Gaussian free field

The Gaussian free field (GFF) is a centered Gaussian random field defined on a domain in \mathbb{R}^d (d \geq 2), characterized by a covariance kernel given by the Green's function of the Laplacian operator with Dirichlet boundary conditions, serving as a multidimensional analogue of Brownian motion where the one-dimensional time parameter is replaced by spatial coordinates.^[1]^[2] In the continuum setting, it is a random generalized function or distribution rather than a pointwise-defined function, due to divergences in variance at individual points, and is often indexed by test functions or measures with finite Dirichlet energy.^[1]^[3] The discrete GFF arises on a finite subset of the integer lattice \mathbb{Z}^d, where it is a Gaussian vector with density proportional to \exp\left(-\frac{1}{4d} \sum_e |\nabla \gamma(e)|^2\right) under zero boundary conditions on the domain's boundary, and its covariance is the discrete Green's function, reflecting a Markov property that allows decomposition into independent fields on subdomains.^[4]^[1] The continuum GFF emerges as a scaling limit of the discrete version as the lattice mesh tends to zero, preserving the Gaussian structure and covariance, though it requires regularization (e.g., via circle averages in two dimensions) to define local values.^[1]^[3] This field can be viewed as a Gaussian perturbation around harmonic functions, as its values "try" to minimize the Dirichlet energy while incorporating random fluctuations aligned with the domain's geometry.^[3] In two dimensions, the GFF is log-correlated—its covariance behaves like -\log|x-y| for nearby points—and exhibits conformal invariance under angle-preserving maps, making it a cornerstone of two-dimensional stochastic geometry.^[1]^[4] Its circle-average processes evolve like Brownian motion in an angular parameter, and level sets or local sets couple with Schramm-Loewner evolution (SLE) curves, such as SLE_4, facilitating the study of random interfaces and scaling limits of lattice models like uniform spanning trees or loop-erased random walks.^[1] The GFF has profound applications across mathematics and physics: in probability theory, it models random surfaces, multiplicative chaos measures, and the geometry of large random graphs or planar maps; in statistical mechanics, it describes massless free fields and critical phenomena; and in quantum field theory, particularly conformal field theory and Liouville quantum gravity, it underlies random metrics of the form e^{\gamma h(z)} |dz|^2 (where h is the GFF and \gamma < 2), enabling the construction of quantum area and length measures via regularization limits.^[1]^[3]^[2] These connections highlight the GFF's role in bridging discrete combinatorics, continuous stochastic processes, and theoretical physics.^[4]

Fundamentals

Definition and overview

The Gaussian free field (GFF) is a centered Gaussian random field \phi on a domain D \subset \mathbb{R}^d (with d \geq 2), characterized by its covariance kernel given by the Green's function G_D(x,y) of the negative Laplacian operator with Dirichlet boundary conditions. Specifically, G_D solves the equation

-\Delta G_D(\cdot, y) = \delta_y \quad \text{in } D, \quad G_D(x,y) = 0 \quad \text{for } x \in \partial D,

yielding \mathbb{E}[\phi(x) \phi(y)] = G_D(x,y). This structure defines the GFF as a random distribution rather than a pointwise function, except in one dimension where it coincides with Brownian motion. The GFF emerges as the scaling limit of discrete height functions from lattice models like the discrete Gaussian free field, capturing large-scale fluctuations in random surfaces or interfaces without a mass term—hence the designation "free."^[3] In mathematical physics, it serves as the prototypical model for a massless scalar field, analogous to the free boson in quantum field theory. The field has zero mean, \mathbb{E}[\phi] = 0, and exhibits stationarity in flat space \mathbb{R}^d, where the covariance depends only on the vector x - y. Among Gaussian fields, the GFF uniquely minimizes the Dirichlet energy \int_D |\nabla \phi|^2 \, dx subject to its covariance constraints.^[4] Formally, it corresponds to the Gaussian measure with quadratic form \frac{1}{2} \int_D |\nabla \phi|^2 \, dx (up to normalization conventions that vary by dimension and context), where the covariance arises as the Green's function, or inverse, of the kinetic operator -\Delta.

