Research

My research program is concerned with analytic and probabilistic questions in mathematical physics and I place particular emphasis on topics in random matrix theory which display intimate connections to mathematical statistical mechanics and the field of integrable differential equations. The application of asymptotic methods, special functions, probability theory, orthogonal polynomials and potential theory is central to this work. Current areas of interest include:

1) Random matrix theory and the theory of random processes. In a nutshell, my published results in this field are concerned with the

  • analysis of gap, distribution and correlation functions in invariant random matrix ensembles and thinned versions thereof.

  • description of extreme values in non-Hermitian random matrix models

  • identification of non-generic universality classes in Hermitian one- or multi-matrix models

  • spectral analysis of integrable integral operators

  • development of Hamiltonian approach to the analysis of gap asymptotics

In the four figures below we plot the eigenvalues for 1000 real Ginibre matrices of varying dimensions n by n in comparison with the unit circle boundary. We display n=4,8,16,32 from top left to bottom right. Observe the saturn effect on the real line, this peculiar phenomenon was quantified in my recent work.

2) Exactly solvable models in statistical mechanics. I have derived results for

  • the six-vertex model with domain wall boundary conditions: computation of free energy and subleading terms for the partition function on the separating lines and resolution of Zinn-Justin's conjecture on phase transitions

  • the 2D Ising model: elementary derivation of the scaling function constant in the short distance expansion of the tau-function associated with the 2-point function

3) Integrable differential equations. Most of my work in this field is concerned with Painleve special functions:

  • unified asymptotic description of two-parameter families to real-valued solutions of Painleve II

  • introduction of Schur/orthogonal polynomial method to the analysis or rational Painleve functions

  • development of nonlinear steepest descent techniques for singular Painleve transcendents

  • total Painleve integral evaluations

  • systematic introduction of integro-differential Painleve equations

In the picture below we plot a certain one-parameter family of solutions to the second Painleve equation. This family is important in random matrix theory and we notice its sensitive dependence on the parameter for large negative values of the independent variable.

Recent preprints as well as published work can be found on arXiv and MathSciNet as well as ORCID, see also the publication list on this homepage.