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Gaussian free fields for mathematicians
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  • Published: 09 May 2007

Gaussian free fields for mathematicians

  • Scott Sheffield1 

Probability Theory and Related Fields volume 139, pages 521–541 (2007)Cite this article

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Abstract

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.

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Authors and Affiliations

  1. Courant Institute, New York University, New York, NY, 10012, USA

    Scott Sheffield

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  1. Scott Sheffield
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Correspondence to Scott Sheffield.

Additional information

Partially supported by NSF grant DMS0403182.

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Sheffield, S. Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139, 521–541 (2007). https://doi.org/10.1007/s00440-006-0050-1

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  • Received: 01 December 2003

  • Revised: 30 October 2006

  • Published: 09 May 2007

  • Issue Date: November 2007

  • DOI: https://doi.org/10.1007/s00440-006-0050-1

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Keywords

  • Hilbert Space
  • Brownian Motion
  • Orthonormal Basis
  • Triangular Lattice
  • Dimensional Hilbert Space
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