Historical development

The Gaussian free field (GFF) traces its origins to the 1960s in axiomatic quantum field theory, where it emerged as the Euclidean version of the massless scalar field, providing a rigorous foundation for non-perturbative constructions. Edward Nelson introduced key concepts through his development of Markov properties for Gaussian processes in Euclidean space, enabling the formulation of free fields as Gaussian random distributions. In the 1970s, Raphael Høegh-Krohn advanced the theory by constructing cutoff-free models of quantum fields in two space-time dimensions based on Gaussian measures, achieving early rigorous results for interacting theories perturbing around the free field.^[5] The 1970s and 1980s saw further progress in constructive quantum field theory, led by James Glimm and Arthur Jaffe, who established the GFF as the central free theory for building interacting models via functional integrals and cluster expansions in dimensions up to three. David Brydges contributed pivotal rigorous techniques, including gradient perturbations and tree expansions for massless Gaussian fields, enhancing the mathematical control over these constructions.^[6] A revival occurred in the 1990s and 2000s within probability theory, shifting focus from perturbative quantum field approaches to non-perturbative tools that revealed the GFF's exact solvability, particularly in two dimensions. Scott Sheffield's work, starting with his 2003 survey and extending to circle average embeddings and scaling limits of discrete approximations around 2005–2010, formalized the continuum GFF as a probabilistic object analogous to Brownian motion.^[7] Jason Miller built on this in the late 2000s and 2010s through collaborations on level lines and boundary data generalizations, solidifying the GFF's role in modern random geometry.^[8] This evolution enabled precise analyses of the GFF's structure without relying on quantum field perturbations.

Discrete Gaussian free field

Lattice construction

The discrete Gaussian free field (GFF) on a finite undirected graph G = (V, E) with designated boundary vertices \partial V \subset V is constructed as a centered multivariate Gaussian random vector \phi = (\phi_v)_{v \in V \setminus \partial V} taking values in \mathbb{R}^{V \setminus \partial V}.^[9] The law of \phi has probability density proportional to

\exp\left( -\frac{1}{2} \sum_{\{u,v\} \in E} (\phi_u - \phi_v)^2 \right)

with respect to Lebesgue measure on \mathbb{R}^{V \setminus \partial V}, subject to the Dirichlet boundary conditions \phi_v = 0 for all v \in \partial V.^[9] This quadratic form in the exponent corresponds to the Dirichlet energy of \phi, ensuring that \phi is a Gaussian vector with mean zero and covariance matrix given by the (pseudo-)inverse of the graph Laplacian restricted to V \setminus \partial V.^[10] The graph Laplacian \Delta_G is defined by its action on functions u: V \to \mathbb{R} as

(\Delta_G u)(v) = \sum_{w \sim v} (u(v) - u(w)) = \deg(v) \, u(v) - \sum_{w \sim v} u(w)

for v \in V \setminus \partial V, where \sim denotes adjacency in E and \deg(v) is the degree of v; on the boundary, (\Delta_G u)(v) = 0.^[9] This operator is symmetric and positive semi-definite on the subspace orthogonal to constants, and the covariance kernel is the Green's function (\Delta_G^{-1})(v, w) solving \Delta_G (\Delta_G^{-1}(\cdot, w)) = \delta_w on V \setminus \partial V with zero boundary conditions, i.e., \mathbb{E}[\phi_v \phi_w] = (\Delta_G^{-1})(v, w).^[3] To sample \phi, one directly draws from the multivariate normal distribution \mathcal{N}(0, \Delta_G^{-1}), which can be computed numerically via Cholesky decomposition or other matrix inversion methods for finite graphs.^[10] An explicit integral representation of the Green's function arises from the associated continuous-time random walk on G, where the generator is -\Delta_G:

(\Delta_G^{-1})(v, w) = \int_0^\infty (e^{-t \Delta_G})_{v,w} \, dt.

