scispace - formally typeset
Search or ask a question
Topic

Gaussian free field

About: Gaussian free field is a research topic. Over the lifetime, 568 publications have been published within this topic receiving 12029 citations.


Papers
More filters
Journal ArticleDOI
TL;DR: In this paper, the authors considered the limits of the uniform spanning tree and the loop-erased random walk (LERW) on a fine grid in the plane, as the mesh goes to zero.
Abstract: The uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of these subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane.

1,209 citations

Journal ArticleDOI
TL;DR: The d-dimensional Gaussian free field (GFF) as mentioned in this paper is a generalization of the simple random walk (when time and space are appropriately scaled), which is the limit of many incrementally varying random functions on a ddimensional grid.
Abstract: 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 appropriately scaled), the GFF is the limit of many incrementally varying random functions on d-dimensional grids. We present an overview of the GFF and some of the properties that are useful in light of recent connections between the GFF and the Schramm–Loewner evolution.

500 citations

Journal ArticleDOI
TL;DR: The theory of Gaussian multiplicative chaos was introduced by Kahane's seminal work in 1985 as discussed by the authors, and it has been applied in many applications, ranging from finance to quantum gravity.
Abstract: In this article, we review the theory of Gaussian multiplicative chaos initially introduced by Kahane’s seminal work in 1985. Though this beautiful paper faded from memory until recently, it already contains ideas and results that are nowadays under active investigation, like the construction of the Liouville measure in $2d$-Liouville quantum gravity or thick points of the Gaussian Free Field. Also, we mention important extensions and generalizations of this theory that have emerged ever since and discuss a whole family of applications, ranging from finance, through the Kolmogorov-Obukhov model of turbulence to $2d$-Liouville quantum gravity. This review also includes new results like the convergence of discretized Liouville measures on isoradial graphs (thus including the triangle and square lattices) towards the continuous Liouville measures (in the subcritical and critical case) or multifractal analysis of the measures in all dimensions.

469 citations

Journal ArticleDOI
TL;DR: In this article, a general quadratic relation between these two dimensions was derived, which they view as a probabilistic formulation of the Knizhnik, Polyakov, Zamolodchikov (Mod. Phys. Lett. A, 3:819-826, 1988) relation from conformal field theory.
Abstract: Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1∫ D ∇h(z)⋅∇h(z)dz, and a constant 0≤γ<2. The Liouville quantum gravity measure on D is the weak limit as e→0 of the measures $$\varepsilon^{\gamma^2/2} e^{\gamma h_\varepsilon(z)}dz,$$ where dz is Lebesgue measure on D and h e (z) denotes the mean value of h on the circle of radius e centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the Knizhnik, Polyakov, Zamolodchikov (Mod. Phys. Lett. A, 3:819–826, 1988) relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of ∂D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.

461 citations

Posted Content
TL;DR: In this article, a general quadratic relation between these two dimensions, which is viewed as a probabilistic formulation of the KPZ relation from conformal field theory, is derived.
Abstract: Consider a bounded planar domain D, an instance h of the Gaussian free field on D (with Dirichlet energy normalized by 1/(2\pi)), and a constant 0 < gamma < 2. The Liouville quantum gravity measure on D is the weak limit as epsilon tends to 0 of the measures \epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz, where dz is Lebesgue measure on D and h_\epsilon(z) denotes the mean value of h on the circle of radius epsilon centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the KPZ relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of the boundary of D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.

351 citations


Network Information
Related Topics (5)
Stochastic differential equation
20.3K papers, 518.6K citations
84% related
Random walk
21.4K papers, 520.5K citations
81% related
Random variable
29.1K papers, 674.6K citations
79% related
Stochastic partial differential equation
21.1K papers, 707.2K citations
79% related
Asymptotic distribution
16.7K papers, 564.9K citations
77% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202315
202236
202159
202055
201953
201873