1 Introduction

In the theory of critical phenomena it is usually assumed that a physical system near a continuous phase transition is characterized by a single length scale (the correlation length) in terms of which all other lengths should be measured. When combined with the experimental observation that the correlation length diverges at the phase transition, this simple but strong assumption, known as the scaling hypothesis, leads to the belief that at criticality the system has no characteristic length, and is therefore invariant under scale transformations. This implies that all thermodynamic functions at criticality are homogeneous functions, and predicts the appearance of power laws.

It also suggests that for models of critical systems realized on a lattice, one can attempt to take a continuum scaling limit in which the mesh of the lattice is sent to zero while focus is kept on “macroscopic” observables that capture the large scale behavior. In the limit, the discrete model should converge to a continuum one that encodes the large scale properties of the original model, containing at the same time more symmetry. In many cases, this allows to derive extra insight by combining methods of discrete mathematics with considerations inspired by the continuum limit picture. The simplest example of such a continuum random model is Brownian motion, which is the scaling limit of the simple random walk. In general, though, the complexity of the discrete model makes it impossible even to guess the nature of the scaling limit, unless some additional feature can be shown to hold, which can be used to pin down properties of the continuum limit. Two-dimensional critical systems belong to the class of models for which this can be done, and the additional feature is conformal invariance.

In the eighties, physicists started exploring the consequences of conformal invariance. The constraints it poses on the continuum limits are particularly stringent in two dimensions, where every analytic function defines a conformal transformation (at points where its derivative is non-vanishing). A large number of critical problems in two dimensions were analyzed using conformal methods, including the Ising and Potts models, Brownian motion, the Self-Avoiding Walk (SAW), percolation, and Diffusion Limited Aggregation (DLA). The large body of knowledge and techniques that resulted, starting with the work of Belavin, Polyakov and Zamolodchikov [6, 7] in the early eighties, goes under the name of Conformal Field Theory (CFT). One of the main goals of CFT, and its most important application to statistical mechanics, is a complete classification of 2D critical systems in universality classes according to their scaling limit.

At the hands of theoretical physicists, CFT has been very successful in producing many interesting results. However, conformal invariance remains a conjecture for most models describing critical systems. Moreover, despite its success, CFT leaves a number of open problems. First of all, the theory deals primarily with correlation functions and quantities that are not directly expressible in terms of those are not easily accessible. Secondly, given some critical lattice model, there is no way, within the theory itself, of deciding to which CFT it corresponds. A third limitation, of a different nature, is due to the fact that the methods of CFT, although very powerful, are generally speaking not completely rigorous from a mathematical point of view.

Partly because of the success of CFT, in recent years work on critical phenomena had slowed down somewhat, probably due to the feeling that most of the leading problems had been resolved. In a somewhat surprising twist, the most recent developments in the area of two-dimensional critical systems have emerged in the mathematics literature, and have followed a completely new direction, which has provided new tools and a way of surpassing at least some of the limitations of CFT.

These developments came on the heels of interesting results on the scaling limits of discrete models (see, e.g., the work of Aizenman [1, 2], Benjamini-Schramm [8], Aizenman-Burchard [3], Aizenman-Burchard-Newman-Wilson [4], Aizenman- Duplantier-Aharony [5] and Kenyon [18, 19]) but they differ greatly from those because they are based on a radically new approach whose main tool is the Stochastic Loewner Evolution (SLE), or Schramm-Loewner Evolution, as it is also known, introduced by Schramm [28]. The new approach, which is probabilistic in nature, focuses directly on non-local structures that characterize a given system, such as cluster boundaries in Ising, Potts and percolation models, or loops in the O(n) model. At criticality, these non-local objects become, in the continuum limit, random curves whose distributions can be uniquely identified thanks to their conformal invariance and a certain “Markovian” property. There is a one-parameter family of SLEs, indexed by a positive real number κ, and they appear to be essentially the only possible candidates for the scaling limits of interfaces of two-dimensional critical systems that are believed to be conformally invariant.

