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Gaussian Free Field

A Gaussian free field is a Gaussian random generalized function whose covariance is the Green function of a Laplacian. For the real zero-Dirichlet field Phi_D on a bounded planar domain D, one convention is

E[Phi_D(f)Phi_D(g)]=int_Dint_Df(z)G_D(z,w)g(w)dzdw,

where f,g are compactly supported smooth test functions and G_D=(-Delta_D)^(-1) is the Green function with Dirichlet boundary conditions. The field is centered, and these covariances determine its joint Gaussian laws. In two dimensions its pointwise variance diverges, so evaluation against test functions is essential. With the Euclidean Laplacian, the logarithmic singularity is (2pi)^(-1)ln(1/|z-w|).

A different frequently used normalization multiplies this field by sqrt(2pi), giving logarithmic singularity ln(1/|z-w|). That normalization is used to define the parameter gamma in Liouville quantum gravity (Duplantier and Sheffield 2008).

The whole-plane Gaussian free field is naturally defined modulo additive constants. It acts on compactly supported test functions of integral zero, using the covariance kernel -(2pi)^(-1)ln||x-y||.

See also

Ashkin-Teller Model, Brownian Motion, Dirichlet Boundary Conditions, Discrete Gaussian Model, Gaussian Process, Green's Function, Imaginary Gaussian Multiplicative Chaos, Integer Lipschitz Height Function, Liouville Quantum Gravity, Real Lipschitz Height Function, Six-Vertex Model, Two-Valued Local Set

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References

Duplantier, B. and Sheffield, S. "Liouville Quantum Gravity and KPZ." 11 Aug 2008. https://arxiv.org/abs/0808.1560.

Cite this as:

Weisstein, Eric W. "Gaussian Free Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GaussianFreeField.html

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