A characterisation of the Gaussian free field
Nathanael Berestycki, Ellen Powell, Gourab Ray
TL;DR
This paper proves that any conformally invariant, domain Markov property-satisfying random distribution in two dimensions is essentially a scaled Gaussian free field, under a mild moment condition.
Contribution
It characterizes the Gaussian free field as the unique conformally invariant distribution with the domain Markov property in two dimensions, given a fourth moment assumption.
Findings
Establishes the uniqueness of the Gaussian free field under specified conditions.
Shows the importance of conformal invariance and the domain Markov property in characterizing the GFF.
Provides a minimal moment condition for the characterization.
Abstract
We prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain Markov property is a multiple of the Gaussian free field. This result holds subject only to a fourth moment assumption.
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