This integral equals the expected total time the walk, starting at v, spends at w before hitting \partial V, providing a probabilistic interpretation of the covariance as an occupation measure.^[3] To extend the construction to infinite lattices such as \mathbb{Z}^d, one approximates via exhaustion by taking a sequence of large finite subdomains V_n \uparrow \mathbb{Z}^d with \partial V_n receding to infinity, defining \phi^{(n)} on each V_n \setminus \partial V_n, and passing to the weak limit of the laws if it exists (which holds for d \geq 3, but requires regularization in d=2 due to logarithmic divergence). Periodization on tori can also define a zero-boundary analog by identifying boundaries periodically, though this yields a Neumann-like field without fixed zeros.^[10] A concrete example occurs on the two-dimensional square lattice \mathbb{Z}^2, where G is the infinite grid graph restricted to a large finite box V = \{0, \dots, N\}^2 \cap \mathbb{Z}^2 with \partial V the outer edges set to zero. Here, the Green's function relates directly to the simple symmetric random walk on \mathbb{Z}^2, and the variance \mathbb{E}[\phi_0^2] = (\Delta_G^{-1})(0,0) grows as \frac{2}{\pi} \log N + O(1) for points near the origin as N \to \infty, reflecting the recurrent nature of random walks in two dimensions.^[10]

Covariance and correlation functions

The covariance kernel of the discrete Gaussian free field (GFF) on a graph G with Laplacian \Delta_G is given by the inverse \Delta_G^{-1}, which serves as the precision matrix's inverse and defines the pairwise covariances \operatorname{Cov}(\phi_v, \phi_w) = (\Delta_G^{-1})_{v,w} for vertices v, w \in V(G).^[3] This kernel is positive definite on finite domains with Dirichlet boundary conditions, ensuring the field is a well-defined Gaussian process with non-negative variances and valid correlation structure.^[11] On the integer lattice \mathbb{Z}^d for d \geq 3, the kernel exhibits power-law decay, with |(\Delta_{\mathbb{Z}^d}^{-1})_{v,w}| \sim c_d / |v - w|^{d-2} for large separations |v - w| and some constant c_d > 0, reflecting long-range correlations analogous to the continuous case. The two-point correlation function is precisely the covariance E[\phi_v \phi_w] = (\Delta_G^{-1})_{v,w}, which quantifies the expected product of field values at distinct sites and decays according to the kernel's properties.^[11] For higher-point correlation functions, the Gaussian nature of the field allows application of Wick's theorem: odd moments vanish, while even moments E[\phi_{v_1} \cdots \phi_{v_{2n}}] equal the sum over all perfect matchings of the vertices, where each pairing contributes the product of the corresponding two-point covariances.^[11] This factorization simplifies computations of joint distributions and underpins many probabilistic analyses of the field. Local variances E[\phi_v^2] = (\Delta_G^{-1})_{v,v} measure pointwise fluctuations and diverge logarithmically with domain size in two dimensions; for a box of side length N, E[\phi_v^2] \sim (2/\pi) \log N for interior points v, indicating growing variability as the lattice expands.^[12] In higher dimensions d > 2, variances remain finite and bounded independently of domain size. Averages of the field over subsets satisfy a central limit theorem, with the normalized sum converging in distribution to a standard Gaussian under mild regularity conditions on the averaging set.^[11] Harnack inequalities apply to the covariance kernel, providing bounds on ratios of variances such as E[\phi_v^2] / E[\phi_w^2] \leq C for vertices v, w within comparable distances from the boundary, where C depends only on graph geometry and not on the specific domain size; this follows from the harmonic properties of the Green's function away from singularities.^[13]

