---
tag: EGCDMWYKQB
kind: concept
name: Gaussian free field
msc: [60G60, 60G15, 81T40]
specializes: [22WEM6RTW0]
sources:
  - url: https://en.wikipedia.org/wiki/Gaussian_free_field
same_as:
  wikidata: Q5527838
  wikipedia: Gaussian free field
---

## Summary

The Gaussian free field is a random function whose law is Gaussian, with covariance given by the Green function of the [[Laplacian|0E0B6RA8K0]].
In the continuum it is a random generalised function rather than a function.
It is the natural Gaussian analogue of a [[harmonic function|0DEB0JMN0W]] with random fluctuations.
In two dimensions the continuum field is conformally invariant and log-correlated.
It appears in statistical physics, in random surfaces and in the theory of the Schramm–Loewner evolution.

## Explanations

### L1 human

Think of a random landscape over a grid whose height is pinned to $0$ at the boundary.
Each height is pulled towards the average height of its neighbours, and randomness pushes it away.
Landscapes with large differences between neighbours are unlikely, so the surface is rough but not wild.
The discrete Gaussian free field is this random landscape, and its fine-mesh limit is the continuum free field.