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 be done with care. In particular, in 2D,
the GFF (Gaussian Free Field) is a random Schwartz 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.

Joint work with J. Najnudel