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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). 


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25. November 2021, 16:15-17:55

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TU Darmstadt
FB Mathematik
Schlossgartenstr. 7
64289 Darmstadt

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FB Mathematik Arbeitsgruppe Stochastik

 

25
November
2021
16:15-17:55

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Stochastik