# Gaussian free fields for mathematicians

```@article{Sheffield2003GaussianFF,
title={Gaussian free fields for mathematicians},
author={Scott Sheffield},
journal={Probability Theory and Related Fields},
year={2003},
volume={139},
pages={521-541}
}```
• S. Sheffield
• Published 4 December 2003
• Mathematics
• Probability Theory and Related Fields
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…
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## References

SHOWING 1-10 OF 37 REFERENCES

• Mathematics, Physics
• 1997
We study the continuum scaling limit of some statistical mechanical models defined by convex Hamiltonians which are gradient perturbations of a massless free field. By proving a central limit theorem
: Random surfaces in statistical physics are commonly modeled by a real-valued function phi on a lattice, whose probability density penalizes nearest-neighbor fluctuations. Precisely, given an even
• Mathematics
• 2001
We consider the lattice version of the free field in two dimensions (also called harmonic crystal). The main aim of the paper is to discuss quantitatively the entropic repulsion of the random surface
This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a
• Mathematics
• 2001
This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain \(D\mathop \subset \limits_ \ne \mathbb{C} \) is equal to the radial SLE2 path. In particular, the