The celebrated ‘conformal invariance’ conjecture of Aizenman and Langlands, Pouliot and Saint-Aubin [1994] states, roughly, that if ∧ is a planar lattice with suitable symmetry, and we consider percolation on ∧ with probability p = pc(∧), then as the lattice spacing tends to zero certain limiting probabilities are invariant under conformal maps of the plane ℝ2 ≅ ℂ. This conjecture has been proved for only one standard percolation model, namely independent site percolation on the triangular lattice. The aim of this chapter is to present this remarkable result of Smirnov [2001a; 2001b], and to discuss briefly some of its consequences.

In the next section we describe the conformal invariance conjecture, in terms of the limiting behaviour of crossing probabilities, and present Cardy's explicit prediction for these conformally invariant limits. In Section 2, we present Smirnov's Theorem and its proof; as we give full details of the proof, this section is rather lengthy. Finally, we shall very briefly describe some consequences of Smirnov's Theorem concerning the existence of certain ‘critical exponents.’

Crossing probabilities and conformal invariance

Throughout this chapter we identify the plane ℝ2 with the set ℂ of complex numbers in the usual way. A domain D ⊂ ℂ is a non-empty connected open subset of ℂ. If D and D′ are domains, then a conformal map from D to D′ is a bijection f : D → D′ which is analytic on D, i.e., analytic at every point of D. Note that f-1 is then analytic on D′.

Percolation theory was founded by Broadbent and Hammersley [1957] almost half a century ago; by now, thousands of papers and many books have been devoted to the subject. The original aim was to open up to mathematical analysis the study of random physical processes such as the flow of a fluid through a disordered porous medium. These bona fide problems in applied mathematics have attracted the attention of many physicists as well as pure mathematicians, and have led to the accumulation of much experimental and heuristic evidence for many remarkable phenomena. Mathematically, the subject has turned out to be much more difficult than might have been expected, with several deep results proved and many more conjectured.

The first spectacular mathematical result in percolation theory was proved by Kesten: in 1980 he complemented Harris's 1960 lower bound on the critical probability for bond percolation on the square lattice, and so proved that this critical probability is 1/2. To present this result, and numerous related results, Kesten [1982] published the first monograph devoted to the mathematical theory of percolation, concentrating on discrete two-dimensional percolation. A little later, Chayes and Chayes [1986b] came close to publishing the next book on the topic when they wrote an elegant and very long review article on percolation theory understood in a much broader sense.

For nearly two decades, Grimmett's 1989 book (with a second edition published in 1999) has been the standard reference for much of the basic theory of percolation on lattices.

Our aim in this chapter is to show that, for a wide class of percolation models, when p < pH the cluster size distribution has an exponential tail. Such a result certainly implies that pT = pH; in fact, it will turn out that exponential decay is relatively easy to prove when p < pT (at least if ‘size’ is taken to mean radius, rather than number of sites). Thus the tasks of proving exponential decay and of showing that pT = pH are closely related.

Results of the latter type were proved independently by Menshikov [1986] (see also Menshikov, Molchanov and Sidorenko [1986]) and by Aizenman and Barsky [1987] under different assumptions. Here we shall present Menshikov's ingenious argument in detail, and say only a few words about the Aizenman–Barsky approach. As we shall see, Menshikov's proof makes essential use of the Margulis–Russo formula and the van den Berg–Kesten inequality.

The van den Berg–Kesten inequality and percolation

Let us briefly recall the van den Berg–Kesten inequality, Theorem 5 of Chapter 2, which has a particularly attractive interpretation in the context of percolation. In the context of site (respectively bond) percolation, an increasing event E is one which is preserved by changing the states of one or more sites (bonds) from closed to open, and a witness for an increasing event E is just a set W of open sites (bonds) such that the fact that all sites (bonds) in W are open guarantees that E holds.

