Conformal Symmetries in the Extremal Process of Two-Dimensional Discrete Gaussian Free Field

Abstract

We study the extremal process associated with the Discrete Gaussian Free Field on the square lattice and elucidate how the conformal symmetries manifest themselves in the scaling limit. Specifically, we prove that the joint process of spatial positions (x) and centered values (h) of the extreme local maxima in lattice versions of a bounded domain $$D\subset {\mathbb {C}}$$ converges, as the lattice spacing tends to zero, to a Poisson point process with intensity measure $$Z^D(\mathrm{d}x)\otimes \mathrm{e}^{-\alpha h}\mathrm{d}h$$, where $$\alpha $$ is a constant and $$Z^D$$ is a random a.s.-finite measure on D. The random measures $$\{Z^D\}$$ are naturally interrelated; restrictions to subdomains are governed by a Gibbs–Markov property and images under analytic bijections f by the transformation rule $$(Z^{f(D)}\circ f)(\mathrm{d}x)\,\overset{\mathrm{law}}{=}\,|f'(x)|^4\, Z^D(\mathrm{d}x)$$. Conditions are given that determine the laws of these measures uniquely. These identify $$Z^D$$ with the critical Liouville Quantum Gravity associated with the Continuum Gaussian Free Field.