Conformal symmetries in the extremal process of two-dimensional discrete Gaussian Free Field

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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 $Z^D(dx)\otimes e^{-\alpha h}d h$, 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 $(Z^{f(D)}\circ f)(d x)\overset{\text{law}}=|f'(x)|^4\, Z^D(d x)$.
• Conditions are given that determine the laws of these measures uniquely.