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Person# Juhan Aru

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Courses taught by this person (5)

Related publications (10)

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.

En son coeur, c'est un cours d'analyse fonctionnelle pour les physiciens et traite les bases de théorie de mesure, des espaces des fonctions et opérateurs linéaires.

This year we will be looking into probabilistic study of some models of mathematical physics.

A topological space is a space endowed with a notion of nearness. A metric space is an example of a topological space, where the concept of nearness is measured by a distance function. Within this abs

Consider CLE4 in the unit disk, and let l be the loop of the CLE4 surrounding the origin. Schramm, Sheffield and Wilson determined the law of the conformal radius seen from the origin of the domain surrounded by l. We complement their result by determining the law of the extremal distance between and the boundary of the unit disk. More surprisingly, we also compute the joint law of these conformal radius and extremal distance. This law involves first and last hitting times of a one-dimensional Brownian motion. Similar techniques also allow us to determine joint laws of some extremal distances in a critical Brownian loop-soup cluster.

We prove that the set of-y-thick points of a planar Gaussian free field (GFF) with Dirichlet boundary conditions is a.s. totally disconnected for all-y =6 0. Our proof relies on the coupling between a GFF and the nested CLE4. In particular, we show that the thick points of the GFF are the same as those of the weighted CLE4 nesting field introduced in [24] and establish the almost sure total disconnectedness of the complement of a nested CLE & kappa;, & kappa; & ISIN; (8/3, 4]. As a corollary we see that the set of singular points for supercritical LQG metrics is a.s. totally disconnected.

We prove that under certain mild moment and continuity assumptions, the d-dimensional continuum Gaussian free field is the only stochastic process satisfying the usual domain Markov property and a scaling assumption. Our proof is based on a decomposition of the underlying functional space in terms of radial processes and spherical harmonics.