Geometry of the Gaussian multiplicative chaos in the Wiener space

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 GMC-volume of a ball, uniformly around all paths, decays exponentially with an explicit decay rate. The exponential rate reflects the balance between two competing terms, namely the principal eigenvalue of the Dirichlet Laplacian and an energy functional defined over a certain compactification developed earlier in [MV14]. For $d\geq 3$ and high temperature, the underlying Gaussian field is also shown to attain 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 fast uniformly over all paths, the field itself attains atypically large values on all paths when sampled according to the GMC. It is also shown that, despite the exponential decay of volume for any temperature, for small enough temperature, the normalized overlap of two independent paths tends to follow one of only a finite number of independent paths for most of its allowed time horizon, allowing the GMC probability to accumulate most of its mass along such trajectories.