Events Archive

Gaussian multiplicative chaos in the Wiener space

In the classical finite dimensional setting, a Gaussian multiplicative chaos (GMC) is obtained by tilting an ambient measure by the exponential of a centred Gaussian field indexed by a domain in the Euclidean space. In the two-dimensional setting and when the underlying field is "log-correlated", GMC measures share close connection to the 2D Liouville quantum gravity, which has seen a lot of revived interest in the recent years. 
A natural question is to construct a GMC in the infinite dimensional setting, where techniques based on log-correlated fields in finite dimensions are no longer available. In the present context, we consider a GMC on the classical Wiener space, driven by a (mollified) Gaussian space-time white noise. In $d\geq 3$, in a previous work with A. Shamov and O. Zeitouni, we showed that the total mass of this GMC, which is directly connected to the (smoothened) Kardar-Parisi-Zhang equation in $d\geq 3$, converges for small noise intensity to a well-defined strictly positive random variable, while for larger intensity (i.e. for small temperature) it collapses to zero. We will report on joint work with Yannic Bröker (Münster) where we study the endpoint distribution of a Brownian path under the GMC measure and show that, for low temperature, the endpoint GMC distribution localizes in few spatial islands and produces asymptotically purely atomic states. 

Dynamics of strongly correlated spin systems

I will discuss some results on the problem of understanding the long-time behaviour of Glauber and Kawasaki dynamics of spin systems in the regimes of strong correlations. This is joint work with Thierry Bodineau.