Probability Theory and Related Fields

, Volume 139, Issue 3, pp 521–541

Gaussian free fields for mathematicians


DOI: 10.1007/s00440-006-0050-1

Cite this article as:
Sheffield, S. Probab. Theory Relat. Fields (2007) 139: 521. doi:10.1007/s00440-006-0050-1


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.

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Courant InstituteNew York UniversityNew YorkUSA

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