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Gaussian free fields for mathematicians. (English) Zbl 1132.60072
Summary: The \(d\)-dimensional Gaussian free field (GFF), also called the (Euclidean bosonic) massless free field, is a \(d\)-dimensional-time analog of Brownian motion. Just as Brownian motion is the limit of the simple random walk (when time and space are appropriately scaled), the GFF is the limit of many incrementally varying random functions on \(d\)-dimensional grids. We present an overview of the GFF and some of the properties that are useful in light of recent connections between the GFF and the Schramm-Loewner evolution.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J65 Brownian motion
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
81T10 Model quantum field theories
60B11 Probability theory on linear topological spaces
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