Gaussian free field

In probability theory and statistical mechanics, the Gaussian free field (GFF) is a Gaussian random field, a central model of random surfaces (random height functions). gives a mathematical survey of the Gaussian free field. The discrete version can be defined on any graph, usually a lattice in d-dimensional Euclidean space. The continuum version is defined on Rd or on a bounded subdomain of Rd. It can be thought of as a natural generalization of one-dimensional Brownian motion to d time (but still one space) dimensions: it is a random (generalized) function from Rd to R. In particular, the one-dimensional continuum GFF is just the standard one-dimensional Brownian motion or Brownian bridge on an interval. In the theory of random surfaces, it is also called the harmonic crystal. It is also the starting point for many constructions in quantum field theory, where it is called the Euclidean bosonic massless free field. A key property of the 2-dimensional GFF is conformal invariance, w

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Liouville Measure As A Multiplicative Cascade Via Level Sets Of The Gaussian Free Field

Juhan Aru

We provide new constructions of the subcritical and critical Gaussian multiplicative chaos (GMC) measures corresponding to the 2D Gaussian free field (GFF). As a special case we recover E. Aidekon's construction of random measures using nested conformally invariant loop ensembles, and thereby prove his conjecture that certain CLE4 based limiting measures are equal in law to the GMC measures for the GFF. The constructions are based on the theory of local sets of the GFF and build a strong link between multiplicative cascades and GMC measures. This link allows us to directly adapt techniques used for multiplicative cascades to the study of GMC measures of the GFF. As a proof of principle we do this for the so-called Seneta-Heyde rescaling of the critical GMC measure.

ANNALES INST FOURIER2020

Gaussian Multiplicative Chaos Through the Lens of the 2D Gaussian Free Field

Juhan Aru

The aim of this review-style paper is to provide a concise, self-contained and unified presentation of the construction and main properties of Gaussian multiplicative chaos (GMC) measures for log-correlated fields in 2D in the subcritical regime. By considering the case of the 2D Gaussian free field, we review convergence, uniqueness and characterisations of the measures; revisit Kahane's convexity inequalities and existence and scaling of moments; discuss the measurability of the underlying field with respect to the GMC measure and present a KPZ relation for scaling exponents.

POLYMAT2020

The First Passage Sets of the 2D Gaussian Free Field: Convergence and Isomorphisms

Juhan Aru

In a previous article, we introduced the first passage set (FPS) of constant level -aof the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path along which the GFF is greater than or equal to -aThis description can be taken as a definition of the FPS for the metric graph GFF, and in the current article, we prove that the metric graph FPS converges towards the continuum FPS in the Hausdorff distance. We also draw numerous consequences; in particular, we obtain a relatively simple proof of the fact that certain natural interfaces of the metric graph GFF converge to SLE4 level lines. These results improve our understanding of the continuum GFF, by strengthening its relationship with the critical Brownian loop-soup. Indeed, a new construction of the FPS using clusters of Brownian loops and excursions helps to strengthen the known GFF isomorphism theorems, and allows us to use Brownian loop-soup techniques to prove technical results on the geometry of the GFF. We also obtain a new representation of Brownian loop-soup clusters, and as a consequence, we prove that the clusters of a critical Brownian loop-soup admit a non-trivial Minkowski content in the gauge r & x21a6;|logr|1/2r2.

SPRINGER2020