Conformal Invariance and Percolation

The large-scale behavior of two-dimensional critical percolation is expected to be described by a conformal field theory (CFT). Moreover, this putative percolation CFT is believed to be of the logarithmic type, exhibiting logarithmic corrections to the most commonly encountered and studied behavior of CFT correlations. It was recently shown by the first author [Cam24] that critical connection probabilities, when appropriately rescaled, have a well-defined and conformally co...

Crossing probabilities for critical 2-D percolation on large but finite lattices have been derived via boundary conformal field theory. These predictions agree very well with numerical results. However, their derivation is heuristic and there is evidence of additional symmetries in the problem. This contribution gives a preliminary examination some unusual modular behavior of these quantities. In particular, the derivatives of the "horizontal" and "horizontal-vertical" crossi...

Under some general assumptions, we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice and to the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. As an application to Bernoulli percolation, we obtain the scaling...

The aim of this paper is to explore possible ways of extending Smirnov's proof of Cardy's formula for critical site-percolation on the triangular lattice to other cases (such as bond-percolation on the square lattice); the main question we address is that of the choice of the lattice embedding into the plane which gives rise to conformal invariance in the scaling limit. Even though we were not able to produce a complete proof, we believe that the ideas presented here go in th...

These are the notes corresponding to the course given at the IAS-Park City graduate summer school in July 2007.

This paper has been withdrawn by the author due to a crucial sign error in Proposition 3.1.

The aim of the paper is to present numerical results supporting the presence of conformal invariance in three dimensional statistical mechanics models at criticality and to elucidate the geometric aspects of universality. As a case study we study three dimensional percolation at criticality in bounded domains. Both on discrete and continuous models of critical percolation, we test by numerical experiments the invariance of quantities in finite domains under conformal transfor...

Scaling limits of critical percolation models show major differences between low and high dimensional models. The article discusses the formulation of the continuum limit for the former case. A mathematical framework is proposed for the direct description of the limiting continuum theory. The resulting structure is expected to exhibit strict conformal invariance, and facilitate the mathematical discussion of questions related to universality of critical behavior, conformal in...

We prove a formula, first obtained by Kleban, Simmons and Ziff using conformal field theory methods, for the (renormalized) density of a critical percolation cluster in the upper half-plane "anchored" to a point on the real line. The proof is inspired by the method of images. We also show that more general bulk-boundary connection probabilities have well-defined, scale-covariant scaling limits, and prove a formula for the scaling limit of the (renormalized) density of the cri...

This is a review of results obtained by the author concerning the relation between conformally invariant random loops and conformal field theory. This review also attempts to provide a physical context in which to interpret these results by making connections with aspects of the nucleation theory of phase transitions and with general properties of criticality.