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Geometry of the Gaussian multiplicative chaos in the Wiener space

Oberseminar Darmstadt

Date: 25.11.2021

Time: 16:15–17:55 h

Abstract: 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 random field is a space-time white noise integrated against Brownian paths. In this set up, we show that in any dimension $d\geq 1$ and for any inverse temperature, the volume of a GMC ball, uniformly around all paths, decays exponentially with an explicit decay rate. The latter resolves the fight between the principal eigenvalue of the Dirichlet Laplacian and an energy functional defined over a certain compactification developed earlier with Varadhan. For $d\geq 3$ and high temperature  the underlying Gaussian field also attains very high values under the GMC -- that is, all paths are "GMC -thick" in this regime. Both statements are natural infinite dimensional extensions of similar behavior captured by $2d$ Liouville quantum gravity and reflect a certain ``atypical behavior" of the GMC: while the GMC volume decays exponentially uniformly over all paths, the field itself attains atypically large values on all paths when sampled according to the GMC. Joint work with Yannic Bröker (Münster).

Speaker

Chiranjib Mukherjee, Universität Münster

Place

TU Darmstadt | Raum S2|15 401
Schlossgartenstraße 7, 64289 Darmstadt

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Organizers

Technische Universität Darmstadt

Fachbereich Mathematik - Stochastik
Schlossgartenstraße 7
64289 Darmstadt
Telefon: +49 6151 16-23380
Telefax: +49 6151 16-23381
info(at)stochastik-rhein-mainde


Organizing partners

Goethe-Universität Frankfurt am Main, Johannes Gutenberg-Universität Mainz

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