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| 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). {{harvtxt|Sheffield|2007}} gives a mathematical survey of the Gaussian free field.
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| The discrete version can be defined on any [[graph (mathematics)|graph]], usually a [[lattice graph|lattice]] in ''d''-dimensional Euclidean space. The continuum version is defined on '''R'''<sup>''d''</sup> or on a bounded subdomain of '''R'''<sup>''d''</sup>. It can be thought of as a natural generalization of [[Wiener process|one-dimensional Brownian motion]] to ''d'' time (but still one space) dimensions; in particular, the one-dimensional continuum GFF is just the standard one-dimensional Brownian motion or [[Brownian bridge]] on an interval.
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| 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 field|bosonic]] massless free field'''. A key property of the 2-dimensional GFF is [[conformal group|conformal invariance]], which relates it in several ways to the [[Schramm-Loewner Evolution]], see {{harvtxt|Sheffield|2005}} and {{harvtxt|Dubédat|2007}}.
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| Similarly to Brownian motion, which is the [[scaling limit]] of a wide range of discrete [[random walk]] models (see [[Donsker's theorem]]), the continuum GFF is the scaling limit of not only the discrete GFF on lattices, but of many random height function models, such as the height function of [[Uniform distribution (discrete)|uniform random]] planar [[domino tiling]]s, see {{harvtxt|Kenyon|2001}}. The planar GFF is also the limit of the fluctuations of the [[characteristic polynomial]] of a [[random matrix]] model, the Ginibre ensemble, see {{harvtxt|Rider|Virág|2007}}.
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| The structure of the discrete GFF on any graph is closely related to the behaviour of the [[Random_walk#Random_walk_on_graphs|simple random walk on the graph]]. For instance, the discrete GFF plays a key role in the proof by {{harvtxt|Ding|Lee|Peres|2012}} of several conjectures about the cover time of graphs (the expected number of steps it takes for the random walk to visit all the vertices).
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| ==Definition of the discrete GFF==
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| [[File:Discrete_Gaussian_free_field_on_60_x_60_square_grid.png|thumb|This surface plot shows a sample of the discrete Gaussian free field defined on the vertices of a 60 by 60 square grid, with zero boundary conditions. The values of the DGFF on the vertices are linearly interpolated to give a continuous function.]]
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| Let ''P''(''x'', ''y'') be the transition kernel of the [[Markov chain]] given by a [[random walk]] on a finite graph ''G''(''V'', ''E''). Let ''U'' be a fixed non-empty subset of the vertices ''V'', and take the set of all real-valued functions <math>\varphi</math> with some prescribed values on ''U''. We then define a [[Gibbs measure|Hamiltonian]] by
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| : <math>H( \varphi ) = \frac{1}{2} \sum_{(x,y)} P(x,y)\big(\varphi(x) - \varphi(y)\big)^2. </math>
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| Then, the random function with [[probability density function|probability density]] proportional to <math>\exp(-H(\varphi))</math> with respect to the [[Lebesgue measure]] on <math>\R^{V\setminus U}</math> is called the discrete GFF with boundary ''U''.
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| It is not hard to show that the [[expected value]] <math>\mathbb{E}[\varphi(x)]</math> is the discrete [[harmonic function|harmonic]] extension of the boundary values from ''U'' (harmonic with respect to the transition kernel ''P''), and the [[covariance]]s <math>\mathrm{Cov}[\varphi(x),\varphi(y)]</math> are equal to the discrete [[Green's function]] ''G''(''x'', ''y'').
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| So, in one sentence, the discrete GFF is the [[Gaussian random field]] on ''V'' with covariance structure given by the Green's function associated to the transition kernel ''P''.
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| ==The continuum field==
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| The definition of the continuum field necessarily uses some abstract machinery, since it does not exist as a random height function. Instead, it is a random generalized function, or in other words, a [[probability distribution|distribution]] on [[distribution (mathematics)|distribution]]s (with two different meanings of the word "distribution").
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| Given a domain Ω ⊆ '''R'''<sup>''n''</sup>, consider the [[Dirichlet energy|Dirichlet inner product]]
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| : <math>\langle f, g\rangle := \int_\Omega (Df(x), Dg(x)) \, dx </math> | |
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| for smooth functions ''ƒ'' and ''g'' on Ω, coinciding with some prescribed boundary function on <math>\partial \Omega</math>, where <math>Df\,(x)</math> is the [[gradient vector]] at <math>x\in \Omega</math>. Then take the [[Hilbert space]] closure with respect to this [[inner product]], this is the [[Sobolev space]] <math>H^1(\Omega)</math>.
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| The continuum GFF <math>\varphi</math> on <math>\Omega</math> is a [[Gaussian random field]] indexed by <math>H^1(\Omega)</math>, i.e., a collection of [[normal distribution|Gaussian]] random variables, one for each <math>f \in H^1(\Omega)</math>, denoted by <math>\langle \varphi,f \rangle</math>, such that the [[covariance]] structure is <math>\mathrm{Cov}[\langle \varphi,f \rangle, \langle \varphi,g \rangle] = \langle f,g \rangle</math> for all <math>f,g\in H^1(\Omega)</math>.
