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BIRS Workshop Lecture Videos

The Kontsevich matrix integral and Painleve hierarchy; rigorous asymptotics and universality at the soft edges of the spectrum in random matrix theory Bertola, Marco


The Kontsevich integral is a matrix integral (aka "Matrix Airy function") whose logarithm, in the appropriate formal limit, generates the intersection numbers on $\mathcal M_{g,n}$. In the same formal limit it is also a particular tau function of the KdV hierarchy; truncation of the times yields thus tau functions of the first Painlev\'e\ hierarchy. This, however is a purely formal manipulation that pays no attention to issues of convergence. The talk will try to address two issues: Issue 1: how to make an analytic sense of the convergence of the Kontsevich integral to a tau function for a member of the Painlev\'e I hierarchy? Which particular solution(s) does it converge to? Where (for which range of the parameters)? Issue 2: it is known that (in fact for any $\beta$) the correlation functions of K points in the $GUE_\beta$ ensemble of size N are dual to the correlation functions of N points in the $GUE_{4/\beta}$ of size $K$. For $\beta=2$ they are self-dual. Consider $\beta=2$: this duality is lost if the matrix model is not Gaussian; however we show that the duality resurfaces in the scaling limit near the edge (soft and hard) of the spectrum. In particular we want to show that the correlation functions of $K$ points near the edge of the spectrum converge to the Kontsevich integral of size $K$ as $N\to\infty$. This line of reasoning was used by Okounkov in the $GUE_2$ for his "edge of the spectrum model". This is based on joint work with Mattia Cafasso (Angers).

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