Shortly after Broadbent and Hammersley started percolation theory and Erdős and Rényi [1960; 1961a], together with Gilbert [1959], founded the theory of random graphs, Gilbert [1961] started a closely related area that is now known as continuum percolation. The basic objects of study are random geometric graphs, both finite and infinite. Such graphs model, for example, a network of transceivers scattered at random in the plane or a planar domain, each of which can communicate with those others within a fixed distance.

Although this field has attracted considerably less attention than percolation theory, its importance is undeniable; in this single chapter, we cannot do justice to these topics. Indeed, this area has been treated in hundreds of papers and several monographs, including Hall [1988] on coverage processes, Møller [1994] on random Voronoi tessellations, Meester and Roy [1996] on continuum percolation, and Penrose [2003] on random geometric graphs. These topics are also touched upon in the books by Matheron [1975], Santaló [1976], Stoyan, Kendall and Mecke [1987; 1995], Ambartzumian [1990] and Molchanov [2005].

In the first section we present the most basic model of continuum percolation, the Gilbert disc model or Boolean model, and give some fundamental results on it, including bounds on the critical area. In the second section we take a brief look at finite random geometric graphs, with emphasis on their connectedness. The most important part of the chapter is the third section, in which we shall sketch a proof of the analogue of the Harris–Kesten result for continuum percolation: the critical probability for random Voronoi percolation in the plane is 1/2.

Informally, imagine an object that falls apart randomly as time passes. The state of the system at some given time consists in the sequence of the sizes of the pieces, which are often called fragments or particles. Suppose that the evolution is Markovian and obeys the following rules. First, different particles evolve independently of each other, that is the so-called branching property is fulfilled. Second, there is a parameter α ∈ ℝ, which will be referred to as the index of self-similarity, such that each fragment with size s is stable during an exponential time with parameter proportional to sα. In other words, a particle with size s > 0 has an exponential lifetime with mean cs–α, where c > 0 is some constant. At its death, this particle splits and there results a family of fragments, say with sizes (si, i ∈ ℕ), where the sequence of ratios (si/s, i ∈ ℕ) has the same distribution for all particles. The purpose of this chapter is to construct such self-similar fragmentation chains, to shed light on their genealogical structure, and to establish some of their fundamental properties.

Construction of fragmentation chains

In this section, we briefly present some basic elements on Markov chains and branching Markov chains in continuous time which are then used for the construction and the study of fragmentation chains.

Exchangeable coalescents form a natural family of Markov processes with values in the space of partitions of ℕ, in which blocks coagulate as time passes. We shall first present the celebrated coalescent of Kingman, which is often used as a model for the genealogy of large populations. Then we shall introduce a more general family of coalescents, called here exchangeable coalescents, in which coagulations may involve several blocks simultaneously. We shall also investigate the coagulation processes of mass-partitions associated with such exchangeable coalescents. The last section of this chapter is devoted to a representation of these exchangeable coalescents in terms of certain stochastic flows on the unit interval. Many ideas and techniques based on exchangeability, which were useful for investigating fragmentations, can also be fruitfully applied to coagulations.

Kingman's coalescent

Coalescence naturally arises when one studies the genealogy of populations; we first briefly explain why. Following Kingman [139], this will lead us to introduce a natural Markov process with values in the space P∞ of partitions of ℕ.

Genealogy of populations in the Wright-Fisher model

Imagine at time T > 0 a population with size n which can be identified with the set [n] = {1, …, n}. Assume the population is haploid, meaning that each individual has exactly one parent at the previous generation, so we may follow its ancestral lineage backwards in time.

Fragmentation chains which have been discussed in the preceding chapter, only form a special and rather simple sub-class of fragmentation processes which enjoy the self-similar and branching properties. The construction and the study of the general family are harder, since we can no longer rely on a discrete genealogy. We shall now prepare material to circumvent this fundamental difficulty, at least in the conservative or dissipative case. In this direction, we shall first introduce several notions of partitions (for masses, for intervals, and for the set of natural integers) and develop their connections. Mass-partitions induced by Poisson random measures, and in particular the so-called Poisson-Dirichlet partitions, will receive special attention. Finally, Kingman's theory for exchangeable random partitions of ℕ will be presented in the ultimate section.

