Gaussian random field

A Gaussian random field is a spatially indexed Gaussian process; its law is determined by its mean and covariance kernel.

A centered Gaussian random field with the displayed covariance. It has independent increments over disjoint rectangles. At fixed positive $T$, $W(t,T)/\sqrt{T}$ is standard Brownian motion in $t$. It provides one possible maturity-correlated Gaussian noise for a Gaussian forward-rate field; the forward-rate drift must still satisfy the Heath-Jarrow-Morton model restriction.

Let $X$ be a standard Gaussian random variable and $S_j$ independent symmetric signs, independent of $X$. The stationary sequence $W_j=S_{j}X$ has standard Gaussian marginals and identity covariance, but is not jointly Gaussian. For two coordinates its moment-generating function is $(1+e^{2t^2})/2$, rather than $e^{t^2}$. Consequently a field's Gaussian one-point densities and two-point covariance alone do not justify an exponential moment of a Gaussian linear functional.

If $G$ is a centered Gaussian random field, the transformation $G+\alpha (G^2-⟨G^2⟩)$ retains zero mean and introduces a three-point connected correlation function at first order in $\alpha$. Wick contractions give $2\alpha$ times the sum of products of two covariances. The corresponding local cosmological model uses $\alpha =3f_{\mathrm{NL}}/5$.

If a linear functional $Z=\int a(\mathbf{r})V(\mathbf{r})d\mathbf{r}$ of a real Gaussian random field exists in mean square, it is a Gaussian random variable. Completing the square in its density givesFor a complex linear functional $\varphi$, apply this to $Z=2\mathrm{Re}\varphi$; then $\frac{1}{2}\mathrm{Var}Z=\mathbb{E}|\varphi -\mathbb{E}\varphi {|}^2+\mathrm{Re}\mathbb{E}(\varphi -\mathbb{E}\varphi {)}^2$.

For a real standard Gaussian random vector $Z$, a real symmetric matrix $Q$, and $I-2Q$ strictly positive,Diagonalize $Q$ and complete the square in each coordinate. Failure of strict positivity makes the positive exponential integral divergent. The same formula extends to a self-adjoint trace-class quadratic form of an isonormal Gaussian process, with square-summable linear coefficients and a Fredholm determinant. A linear-plus-quadratic Gaussian random field functional is reduced to this form by its covariance square root.

A stationary Gaussian random field has Gaussian finite-dimensional distributions invariant under common spatial translations. Its mean is constant and its two-point covariance depends only on the separation of the points.

For a zero-mean stationary Gaussian random field $V$, the unnormalized autocorrelation function of a random field isIt depends only on the separation $h$, and $C(0)$ is the pointwise variance. If $C(0)>0$, the normalized correlation is $\rho (h)=C(h)/C(0)$; the two conventions have different units.

For a mean-square continuous field, its spectral measure of a stationary random field $\nu$ obeys $C(h)=\int e^{ik\cdot h}\nu (dk)$. If $C$ is integrable, an ordinary power spectral density exists and, in the angular-wavevector convention,Thus $\nu$ has density $S(k)/(2\pi {)}^d$. A spectral measure need not have a density: a spatially constant centered Gaussian field of variance $a>0$ has $C(h)=a$ and spectral measure $a{\delta }_0$, concentrated at zero wavevector. Calling every spectrum an ordinary function would miss this case.^[ref]

For $W=n^2-⟨n^2⟩$, the covariance function is $⟨n^2{n}^{\prime 2}⟩-⟨n^2⟩⟨n^{\prime 2}⟩$. It involves fourth moments and is not determined by the two-point correlation of $n$ in general. For a centered Gaussian pair the fourth-moment identity reduces it to $2⟨n{n}^{\prime }{⟩}^2$. This extra Gaussian hypothesis must be stated before using that reduction.

For a centered Gaussian random field $W$ with covariance $C$ and a sufficiently regular functional $F[W]$, the Furutsu--Novikov formula is the functional integration by parts identityIt closes moment equations for systems driven multiplicatively by Gaussian fluctuations.

When a random medium's longitudinal correlation length is much shorter than the scale on which a propagating envelope changes, the Markov approximation replaces the longitudinal covariance by white noise with the same integrated covariance. Field-moment equations then become local in propagation distance.

