Guido Mazzuca: Central limit theorem for the real beta ensemble in the high temperature regime, and the Toda lattice

Speaker
Guido Mazzuca, Tulane University

Abstract
In this talk, we present a central limit theorem (CLT) for the real beta ensemble in the high-temperature regime, as well as for the Toda lattice. Specifically, we consider the real beta ensemble with a polynomial potential in the regime β= 2α/N, where N represents the number of particles, and we examine the limit as N goes to infinity. In this setting, we establish a polynomial central limit theorem for the ensemble. Additionally, we study the Toda lattice, a well-known integrable system, with initial data sampled randomly according to a Generalized Gibbs Ensemble. In this regime, we also prove a polynomial central limit theorem for the Toda lattice. Importantly, we demonstrate a connection between the two central limit theorems, allowing us to explicitly compute several key quantities for the Toda lattice, including the expected values of conserved quantities, their currents, and spatial correlations. These results are essential for applying the theory of generalized hydrodynamics (GHD) to this model, a new framework that describes the space-time correlations of discrete integrable systems like the Toda lattice.
This talk is based on recent work in collaboration with Ronan Memin: CLT for β ensembles at high temperature, and for integrable systems: a transfer operator approach, Ann. Henri Poincaré (2024). DOI:10.1007/s00023-024-01435-0.