Inprobability theory andstatistical mechanics, theGaussian free field (GFF) is aGaussian random field, a central model of random surfaces (random height functions).

The discrete version can be defined on anygraph, usually alattice ind-dimensional Euclidean space. The continuum version is defined onR^d or on a bounded subdomain ofR^d. It can be thought of as a natural generalization ofone-dimensional Brownian motion tod time (but still one space) dimensions: it is a random (generalized) function fromR^d toR. In particular, the one-dimensional continuum GFF is just the standard one-dimensional Brownian motion orBrownian bridge on an interval.

In the theory of random surfaces, it is also called theharmonic crystal. It is also the starting point for many constructions inquantum field theory, where it is called theEuclideanbosonic massless free field. A key property of the 2-dimensional GFF isconformal invariance, which relates it in several ways to theSchramm–Loewner evolution, seeSheffield (2005) andDubédat (2009).

Similarly to Brownian motion, which is thescaling limit of a wide range of discreterandom walk models (seeDonsker'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 ofuniform random planardomino tilings, seeKenyon (2001). The planar GFF is also the limit of the fluctuations of thecharacteristic polynomial of arandom matrix model, the Ginibre ensemble, seeRider & Virág (2007).

The structure of the discrete GFF on any graph is closely related to the behaviour of thesimple random walk on the graph. For instance, the discrete GFF plays a key role in the proof byDing, 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).

LetP(x, y) be the transition kernel of theMarkov chain given by arandom walk on a finite graph G(V, E). LetU be a fixed non-empty subset of the verticesV, and take the set of all real-valued functions${\displaystyle \phi }$ with some prescribed values on U. We then define aHamiltonian by

${\displaystyle H(\phi )=\frac{1}{2}\sum _{(x,y)}P(x,y)(\phi (x)-\phi (y){)}^2.}$

Then, the random function withprobability density proportional to${\displaystyle \exp (-H(\phi ))}$ with respect to theLebesgue measure on${\displaystyle {\mathbb{R}}^{V\setminus U}}$ is called the discrete GFF with boundary U.

It is not hard to show that theexpected value${\displaystyle \mathbb{E}[\phi (x)]}$ is the discreteharmonic extension of the boundary values from U (harmonic with respect to the transition kernel P), and thecovariances${\displaystyle \mathrm{C}\mathrm{o}\mathrm{v}[\phi (x),\phi (y)]}$ are equal to the discreteGreen's function G(x, y).

So, in one sentence, the discrete GFF is theGaussian random field onV with covariance structure given by the Green's function associated to the transition kernel P.

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, aprobability distribution ondistributions (with two different meanings of the word "distribution").

Given a domain Ω⊆Rⁿ, consider theDirichlet inner product

${\displaystyle ⟨f,g⟩:={\int }_{\mathrm{\Omega }}(Df(x),Dg(x))dx}$

for smooth functionsƒ andg on Ω, coinciding with some prescribed boundary function on${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$, where${\displaystyle Df(x)}$ is thegradient vector at${\displaystyle x\in \mathrm{\Omega }}$. Then take theHilbert space closure with respect to thisinner product, this is theSobolev space${\displaystyle H^1(\mathrm{\Omega })}$.

The continuum GFF${\displaystyle \phi }$ on${\displaystyle \mathrm{\Omega }}$ is aGaussian random field indexed by${\displaystyle H^1(\mathrm{\Omega })}$, i.e., a collection ofGaussian random variables, one for each${\displaystyle f\in H^1(\mathrm{\Omega })}$, denoted by${\displaystyle ⟨\phi ,f⟩}$, such that thecovariance structure is${\displaystyle \mathrm{C}\mathrm{o}\mathrm{v}[⟨\phi ,f⟩,⟨\phi ,g⟩]=⟨f,g⟩}$ for all${\displaystyle f,g\in H^1(\mathrm{\Omega })}$.

