$\beta$-ensembles and higher general Catalan numbers
A talk in the Seminar Zufallsmatrizen series by
Luca Cassia
| Abstract: | In this talk I will show how to derive formulas for the large $N$
expansion of the generating function of connected correlators of the
$\beta$-deformed Gaussian and Wishart-Laguerre matrix models. I will
show that these formulas satisfy the known transformation
properties under the exchange of $\beta$ with $1/ \beta$ and, using Virasoro
constraints, I will derive a recursion formula for the coefficients of
the expansion. In the undeformed limit $\beta=1$, these coefficients are
integers and they have the combinatorial interpretation of
generalized Catalan numbers. For generic $\beta$, we are led to define the
notion of higher genus Catalan polynomials whose coefficients are
integer numbers. Based on arXiv:2310.05626. Please contact Lucas Hackl (Lucas.Hackl@unimelb.edu.au) for details regarding access Within the CRC this talk is associated to the project(s): C6 |