Continuum Gaussian free field

General formulation

The continuum Gaussian free field (GFF) on a domain D \subset \mathbb{R}^d is defined in the distributional sense as a random generalized function \phi, acting on test functions f, g \in C_c^\infty(D), which are smooth with compact support in D. Specifically, \phi is a centered Gaussian random variable on the space of distributions such that \mathbb{E}[\phi(f) \phi(g)] = \int_D \int_D f(x) G_D(x,y) g(y) \, dx \, dy, where G_D(x,y) is the Dirichlet Green's function on D.^[14] The Dirichlet Green's function G_D(x,y) satisfies the equation -\Delta_x G_D(x,y) = \delta(x-y) for x,y \in D, with boundary condition G_D(x,y) = 0 when x or y lies on \partial D. In the whole space \mathbb{R}^d, the Green's function can be expressed via Fourier transform as G(x,y) = (2\pi)^{-d} \int_{\mathbb{R}^d} |\xi|^{-2} e^{i \xi \cdot (x-y)} \, d\xi, which asymptotically behaves as G(x,y) \sim c_d |x-y|^{2-d} for d \neq 2, where c_d is a dimension-dependent constant.^[14] The GFF can also be characterized through its energy form: it is the unique (in law) Gaussian field that minimizes the Dirichlet energy \mathbb{E}[\|\nabla \phi\|^2_{L^2(D)}] subject to the fixed covariance structure given by G_D. Formally, the GFF corresponds to a Gaussian measure with density proportional to \exp\left( -\frac{1}{2} \int_D |\nabla \phi|^2 \, dx \right), though this expression requires careful renormalization in the continuum limit (constants may vary with normalization conventions, especially in 2D).^[14] Due to ultraviolet (UV) divergences, the GFF is not a pointwise-defined function but requires regularization, such as circle averages \phi_\varepsilon(z) over balls of radius \varepsilon around z \in D or mollification with smooth kernels. Zero-boundary conditions are imposed by conditioning the regularized field on vanishing values near \partial D. As \varepsilon \to 0, these approximations converge in distribution to the GFF on suitable test function spaces.^[14]

One-dimensional case

In the one-dimensional continuum setting, the Gaussian free field (GFF) is defined on an interval (0, L) with Dirichlet boundary conditions \phi(0) = \phi(L) = 0. It is a centered Gaussian process \{\phi(x)\}_{x \in (0,L)} whose covariance kernel is given by the Green's function of the negative Laplacian operator -\frac{d^2}{dx^2} with zero boundary conditions, explicitly

G(x,y) = \min(x,y) - \frac{xy}{L}.

This kernel satisfies -\frac{\partial^2}{\partial x^2} G(x,y) = \delta(x-y) for x,y \in (0,L), with G(0,y) = G(L,y) = 0.^[15] The one-dimensional GFF is equivalent to a Brownian bridge from 0 to 0 over the time interval [0, L], scaled such that the underlying Brownian motion has variance parameter 1. As a consequence, \phi has continuous sample paths almost surely, distinguishing it from the distribution-valued nature of the GFF in higher dimensions. The process exhibits the Markov property, and its finite-dimensional distributions are multivariate Gaussian with the specified covariance. In this sense, the one-dimensional GFF is considered "trivial" compared to higher-dimensional cases, as its roughness is solely that of Brownian motion, without additional logarithmic divergences in the covariance structure near the diagonal—specifically, G(x,x) = x(L - x)/L < \infty for all x \in (0,L).^[15]^[16] The variance of the field at position x is \mathbb{E}[\phi(x)^2] = G(x,x) = x(L - x)/L, which achieves a maximum of L/4 at x = L/2 and exhibits linear growth near the boundaries before vanishing there. Moments and simulations can be obtained explicitly via the Karhunen–Loève expansion, which diagonalizes the covariance operator using the eigenfunctions of -\frac{d^2}{dx^2} on (0,L) with Dirichlet boundaries: \phi_j(x) = \sqrt{2/L} \sin(j \pi x / L) for j = 1, 2, \dots, with corresponding eigenvalues \lambda_j = (j \pi / L)^2. Thus,