The Gaussian free field is a centered Gaussian random field whose covariance is the Green kernel of a Laplace operator. It has discrete versions on graphs and continuum versions as a random generalized function on a domain.

The discrete Gaussian free field on a transient weighted graph is the centered Gaussian random field whose covariance kernel is the random-walk Green function of a transient weighted graph.

A Gaussian free field with Dirichlet boundary condition is fixed to zero on a boundary set and has density proportional to the exponential of minus one half of its discrete Dirichlet energy on the remaining vertices.

Conditionally on all other values, a discrete Gaussian free field value is Gaussian with mean equal to the weighted average of its neighbors and variance equal to the reciprocal local precision.

Conditioning a Gaussian free field on its values along a boundary leaves an independent zero-boundary Gaussian free field plus the harmonic extension of those boundary values in each remaining component.

The Continuum Gaussian free field on a planar domain $D$ is a centered Gaussian process indexed by finite-energy measures or test functions, with covariance given by the Green function of the Laplacian on $D$.

If $\varphi :D\to \tilde{D}$ is a conformal map, pulling back a zero-boundary Gaussian free field on $\tilde{D}$ gives a zero-boundary Gaussian free field on $D$. This follows from the conformal invariance of the planar Dirichlet Green function.

For finite Borel measures $\rho ,\nu$ of finite Green energy, the zero-boundary Gaussian free field $h$ on $D$ is characterized bywhere $G_D$ is the zero-Dirichlet Green function of the Laplacian.

For a Gaussian free field with boundary heights $-\lambda$ and $+\lambda$ on the two sides of the starting point, its distinguished zero-height interface has chordal ${\mathrm{SLE}}_4$ law. Conditional on an initial curve segment, the field is a zero-boundary field in the slit domain plus $m_t(z)=\lambda -(2\lambda /\pi )\arg (g_t(z)-W_t)$. Its mean bracket equals minus the variation of its conditional Green covariance. Subtracting the initial harmonic function gives the corresponding coupling to a zero-boundary field.

Let $M$ be a continuous local martingale on $[0,T]$, with deterministic $M_0=m$, and let $V$ be a continuous nonnegative adapted finite-variation process with deterministic $V_0=v<\infty$. Suppose $d[M{]}_t=-d{V}_t$ and $V_T=0$ almost surely. For each real $u$, the Itô formula showsis a complex local martingale: its two finite-variation terms cancel. Its absolute value is at most one, so it is a true bounded martingale. Consequentlywhich identifies the entire normal distribution $N(m,v)$ by its characteristic function.

For example, $M_t=B_t$ and $V_t=T-t$ give $B_T\sim N(0,T)$. The residual-variance condition $V_T=0$ is essential for this conclusion. With $M_t=0$ and $V_t=1$ throughout, the displayed exponential is still a bounded martingale, but its terminal mean variable is the constant zero, not $N(0,1)$. In a field-revelation construction, $M_t$ represents a conditional mean and $V_t$ the remaining conditional variance; only after the latter has vanished does the martingale expectation identify the terminal sample law.

A continuum field level line is specified by the harmonic boundary heights it creates and the conditional zero-boundary field remaining on each side. Since a Continuum Gaussian free field is a random distribution, this is not the pointwise set of its zeros. For the height jump appropriate to the Green-function normalization, the zero-height line between opposite Dirichlet heights is chordal ${\mathrm{SLE}}_4$.

For a test function $\varphi$ on $D$, let $f_{\varphi }=-2\pi {\Delta }^{-1}\varphi \in H_0^1(D)$. The pairingis a centered Gaussian random variable with variance

Fix a bounded planar domain $D$ and explicitly choose the Dirichlet form normalizationIts Dirichlet Green function of the Laplacian satisfies $-{\Delta }_x{G}_D(x,y)=2\pi {\delta }_y(x)$. Let $(e_n)$ be an orthonormal basis of the corresponding Sobolev space $H_0^1(D)$. The formal expansionis interpreted as a weak kernel identity. Concretely, for smooth compactly supported test functions $\varphi ,\psi$,The identity follows by applying Parseval's identity to the two Riesz representatives for the integration functionals. It is not a claim that arbitrary Sobolev space basis functions have pointwise values everywhere or that the series converges on the singular diagonal.

