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∼L^{2}/sqrt[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.