Level Set Percolation in Two-Dimensional Gaussian Free Field

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.

arXiv comment8 pages, 4 figures; v3: accepted version, minor changes, Supplemental Material included

References in corpus (24)

• Recent advances in percolation theory and its applications
• Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and SinhGordon models
• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Recent advances and open challenges in percolation
• Gaussian field theories, random Cantor sets and multifractality
• Conformal radii for conformal loop ensembles
• Conformal Random Geometry
• Duality and KPZ in Liouville Quantum Gravity
• Percolation with long-range correlated disorder
• Conformal field theory on the plane
• Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries
• Percolation thresholds and fractal dimensions for square and cubic lattices with long-range correlated defects
• Exact distributions of the number of distinct and common sites visited by N independent random walkers
• Correlation functions in loop models
• The Fyodorov-Bouchaud formula and Liouville conformal field theory
• Logarithmic CFT at generic central charge: from Liouville theory to the $Q$-state Potts model
• The sign clusters of the massless Gaussian free field percolate on ${\mathbb{Z}}^d$, $d⩾3$ (and more)
• Local logarithmic correlators as limits of Coulomb gas integrals
• Lecture notes on the Gaussian Free Field
• Level-set percolation for the Gaussian free field on a transient tree
• Two-point connectivity of two-dimensional critical $Q-$ Potts random clusters on the torus
• Liouville field theory and log-correlated Random Energy Models
• Topological effects and conformal invariance in long-range correlated random surfaces
• Operator Product Expansion in Liouville Field Theory and Seiberg type transitions in log-correlated Random Energy Models

Cited by in corpus (1)

• Percolation for two-dimensional excursion clouds and the discrete Gaussian free field