Level Set Percolation in the Two-Dimensional Gaussian Free Field

The nature of level set percolation in the two-dimensional 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 similar to L-2/root lnL. The two-point connectivity also decays as the log of the distance. We corroborate our theory by numerical simulations. Possible conformal field theory interpretations are discussed.

Automated summary

A nontrivial percolation transition in level set percolation within the two-dimensional Gaussian free field has been identified, with the critical point characterized and properties such as exponentially diverging correlation length and critical clusters exhibiting logarithmic fractals. The area of these clusters scales with linear size as A similar to L-2/root lnL, while two-point connectivity decays logarithmically with distance. These findings are supported by numerical simulations, with potential interpretations in conformal field theory discussed.