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On the asymptotic analysis of the Calogero-Painleve systems and the Tracy-Widom$_\beta$ distribution for $\beta=6$
Probability| Speaker: | Alexander R. Its, IUPUI |
| Related Webpage: | https://math.iupui.edu/people/its-alexander |
| Location: | Zoom lecture |
| Start time: | Wed, Oct 14 2020, 4:10PM |
The Calogero-Painleve systems were introduce in 2001 by K.Takasaki as
a natural generalization of the classical Painleve equations to the case
of the several Painleve "particles'' coupled via the Calogero
type interactions. In 2014, I. Rumanov discovered a remarkable fact
that a particular case of the Calogero--Painleve II equation describes
the Tracy-Widom distribution function for the general
$\beta$-ensembles with the even values of parameter $\beta$. Recently,
in 2017 work of M. Bertola, M. Cafasso , and V. Rubtsov, it was proven that all
Calogero-Painleve systems are Lax integrable, and hence their
solutions admit a Riemann-Hilbert representation. This important
observation has opened the door to rigorous asymptotic analysis of the Calogero-Painleve
equations which in turn yields the possibility of rigorous
evaluation of the asymptotic behavior of the Tracy-Widom distributions
for the values of $\beta$ beyond the classical $\beta =1, 2, 4$.
In the talk these recent developments will be outlined with a special
focus on the Calogero-Painleve system corresponding to $\beta = 6$.
This is a joint work with Andrei Prokhorov.