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| Such a random field indeed exists, and its distribution is unique. Given any [[orthonormal basis]] <math>\psi_1, \psi_2, \dots</math> of <math>H^1(\Omega)</math> (with the given boundary condition), we can form the formal infinite sum
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| : <math> \varphi := \sum_{k=1}^\infty \xi_k \psi_k,</math> | |
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| where the <math>\xi_k</math> are [[i.i.d.]] [[standard normal variable]]s. This random sum almost surely will not exist as an element of <math>H^1(\Omega)</math>, since its [[variance]] is infinite. However, it exists as a random [[distribution (mathematics)|generalized function]], since for any <math>f \in H^1(\Omega)</math> we have
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| : <math>f=\sum_{k=1}^\infty c_k \psi_k,\text{ with }\sum_{k=1}^\infty c_k^2 < \infty,</math>
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| hence
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| : <math>\langle \varphi,f \rangle := \sum_{k=1}^\infty \xi_k c_k</math>
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| is a well-defined finite random number.
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| ===Special case: ''n'' = 1===
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| Although the above argument shows that <math> \varphi </math> does not exist as a random element of <math>H^1(\Omega)</math>, it still could be that it is a random function on <math>\Omega</math> in some larger function space. In fact, in dimension <math>n=1</math>, an orthonormal basis of <math>H^1[0,1]</math> is given by
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| : <math>\psi_k (t):= \int_0^t \varphi_k(s) \, ds\,,</math> where <math>(\varphi_k)</math> form an orthonormal basis of <math>L^2[0,1]\,,</math>
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| and then <math>\varphi(t):=\sum_{k=1}^\infty \xi_k \psi_k(t)</math> is easily seen to be a one-dimensional Brownian motion (or Brownian bridge, if the boundary values for <math>\varphi_k</math> are set up that way). So, in this case, it is a random continuous function. For instance, if <math>(\varphi_k)</math> is the [[Haar basis]], then this is Lévy's construction of Brownian motion, see, e.g., Section 3 of {{harvtxt|Peres|2001}}.
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| On the other hand, for <math>n \geq 2</math> it can indeed be shown to exist only as a generalized function, see {{harvtxt|Sheffield|2007}}.
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| ===Special case: ''n'' = 2===
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| In dimension ''n'' = 2, the conformal invariance of the continuum GFF is clear from the invariance of the Dirichlet inner product.
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| {{Expand section|date=November 2010}}
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| ==References==
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| *{{Citation | last1=Ding | first1=J. | last2=Lee | first2=J. R. | last3=Peres | first3=Y. | title=Cover times, blanket times, and majorizing measures|url=http://front.math.ucdavis.edu/1004.4371| journal = Annals of Mathematics |volume=175 | year=2012|pages=1409–1471}}
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| *{{Citation | last1=Dubédat | first1=J. | title=SLE and the free field: Partition functions and couplings|url=http://front.math.ucdavis.edu/0712.3018| journal = J. Amer. Math. Soc. |volume= 22| pages= 995–1054 | year=2009}}
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| *{{Citation | last1=Kenyon | first1=R. | title= Dominos and the Gaussian free field| url=http://uk.arxiv.org/abs/math-ph/0002027 | mr=1872739 | year=2001 | journal=Annals of Probability | volume=29, no. 3 | pages=1128–1137}}
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| *{{Citation | last1=Peres | first1=Y. | title=An Invitation to Sample Paths of Brownian Motion|url=http://www.stat.berkeley.edu/~peres/bmall.pdf| journal = Lecture notes at UC Berkeley| year=2001}}
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| *{{Citation | last1=Rider | first1=B. | last2=Virág | first2=B. | title= The noise in the Circular Law and the Gaussian Free Field| url=http://front.math.ucdavis.edu/0606.5663 | mr=2361453 | year=2007 | journal=International Mathematics Research Notices | pages=article ID rnm006, 32 pages}}
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| *{{Citation|last=Sheffield|first=S.|title=Local sets of the Gaussian Free Field|url=http://www.fields.utoronto.ca/audio/05-06/#percolation_SLE|journal=Talks at the Fields Institute, Toronto, on September 22–24, 2005, as part of the "Percolation, SLE, and related topics" Workshop.|year=2005}}
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| *{{Citation|last=Sheffield|first=S.|title=Gaussian free fields for mathematicians|arxiv=math.PR/0312099|journal=Probability Theory and Related Fields|volume=139|year=2007|pages=521–541| mr=2322706}}
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| {{DEFAULTSORT:Gaussian Free Field}}
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| [[Category:Statistical mechanics]]
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| [[Category:Stochastic processes]]
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