For a finite measure $\rho$ with finite Green energy, the same Hilbert-space identity gives ${\sum }_n(\int e_{n}d\rho {)}^2=\iint G_D(x,y)\rho (dx)\rho (dy)$. This is the square-summability condition for a Finite-Green-energy measure pairing with a Gaussian free field. An alternative energy normalization rescales both the basis and the Green kernel; specifying the factor prevents an unnoticed $2\pi$ mismatch.^[ref]

Let $\rho$ be a finite measure on $D$ with finite Green energy, in the normalization of the Green-kernel expansion in the Dirichlet space. For an orthonormal basis $(e_n)$ and independent standard Gaussian random variables ${\xi }_n$, defineThe sum converges in L2 space, since the sum of the coefficient squares equals the Green energy. It is centered Gaussian withFor several measures, the same coefficients give their cross-covariances by the corresponding mixed Green integrals.

The joint law is independent of the chosen basis. A stronger pathwise statement needs consistent coefficients: realize the field through one isonormal Gaussian process $W$ on the Dirichlet Hilbert space and take ${\xi }_n=W(e_n)$ for each basis. Then Parseval's identity makes both expansions equal to the same $W$-pairing in L2 space. Drawing a fresh independent coefficient family for a different basis produces another field with the same law, not the same outcome-by-outcome field. This distinguishes basis invariance of a construction from an unspecified coupling of two constructions.^[ref]

For an open $U\subset D$, the field decomposes as $h=h_U+h^{D\setminus U}$, where $h_U$ is an independent zero-boundary Gaussian free field on $U$ and $h^{D\setminus U}$ is harmonic on $U$ and agrees with the outside field in the distributional sense.

A random set is local for a Gaussian free field if, conditionally on the set and its associated harmonic function, the remaining field is an independent zero-boundary field in its complement. This extends the deterministic Domain Markov property of the Gaussian free field to appropriate random revealed domains. Arbitrary field-dependent random sets need not have this property.

For the zero-boundary Gaussian free field in the unit disc, choose the logarithmic Green normalizationFor $t>0$, let ${\rho }_t$ be the uniform probability measure on the circle of radius $e^{-t}$ about zero. The Circle-average process of the Gaussian free field is the measure pairing $X_t=(h,{\rho }_t)$. This is meaningful even though the continuum field has no ordinary pointwise values: the circular measure has finite Green energy, because its diagonal logarithmic singularity is integrable.

Decreasing radius means increasing logarithmic time. Angular averaging of the Green kernel gives $\mathbb{E}[X_s{X}_t]=\min (s,t)$. Thus $X_t\to 0$ in L2 space as $t\downarrow 0$, and define $X_0=0$ by this limit; it is not a pointwise field restriction to the unit-disc boundary. The Circle-average process of the Gaussian free field is Brownian motion article verifies the covariance and continuity. For another positive Green normalization the same process is rescaled by its square root.^[ref]

In the logarithmic normalization of the Circle-average process of the Gaussian free field, the circular averages form standard Brownian motion in logarithmic radius. Let $r_s=e^{-s}$ and $r_t=e^{-t}$. Averaging $\log |1-z\overline{w}|$ over the two circles gives zero by the mean value property for harmonic functions. The average of $\log |z-w|$ is $\log \max (r_s,r_t)$, including equal radii by integrability of the logarithmic singularity. HenceThe pairings are jointly Gaussian and centered. Their increments on disjoint logarithmic-time intervals have zero covariance, hence are independent. Also $\mathbb{E}|X_t-X_s{|}^4=3|t-s{|}^2$, so the Kolmogorov continuity theorem supplies a continuous modification. Together with the limiting value $X_0=0$, this proves the Brownian motion law.

The result describes one nested family of circle averages about a fixed center, not the field on every spatial curve as a single Brownian path. If the Green kernel is multiplied by $c>0$, the covariance becomes $c\min (s,t)$ and $X/\sqrt{c}$ is standard Brownian motion.

For a measure $\mu$ and Green kernel $G$, its Green energy isFinite Green energy is the variance condition needed to pair a Gaussian free field with $\mu$.

Pinning a Gaussian free field at a vertex $o$ gives the field with value zero at $o$ and covarianceIt is independent of the removed scalar Gaussian mode at $o$.