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Session S04 - Random Walks and Related Topics

Monday, July 19, 16:00 UTC-3

Geometry of Gaussian multiplicative chaos in the Wiener space

Chiranjib Mukherjee

Universität Münster, Germany   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We report on a recent joint work with Y. Bröker (Münster) where we develop an approach for investigating geometric properties of Gaussian multiplicative chaos (GMC) in an infinite dimensional set up. The base space is chosen to be the space of continuous functions endowed with Wiener measure, and the Gaussian field is a space-time Gaussian noise integrated against Brownian paths. In this set up, we show that in any dimension $d\geq 1$ and for any inverse temperature $\gamma>0$, the volume of a GMC ball, uniformly around all paths, decays exponentially with an explicit decay rate. For $d\geq 3$ and high temperature, the decay rate is just the principal eigenvalue of the Dirichlet Laplacian of the ball, which reflects a similar behavior of the free Brownian path. Incidentally, this is also the regime when our GMC attains very high values on all paths, making these points thick under the GMC measure. For any $d\in \mathbb N$, and small temperatures, the rate is given by an additional energy functional minimized over (probability measures on) a translation- invariant compactification constructed together with Varadhan. Quantifying exponential decay rates of GMC balls are natural infinite dimensional extensions of similar behavior captured by thescaling exponents of $2d$ Liouville quantum gravity, which has been studied intensively over the last few years.

Joint work with Yannic Bröker (University of Münster, Germany).

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