrary untwisted affine...     

Thermodynamic properties and ordering in liquid NaK alloys

The thermodynamic activities of sodium and potassium were determined by vapour-phase absorption spectrophotometry of atomic resonance lines. Both components exhibit significant positive deviations from ideality. Excess Gibbs energies computed from the activity data were combined with available heat-of-mixing data to yield values for excess entropies. The entropy of mixing which was found to be ideal in the sodium-rich liquid alloys provides no support for the earlier speculations about the existence of Na₂K molecules therein. The partial molar excess entropy of sodium in dilute solution in potassium was found to be surprisingly large and negative for a system of this type. A plausible model, involving sodium-sodium pairing, is proposed to account for the loss of configurational entropy.     

Block Toeplitz Determinants,Constrained KP and Gelfand-Dickey Hierarchies

We propose a method for computing any Gelfand-Dickey τ function defined on the Segal-Wilson Grassmannian manifold as the limit of block Toeplitz determinants associated to a certain class of symbols . Also truncated block Toeplitz determinants associated to the same symbols are shown to be τ functions for rational reductions of KP. Connection with Riemann-Hilbert problems is investigated both from the point of view of integrable systems and block Toeplitz operator theory. Examples of applications to algebro-geometric solutions are given.