\phi(x) = \sum_{j=1}^\infty Z_j \frac{\sqrt{2/L} \sin(j \pi x / L)}{j \pi / L} = \sqrt{2 L} \sum_{j=1}^\infty \frac{Z_j}{j \pi} \sin\left( j \pi \frac{x}{L} \right),

where the Z_j are i.i.d. standard normal random variables. This series converges almost surely and uniformly on compact subintervals of (0,L), enabling efficient numerical sampling by truncating at large j.^[4]

Two-dimensional case

The two-dimensional continuum Gaussian free field (GFF) on a domain D \subset \mathbb{C} with Dirichlet boundary conditions is defined as a centered Gaussian random distribution whose covariance kernel is the Green's function G_D(x,y) of the Laplacian -\Delta on D with zero boundary values.^[17] On the whole plane, this kernel takes the form G(x,y) = \frac{1}{2\pi} \log \frac{1}{|x-y|}, exhibiting the characteristic logarithmic singularity.^[17] For bounded domains such as the unit disk, G_D(x,y) asymptotically behaves as \frac{1}{2\pi} \log \frac{1}{|x-y|} near x = y, plus additional harmonic terms that ensure the boundary conditions are satisfied.^[17] Due to the logarithmic divergence of the covariance, the 2D GFF cannot be realized pathwise as a random function but exists almost surely only as a random distribution in the sense of Schwartz.^[2] To make sense of local values, regularization is necessary; a common approach is the circle average process, defined for \varepsilon > 0 small by

\phi_\varepsilon(x) = \frac{1}{2\pi \varepsilon} \int_{|z-x|=\varepsilon} \phi(z) \, |dz|,

where the integral is taken over the circle of radius \varepsilon centered at x \in D with \varepsilon small enough to stay inside D. The variance of this process satisfies \mathrm{Var}(\phi_\varepsilon(x)) \sim \log(1/\varepsilon) as \varepsilon \to 0, reflecting the growing fluctuations at finer scales. An alternative whole-plane regularization involves the massive GFF, obtained by adding a small mass term m^2 \phi to the field equation and taking the limit m \to 0, which approximates the massless case while remaining a well-defined random function.^[17] The family of circle average processes \{\phi_\varepsilon(x)\}_{\varepsilon > 0} admits a locally Hölder continuous modification in the joint variables (x, -\log \varepsilon) for any exponent \gamma < 1/2, providing a pathwise realization of the field's local behavior despite the underlying distributional nature. The 2D continuum GFF arises as the scaling limit of the discrete GFF on the \mathbb{Z}^2 lattice, suitably rescaled, as proven by Sheffield.^[2] This log-correlated structure also underpins its connection to Gaussian multiplicative chaos measures, constructed via limits of approximations like e^{\gamma \phi_\varepsilon - \frac{\gamma^2}{2} \mathrm{Var}(\phi_\varepsilon)} \, d\lambda for \gamma \in (0,2), yielding random measures supported on D.^[18]

Properties and applications

Conformal invariance and scaling limits

The two-dimensional Gaussian free field exhibits conformal covariance, a property that describes how its distribution transforms under conformal mappings of the domain. Specifically, for a simply connected domain D and a conformal map f: D \to D', the field \phi' on D' satisfies \phi' = \phi \circ f^{-1} + \frac{2}{\sqrt{\pi}} \arg f' \circ f^{-1}, where \arg f' denotes the argument of the complex derivative, ensuring the law remains consistent up to this deterministic shift. This transformation law preserves the Gaussian structure and covariance kernel, reflecting the logarithmic nature of the Green's function in two dimensions. The property was rigorously established in the context of contour lines and their scaling behavior. Scaling limits provide a bridge between discrete and continuum models of the Gaussian free field. The discrete Gaussian free field on a lattice \varepsilon \mathbb{Z}^2 \cap D, with Dirichlet boundary conditions, converges in distribution to the continuum Gaussian free field as the mesh size \varepsilon \to 0, measured in negative Sobolev spaces H^{-\epsilon} for \epsilon > 0. This convergence is achieved by establishing tightness of the measures on the space of distributions, leveraging techniques such as Dubédat's conformal welding, which couples boundary arcs via SLE processes to reconstruct the field consistently across scales. The limit mechanism relies on the uniform control of the discrete field's variances and correlations, ensuring the logarithmic kernel emerges in the continuum. The conformal invariance of the Gaussian free field is dimension-dependent, arising fundamentally from the form of the covariance kernel. In two dimensions, the logarithmic kernel -\log |z - w| (up to constants and harmonic corrections) aligns with the invariance under full conformal transformations of the plane, enabling the precise covariance law described above. In higher dimensions d > 2, the kernel |z - w|^{2-d} supports only Weyl rescalings of the metric, limiting invariance to dilation-type transformations rather than general conformals, as the full conformal group is smaller. In one dimension, the field reduces to a Brownian motion (or bridge), whose scaling limit from discrete random walks inherits pathwise invariance under linear maps, but lacks the richer two-dimensional structure. A key aspect of these scaling limits in two dimensions is captured by the Dubins-Schwartz theorem adapted to the radial maximum of the field, where the maximum value around the origin behaves asymptotically as \sqrt{\log (1/r)} (with leading deterministic term), highlighting the logarithmic fluctuations inherent to the limit process. This result underscores the tightness mechanisms by embedding the field's extrema into a Brownian-like structure, facilitating convergence proofs.

