Gaussian free fields for mathematicians

By Scott Sheffield. 2003

511 citations recorded by the source corpus.

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.

Subfields: Finance, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1fa66d0642095793fb753f90358cbc284629966e

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Citations in the corpus

An elementary approach to Gaussian multiplicative chaos (2015, background): “This is the particular case of the theory when d = 2 and h is the massless Gaussian free field or GFF (see [13]) with appropriate boundary conditions.”
The Noise in the Circular Law and the Gaussian Free Field (2006, background): “1 Planar GFF harmonic outside U and Corollary 2 The space H(1)(C) decomposes into three orthogonal parts (see [19]): H(1)(C) = AI ⊕AO⊕AH .”
SLE and the free field: partition functions and couplings (2007, background): “See [37, 14] for background on the free field, [15] for Gaussian Hilbert spaces; also the survey [36].”
Conformally invariant scaling limits: an overview and a collection of problems (2006, background): “See [85] or [81] for further details, and see Figure 7 for a simulation of the harmonic explorer, and therefore an approximation of SLE4.”
Liouville quantum gravity and KPZ (2008, methodology): “n αnfn where fn are any orthonormal basis for H(D)) — the sum converges almost surely in the space of distributions on D, see [She07].”
Discretization-invariant Bayesian inversion and Besov space priors (2009, unknown): “This fact is extensively used in quantum field theory in the study of the free Gaussian field, a random field that is very similar to the Gaussian smoothness prior [58].”
Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (2013, background): “As in the case of the GFF on a compact domain, the convergence almost surely holds for all such φ, and the law of h turns out not to depend on the choice of (fn) [27].”
Freezing and Decorated Poisson Point Processes (2014, background): “We also mention the 2-dimensional Gaussian free field which plays an important role in statistical physics, the theory of random surfaces, and quantum field theory [15, 24, 27, 48].”
Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, background): “Indeed, the definition we adopt in this book is more general than the standard one [110], see Remark 5.”
Localization and delocalization of random interfaces (2005, background): “A more detailed introduction to the Gaussian free field can be found in [87].”
Gaussian multiplicative chaos and applications: A review (2013, background): “In the situations of interest, X is rather badly behaved and cannot be defined as a random function: it is a random distribution (in the sense of Schwartz), like the Gaussian Free Field (GFF for short, see [134] for an overview about the GFF) for instance.”