The identification of the scaling limit of interfaces of critical lattice models with SLE curves has led to tremendous progress in recent years. The main power of SLE stems from the fact that it allows to compute different quantities; for example, percolation crossing probabilities and various percolation critical exponents. Therefore, relating the scaling limit of a critical lattice model to SLE allows for a rigorous determination of some aspects of the large scale behavior of the lattice model. For the mathematician, the biggest advantage of SLE over CFT lies maybe in its mathematical rigor. But many physicists working on critical phenomena and CFT have promptly recognized the importance of SLE and added it to their toolbox.

In the context of the Ising, Potts and O(n) models, as well as percolation, an SLE curve is believed to describe the scaling limit of a single interface, which can be obtained by imposing special boundary conditions. A single SLE curve is therefore not in itself sufficient to immediately describe the scaling limit of the unconstrained model without boundary conditions in the whole plane (or in domains with boundary conditions that do not determine a single interface), and contains only limited information concerning the connectivity properties of the model.

A more complete description can be obtained in terms of loops, corresponding to the scaling limit of cluster boundaries. Such loops should also be random and have a conformally invariant distribution, closely related to SLE. This observation led Wendelin Werner [35, 36] and Scott Sheffiled [29] to the definition of Conformal Loop Ensembles (CLEs), which are, roughly speaking, random collections of conformally invariant fractal loops. As for SLE, there is a one parameter family of CLEs.

The study of SLE and its connections to statistical mechanics is currently among the most active and rapidly growing areas of probability theory, and, especially in the work of Lawler, Scrhamm and Werner [20–24, 26, 25] has already provided many impressive (rigorous) results concerning various two-dimensional models such as Brownian motion, the loop erased random walk and the uniform spanning tree.

At present, there are at least three distinct directions into which research is developing. The first concerns the study of SLE itself (see [27]), a task that is being carried out by a growing number of mathematicians. Physicists on their part have used SLE to obtain information about CFT, and have also been active in reproducing certain SLE results with “traditional” CFT methods, thus pushing the boundaries of CFT. The hope in both the mathematics and physics community is that the interaction between SLE and CFT will continue to shed new light on both approaches and provide a deeper understanding of critical phenomena.

The third direction, which has been explored most notably by Lawler, Schramm and Werner [24, 26], Smirnov [30, 33, 31, 32], Smirnov and Werner [34], and Camia and Newman [12–15], consists in proving conformal invariance of the scaling limit for individual models of statistical mechanics, proving the convergence of interfaces in those models to SLE and CLE curves, and deriving information about the discrete models from their scaling limits.

The case of percolation is particularly illuminating because the connection with both SLE and CLE has been fully established and successfully exploited to derive rigorously several important results previously obtain by physicists using CFT methods. Moreover, the description of the “full” scaling limit, i.e., the scaling limit of all cluster boundaries, represents an important first step in understanding the near-critical scaling limit, when the system is slightly off-critical and it approaches criticality at an appropriate speed in the scaling limit.

1.2 Percolation

Percolation as a mathematical theory was introduced by Broadbent and Hammersley [10, 11] to model the spread of a gas or a fluid through a porous medium. To mimic the randomness of the medium, they declared the edges of the d-dimensional cubic lattice independently open (to the passage of the gas or fluid) with probability p or closed with probability 1 − p. Since then, many variants of this simple model have been studied, attracting the interest of both mathematicians and physicists.

Mathematicians are interested in percolation because of its deceiving simplicity which hides difficult and elegant results. From the point of view of physicists, percolation is maybe the simplest statistical mechanical model undergoing a continuous phase transition as the value of the parameter p is varied, with all the standard features typical of critical phenomena (scaling laws, a conformally invariant scaling limit, universality). On the applied side, percolation has been used to model the spread of a disease, a fire or a rumor, the displacement of oil by water, the behav-