On Fredholm Pfaffians and Riemann-Hilbert problems

• Amari Jaconelli (University of Bristol)

In this talk, I will illustrate how classes of Fredholm Pfaffians, which can describe the eigenvalue distributions for orthogonal and symplectic matrix ensembles, can be computed in terms of canonical, auxiliary Riemann-Hilbert problems. This is done by first algebraically manipulating the kernel of the Fredholm Pfaffian and then relating the resulting Pfaffian to a Riemann-Hilbert problem, assuming its main kernel is either of additive Hankel composition or truncated Wiener-Hopf type. One can then find asymptotic results for the Fredholm Pfaffian as a natural consequence of the Rieman-Hilbert problem characterisation.'

Discrete N-particle ensembles at high temperature and Jack polynomials

• Cesar Cuenca (Ohio State University)

Following a brief discussion of the Gaussian beta ensemble and the classical LLN of its empirical measures in the fixed and high temperature limit regimes, we switch to the discrete setting. By using Fourier transforms based on Jack symmetric polynomials, we study discrete particle ensembles $x_1>\dots>x_N$ with the inverse temperature theta in the regime where theta tends to zero, as the number N of particles tends to infinity. We prove the LLN and characterize the limiting measure in terms of a moment problem. For fixed-time distributions of multiparameter families of Markov chains of N non-intersecting particles (discrete versions of the Dyson Brownian motion), the density of the limiting measures is calculated and expressed in terms of the eigenvalues of certain Jacobi operators, or the countable real zeros of certain special functions. The talk is based on joint work with Maciej Dolega.