Our monthly meeting will take place on Monday the 26th of April, from 4pm to 6pm (Central European Time). Our speakers will be Guido Mazzuca and Andrei Prokhorov. The zoom link will be sent to the members of the project through the mailing list. If you are interested in participating and you are not a PIICQ participant, please register sending us an email!
First talk by Guido Mazzuca at 4pm
Title: High temperature regime for the classical beta ensembles: new connection with Integrable Systems.
Abstract: In this talk I will mainly focus on the Gaussian and Circular beta ensembles, two classical random matrix models. After a brief introduction, I will define the high temperature regime for these two systems. We will see that in this regime they are related to the Gibbs ensembles of the Toda and the Ablowitz-Ladik systems. Specifically, I will show that the mean density of states of the Toda and the Ablowitz-Ladik lattices can be computed explicitly from the one of the Gaussian and Circular beta ensembles in the high temperature regime. This talk is partially based on an ongoing project with Tamara Grava.
Second talk by Andrei Prokhorov at 5pm
Title: On beta=6 Tracy-Widom distribution and the second Calogero-Painlevé system.
Abstract: Painlevé equations can be interpreted as equations of motion for a particle in the time dependent potential. The Calogero-Painlevé systems are the generalization of the classical Painlevé equations to the case of the several particles in Painlevé potential coupled with the Calogero interaction. They are related to the quantization of Painlevé equations. Particular case of the second Calogero-Painlevé equation describes the Tracy-Widom distribution function for the general beta-ensembles with the even values of beta. Calogero-Painlevé systems admit a Riemann-Hilbert representation. It gives the possibility of rigorous analysis of the asymptotic behavior of the Tracy-Widom distributions for the values of beta beyond the classical beta=1, 2, 4. We present an asymptotic analysis of the Calogero-Painlevé system corresponding to beta = 6 Tracy-Widom distribution function. This is a joint work with Alexander Its.