Related papers: Lecture notes on the Gaussian Free Field
The d-dimensional Gaussian free field (GFF), also called the (Euclidean bosonic) massless free field, is a d-dimensional-time analog of Brownian motion. Just as Brownian motion is the limit of the simple random walk (when time and space are…
The Gaussian Free Field (GFF) is a canonical random surface in probability theory generalizing Brownian motion to higher dimensions. In two dimensions, it is critical in several senses, and is expected to be the universal scaling limit of a…
We point out a new simple way to couple the Gaussian Free Field (GFF) with free boundary conditions in a two-dimensional domain with the GFF with zero boundary conditions in the same domain: Starting from the latter, one just has to sample…
In a previous article, we introduced the first passage set (FPS) of constant level $-a$ of 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…
We further investigate properties of the Gaussian free field (GFF) on the metric graph associated to a discrete weighted graph (where the edges of the latter are replaced by continuous line-segments of appropriate length) that has been…
We establish a martingale-type characterisations for the continuum Gaussian free field (GFF) and for fractional Gaussian free fields (FGFs), using their connection to the stochastic heat equation and to fractional stochastic heat equations.…
The main topic of these lecture notes is the continuum scaling limit of planar lattice models. One reason why this topic occupies an important place in the theory of probability and mathematical statistical physics is that scaling limits…
We study the level lines of a Gaussian free field in a planar domain with general boundary data $F$. We show that the level lines exist as continuous curves under the assumption that $F$ is regulated (i.e., admits left and right limits at…
These are lecture notes from a course given at the CRM in Montreal in 1992. They survey the author's attempts to find and understand canonical probabilistic entities in a local field (e.g. p-adic) setting. We propose answers to the related…
We introduce the group field theory (GFT) formalism for non-perturbative quantum gravity, and present it as a potential unifying framework for several other quantum gravity approaches, i.e. loop quantum gravity and simplicial quantum…
We study the level lines of GFF starting from interior points. We show that the level line of GFF starting from an interior point turns out to be a sequence of level loops. The sequence of level loops satisfies "target-independent"…
In these mostly expository lectures, we give an elementary introduction to conformal field theory in the context of probability theory and complex analysis. We consider statistical fields, and define Ward functionals in terms of their Lie…
Consider a Brownian loop soup $\mathcal{L}_D^\theta$ with subcritical intensity $\theta \in (0,1/2]$ in some 2D bounded simply connected domain. We define and study the properties of a conformally invariant field $h_\theta$ naturally…
Fractional Gaussian fields are scalar-valued random functions or generalized functions on an $n$-dimensional manifold $M$, indexed by a parameter $s$. They include white noise ($s = 0$), Brownian motion ($s=1, n=1$), the 2D Gaussian free…
We consider a class of Gaussian Free Fields denoted by $(g_x)_{x \in {\cal V}_N}$, where $ {\cal V}_N = \{0,1\}^N$ and $N\in \mathbb{Z}_+$. These fields are related to a general class of $N$-dimensional random walks on the hypercube, which…
The objective of the paper is to characterize the Gaussian free field as a stationary solution of the heat equation with additive space-time white noise. In the case of whole space, the investigation leads to other types of Gaussian fields,…
In this note we show that the 2D continuum Gaussian free field (GFF) admits an excursion decomposition that is on the one hand similar to the classical excursion decomposition of the Brownian motion, and on the other hand can be seen as an…
These lecture notes offer a gentle introduction to the two-dimensional Discrete Gaussian Free Field with particular attention paid to the scaling limits of the level sets at heights proportional to the absolute maximum. The bulk of the text…
The linearisation of a second-order formulation of the conformal Einstein field equations (CEFEs) in Generalised Harmonic Gauge (GHG), with trace-free matter is derived. The linearised equations are obtained for a general background and…
A discussion is given of the conformal Einstein field equations coupled with matter whose energy-momentum tensor is trace-free. These resulting equations are expressed in terms of a generic Weyl connection. The article shows how in the…