# On the circle, Kahane’s Gaussian Multiplicative Chaos and Circular Random Matrices match – Reda Chhaibi

Reda Chhaibi
IMT, Université Paul Sabatier Toulouse III
https://www.math.univ-toulouse.fr/~rchhaibi/

Date(s) : 07/06/2019   iCal
11 h 00 min - 12 h 00 min

In this talk, I would like to advertise an equality between two objects from very different areas of mathematical physics. This bridges the Gaussian Multiplicative Chaos, which plays an important role in certain conformal field theories, and a reference model in random matrices.On the one hand, in 1985, J.P Kahane introduced a random measure called the Gaussian Multiplicative Chaos (GMC). Morally, this is the measure whose Radon-Nikodym derivative w.r.t to Lebesgue is the exponential of a log correlated Gaussian field. In the cases of interest, this Gaussian field is a Schwartz distribution but not a function. As such, the construction of GMC needs to bedone with care. In particular, in 2D, the GFF (Gaussian Free Field) is a randomSchwartz distribution because of the logarithmic singularity of the Green kernel in 2D. Here we are interested in the 1D case on the circle. On the other hand, it is known since Verblunsky (1930s) that a probability measure on the circle is entirely determined by the coefficients appearing in the recurrence of orthogonal polynomials. Furthermore, Killip and Nenciu (2000s) have given a realization of the CBE, an important model in random matrices, thanks to random orthogonal polynomials of the circle. I will give the precise statement whose loose form is CBE = GMC.

With J. Najnudel.

https://arxiv.org/abs/1904.00578

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