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.
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