Connections to random geometry and quantum gravity

The two-dimensional Gaussian free field (GFF) provides the underlying structure for Liouville quantum gravity (LQG), a probabilistic model of random conformal metrics on surfaces that formalizes notions from two-dimensional quantum gravity. In this framework, the GFF φ serves as the height function for a random metric informally defined by e^{\gamma \phi} |dz|^2, where γ ∈ (0, 2) parametrizes the field's roughness and governs the geometry's fractal dimension. The corresponding volume form is obtained via Gaussian multiplicative chaos as the almost-sure limit

\mu_\gamma(dz) = \lim_{\epsilon \to 0} e^{\gamma \phi_\epsilon(z) - \frac{\gamma^2}{2} \mathrm{Var}(\phi_\epsilon(z))} \, dz,

where φ_ε denotes a standard mollification of the GFF; this construction holds in the subcritical regime γ < 2 and enables the definition of LQG measures on domains and surfaces. LQG with parameter γ corresponds to a central charge c = 1 for the underlying GFF, while the central charge of the Liouville theory is 26 - c_m, where c_m is the central charge of any coupled matter fields, ensuring consistency with conformal field theory expectations for quantum gravity (with c_m = 25 - 6 Q^2 and Q = \gamma/2 + 2/\gamma).^[19] The GFF's level lines—continuous curves where the field attains constant height values—couple naturally with Schramm-Loewner evolution (SLE) processes, revealing deep ties to random geometry. Specifically, with appropriate boundary conditions, these level lines are distributed as chordal SLE_4 paths, a result established through exact solvability and partition function identities.^[20] Flow lines of the GFF, interpreted via exponentiation with an imaginary angle, yield SLE_κ(ρ) processes for κ = 4 and suitable force points ρ, parameterized by the "height" of the lines; this framework, known as imaginary geometry, extends to space-filling SLE variants and underpins couplings with discrete models.^[21] Conformal invariance of the GFF enables these SLE couplings, allowing random curves to be viewed as deterministic flows in the field's geometry. In random geometry applications, the GFF emerges as the scaling limit of height functions for discrete structures like the uniform spanning tree (UST) on planar lattices. The UST's Peano curve, which traverses the tree's branches to fill the domain, converges in the fine-mesh limit to an SLE_8 path, with the associated height fluctuations converging to the GFF through imaginary geometry constructions.^[22] More broadly, LQG surfaces arise as scaling limits of uniform infinite planar quadrangulations, where discrete embeddings and metrics converge to the LQG metric tensor in the Gromov-Hausdorff sense; this establishes LQG with γ = √(8/3) as the continuum limit of "pure gravity" models without matter fields. These convergences bridge discrete random maps to continuum quantum gravity, with the GFF encoding the field's logarithmic correlations as the universal fluctuation mechanism.