Particles interacting via Coulomb interactions in an external field tend to condensate on a droplet as the number of particles increases. In this talk, I will discuss the boundary conditions that can be imposed on the boundaries of the droplets. While the free boundary condition allows particles to distribute outside the droplet, the hard edge condition forces particles to reside within a specific set, which leads to changes in the equilibrium density or the local statistics at the boundary. I will explain recent results on 2D Coulomb gas ensembles with hard edge boundary conditions.

The 2D Ising model is one of the most celebrated examples of an exactly solvable lattice model. Motivated by problems in statistical mechanics and 2D quantum gravity, in 1986 Vladimir Kazakov considered the Ising model on a random planar lattice using techniques from random matrix theory. He was able to derive a formula for the free energy of this model, and made the first prediction of the Kniznik-Polyakov-Zamolodchikov (KPZ) formula for the shift of the critical exponents of a conformal field theory when coupled to quantum gravity. Unfortunately, his derivation was not mathematically rigorous, and the formula he obtained for the free energy was somewhat unwieldy. In this talk, I will review some of the details regarding both the Ising model and random matrices, and sketch a rigorous proof of Kazakov’s formula for the free energy. I will also present a parametric formula for the free energy, which seems to be new..

Coulomb gases consist of $n$ particles repelling each other via the 2D Coulomb law and subject to the presence of an external potential. In this talk, I will discuss recent results on the probability that a given subset of the plane is free from particles when $n$ is large. I will also discuss the most likely point configurations (“far from equilibrium”) which produce such holes.

The quantum transport problem for $1$ dimensional disordered wires can be modeled by the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation that is similar to the Dyson Brownian motion, and if the time-reversal symmetry is broken, the DMPK equation has a free fermion solution, which is, after taking the metallic limit, a biorthogonal ensemble. The biorthogonal ensemble has the form $$\prod_{1 \leq i<j \leq n} (x_i - x_j)(f(x_i) - f(x_j)) \prod^n_{i = 1} x^{\alpha}_i e^{-nV(x_i)}, \quad f(x) = \sinh^2(\sqrt{x}).$$ It is a determinantal point process, and the correlation kernel can be expressed by biorthogonal polynomials. In this talk we discuss an approach to the Plancherel-Rotach type asymptotics of the biorthogonal polynomials by vector Riemann-Hilbert problems. We also discuss its relation to the universal conductance fluctuations.

We will report on recent progress regarding the universality of the extremal eigenvalues of a large random matrix with i.i.d. entries. Beyond the radius of the celebrated circular law, we will establish a precise three-term asymptotic expansion for the largest eigenvalue (in modulus) with an optimal error term. Based on this result, we will further show that the properly normalized largest eigenvalue converges to a Gumbel distribution as the dimension goes to infinity. Furthermore, we also prove that the argument of the largest eigenvalue is uniform on the unit circle and that the extremal eigenvalues form a Poisson point process. Similar results also apply to the rightmost eigenvalue of the matrix. These results are based on several joint works with Giorgio Cipolloni, Laszlo Erdos, and Dominik Schroder.

In this talk, I will report my recent works on the asymptotic aspects of the Painlevé II hierarchy. Their applications in random matrix theory and integrable differential equations will also be discussed.

A significant advance in random matrix theory in recent years has been the development of a matrix transform theory based on spherical functions from harmonic analysis. One aspect has been the identification of the previously unknown Polya ensembles—intimately related to Polya frequency functions—which exhibit a key closure property of the functional form of their joint eigenvalue probability density function with respect to matrix addition or multiplication. They also form determinantal point processes constructed out of a special class of biorthogonal functions. The latter permit explicit forms in terms of sums or integrals, which moreover allow for the correlation kernel to be written in a double contour integral form, which is a key ingredient in subsequent asymptotic analysis. In this talk I will give a brief review on the theory for Polya ensembles, their constructions, the determinantal structures and their kernels.

As a fundamental concept in modern probability theory, universality asserts that the outcome of a system is largely independent of its specific structural details, provided there are sufficiently many different sources of randomness. In this talk, I will present recent progress on the universality principle in the context of the non-Hermitian random matrix theory. In particular, I will introduce the local universality problem of the planar symplectic ensembles and present my contributions to this topic.