Sung-Soo Byun

Spectral moments and superintegrability of the Gaussian beta ensemble and its (q,t) generalisation

The spectral moments of random matrices play a key role in understanding eigenvalue statistics. In this talk, I will discuss how the spectral moments of the Gaussian beta ensemble or its (q,t) generalization can be evaluated using the theory of symmetric polynomials, typically the Jack or Macdonald polynomials. In particular, I will explain the superintegrability conjecture, which relates to the average over a distinguished basis of symmetric functions, that Peter J. Forrester and I recently proved using a theory of multivariable Al-Salam and Carlitz polynomials based on Macdonald polynomials.