Title :
Gaussian Multiplicative Chaos theory and several extensions
Abstract :
The Gaussian Multiplicative Chaos, introduced by Kahane (1985), is a
random measure defined as the exponential of a log-correlated gaussian
field. It is a crucial element in the recent rigorous probabilistic
construction by David-Kupiainen-Rhodes-Vargas (2014) of Liouville
Conformal Field Theory, introduced by Polyakov (1981) in his theory of
integration over 2d Riemann surfaces.
The first part of this talk is an introduction to Gaussian
Multiplicative Chaos and Liouville Conformal Field Theory on the
Riemann sphere a la David-Kupiainen-Rhodes-Vargas. We also briefly
explain the case of the unit disk, where the presence of a boundary
requires careful analysis of the behaviour of a non-conventional
Gaussian Multiplicative Chaos measure.
In the second part of this talk, we will discuss some extensions of
these theories to cases with complex parameters. We will derive in
particular some results on analytic continuation of correlation
functions using the path-integral formalism.