Mass-partitions

In this section, we introduce some elementary material on the simple notion of partition of a unit mass.

Partitions of a unit mass

A partition of some set E is a collection of disjoint subsets whose union is E. When one focusses on finite sets and their cardinals, this yields the notion of partition of integers in combinatorics: A partition of n ∈ ℕ := {1, 2, …} refers to a finite family of positive integers, say {p1, …, pk}, with p1 + … + pk = n. We stress that this family is unordered; for instance {2, 3} and {3, 2} represent the same partition of 5.

We now resume the study of processes in continuous time that describe the evolution of a unit mass which breaks down randomly into pieces, in such a way that distinct components have independent and self-similar evolutions. The crucial point is that we now allow fragments to split immediately, a situation that could not be handled by the discrete techniques of Chapter 1.

We shall first focus on the homogeneous case when splitting rates are further assumed to be the same for every fragment. The framework of exchangeable random partitions, which has been developed in the preceding chapter, provides a powerful tool for the construction and the analysis of this special class of fragmentation processes. In particular, we shall specify their Poissonian structure and characterize their distributions in terms of an erosion coefficient and rates of sudden dislocations. We shall also point out an important relation between a randomly tagged fragment and a certain subordinator, extending our observations for the evolution of the randomly tagged branch in Chapter 1. Finally, we shall present a transformation of homogeneous fragmentations which enables us to construct general self-similar fragmentations (i.e. with splitting rates proportional to a power function of the mass).

Homogeneous fragmentation processes

We shall now investigate fragmentation processes in the framework of random exchangeable partitions. Let us briefly explain the intuition which will guide us, by presenting an important example that will be used several times to illustrate notions and results in this chapter.

Fragmentation and coagulation are two natural phenomena that can be observed in many sciences, at a great variety of scales. To give just a few examples, let us simply mention first for fragmentation, the studies of stellar fragments in astrophysics, fractures and earthquakes in geophysics, breaking of crystals in crystallography, degradation of large polymer chains in chemistry, DNA fragmentation in biology, fission of atoms in nuclear physics, fragmentation of a hard drive in computer science, … For coagulation, we mention the formation of the large structures (galaxies) in the universe and of planets by accretion in astrophysics, of polymer chains in chemistry, of droplets of liquids in aerosols or clouds, coalescence of ancestral lineages in genealogy of populations in genetics, …

The main purpose of this monograph is to develop mathematical models which may be used in situations where either phenomenon occurs randomly and repeatedly as time passes. For instance, in the case of fragmentation, we can think of the evolution of blocks of mineral in a crusher. The text is intended for readers having a solid background in probability theory. I aimed at providing a rather concise and self-contained presentation of random fragmentation and coagulation processes; I endeavored to make accessible some recent developments in this field, but did not try to be exhaustive.

This chapter is concerned with systems in which particles coagulate pairwise and randomly as time passes. The key assumption is that the rate at which a pair of particles merges only depends on the two particles involved in the coagulation. Although defining rigorously the evolution is easy when there is only a finite number of particles, the extension to systems with an infinite number of particles and finite total mass requires some regularity assumptions on the rates of coagulation. In a different direction, we shall consider the hydrodynamic regime when the total mass of the system tends to infinity. For so-called sub-multiplicative kernels and under appropriate assumptions, the empirical distribution of particles in the system converges after a suitable renormalization to a deterministic measure depending on time. The latter solves a system of non-linear PDEs which is known as Smoluchowski's coagulation equation. The approach relies mainly on combinatorial arguments via a remarkable connection with certain random graphs models. Finally, we pay special attention to the so-called additive coalescent. We shall develop a combinatorial construction due to Pitman, which involves random trees and forests. This provides a key to understanding the asymptotic behavior of such coagulation dynamics when, roughly speaking, the process starts from dust, that is from a large number of small particles.