Discrete N-particle ensembles at high temperature through Jack symmetric functions

Following a 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 $x_1 > x_2 > \dots > x_N$ with the inverse temperature theta in the regime where theta tends to zero, as the number 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 limiting measures can be expressed in terms of the eigenvalues of certain Jacobi operators. The talk is based on joint work with Maciej Dolega.