Such a random field indeed exists, and its distribution is unique. Given anyorthonormal basis${\displaystyle {\psi }_1,{\psi }_2,\dots }$ of${\displaystyle H^1(\mathrm{\Omega })}$ (with the given boundary condition), we can form the formal infinite sum

${\displaystyle \phi :=\sum _{k=1}^{\mathrm{\infty }}{\xi }_k{\psi }_k,}$

where the${\displaystyle {\xi }_k}$ arei.i.d.standard normal variables. This random sumalmost surely will not exist as an element of${\displaystyle H^1(\mathrm{\Omega })}$, since if it did then

${\displaystyle ⟨\phi ,\phi ⟩=\sum _{k=1}^{\mathrm{\infty }}{\xi }_k^2=\mathrm{\infty }\text{a.s.}}$

However, it exists as a randomgeneralized function, since for any${\displaystyle f\in H^1(\mathrm{\Omega })}$ we have

hence

${\displaystyle ⟨\phi ,f⟩:=\sum _{k=1}^{\mathrm{\infty }}{\xi }_k{c}_k}$

is a centered Gaussian random variable with finite variance${\displaystyle \sum _k{c}_k^2.}$

Although the above argument shows that${\displaystyle \phi }$ does not exist as a random element of${\displaystyle H^1(\mathrm{\Omega })}$, it still could be that it is a random function on${\displaystyle \mathrm{\Omega }}$ in some larger function space. In fact, in dimension${\displaystyle n=1}$, an orthonormal basis of${\displaystyle H^1[0,1]}$ is given by

${\displaystyle {\psi }_k(t):={\int }_0^t{\phi }_k(s)ds,}$ where${\displaystyle ({\phi }_k)}$ form an orthonormal basis of${\displaystyle L^2[0,1],}$

and then${\displaystyle \phi (t):=\sum _{k=1}^{\mathrm{\infty }}{\xi }_k{\psi }_k(t)}$ is easily seen to be a one-dimensional Brownian motion (or Brownian bridge, if the boundary values for${\displaystyle {\phi }_k}$ are set up that way). So, in this case, it is a random continuous function (not belonging to${\displaystyle H^1[0,1]}$, however). For instance, if${\displaystyle ({\phi }_k)}$ is theHaar basis, then this is Lévy's construction of Brownian motion, see, e.g., Section 3 ofPeres (2001).

On the other hand, for${\displaystyle n\ge 2}$ it can indeed be shown to exist only as a generalized function, seeSheffield (2007).

In dimensionn = 2, the conformal invariance of the continuum GFF is clear from the invariance of the Dirichlet inner product. The correspondingtwo-dimensional conformal field theory describes amassless free scalar boson.

This sectionneeds expansion. You can help byadding to it.(November 2010)

• Brownian sheet

• Ding, J.; Lee, J. R.; Peres, Y. (2012), "Cover times, blanket times, and majorizing measures",Annals of Mathematics,175 (3):1409–1471,arXiv:1004.4371,doi:10.4007/annals.2012.175.3.8
• Dubédat, J. (2009), "SLE and the free field: Partition functions and couplings",J. Amer. Math. Soc.,22 (4):995–1054,arXiv:0712.3018,Bibcode:2009JAMS...22..995D,doi:10.1090/s0894-0347-09-00636-5,S2CID 8065580
• Kenyon, R. (2001), "Dominos and the Gaussian free field",Annals of Probability,29 (3):1128–1137,arXiv:math-ph/0002027,doi:10.1214/aop/1015345599,MR 1872739,S2CID 119640707
• Peres, Y. (2001),"An Invitation to Sample Paths of Brownian Motion"(PDF),Lecture Notes at UC Berkeley
• Rider, B.; Virág, B. (2007), "The noise in the Circular Law and the Gaussian Free Field",International Mathematics Research Notices: article ID rnm006, 32 pages,arXiv:math/0606663,doi:10.1093/imrn/rnm006,MR 2361453
• Sheffield, S. (2005),"Local sets of the Gaussian Free Field",Talks at the Fields Institute, Toronto, on September 22–24, 2005, as Part of the "Percolation, SLE, and Related Topics" Workshop.
• Sheffield, S. (2007), "Gaussian free fields for mathematicians",Probability Theory and Related Fields,139 (3–4):521–541,arXiv:math.PR/0312099,doi:10.1007/s00440-006-0050-1,MR 2322706,S2CID 14237927
• Friedli, S.; Velenik, Y. (2017).Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction. Cambridge: Cambridge University Press.ISBN 9781107184824.

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