The 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.
Pinning a Gaussian free field at a vertex gives the field with value zero at and covarianceIt is independent of the removed scalar Gaussian mode at .
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The Gaussian free field (GFF) is a mathematical object commonly studied in the fields of probability theory, statistical mechanics, and quantum field theory. It serves as a foundational model for understanding various phenomena in physics and mathematics due to its intrinsic properties and connections to Gaussian processes.