Level Set Percolation in Two-Dimensional Gaussian Free Field

TL;DR

This paper investigates the percolation properties of level sets in the 2D Gaussian Free Field, revealing a nontrivial transition with unique critical behavior and supporting findings through simulations and theoretical analysis.

Contribution

It demonstrates the existence of a nontrivial percolation transition in the 2D Gaussian Free Field and characterizes the critical phenomena using a loop-model mapping.

Findings

Percolation transition is nontrivial with exponential divergence of correlation length.

Critical clusters are logarithmic fractals with specific scaling laws.

Numerical simulations support the theoretical predictions.

Abstract

The nature of level set percolation in the two-dimension Gaussian Free Field has been an elusive question. Using a loop-model mapping, we show that there is a nontrivial percolation transition, and characterize the critical point. In particular, the correlation length diverges exponentially, and the critical clusters are "logarithmic fractals", whose area scales with the linear size as $A \sim L^2 / \sqrt{\ln L}$. The two-point connectivity also decays as the log of the distance. We corroborate our theory by numerical simulations. Possible CFT interpretations are discussed.