Random fractals in Nature arise for a variety of reasons (dynamic chaotic processes, self-organized criticality, etc.) that are the focus of much current research. Percolation is one such chief mechanism. The importance of percolation lies in the fact that it models critical phase transitions of rich physical content, yet it may be formulated and understood in terms of very simple geometrical concepts. It is also an extremely versatile model, with applications to such diverse problems as supercooled water, galactic structures, fragmentation, porous materials, and earthquakes.

The percolation transition

Consider a square lattice on which each bond is present with probability p, or absent with probability 1 − p. When p is small there is a dilute population of bonds, and clusters of small numbers of connected bonds predominate. As p increases, the size of the clusters also increases. Eventually, for p large enough there emerges a cluster that spans the lattice from edge to edge (Fig. 2.1). If the lattice is infinite, the inception of the spanning cluster occurs sharply upon crossing a critical threshold of the bond concentration, p = pc.

The probability that a given bond belongs to the incipient infinite cluster, P∞, undergoes a phase transition: it is zero for p < pc, and increases continuously as p is made larger than the critical threshold pc (Fig. 2.2).

In this chapter the physical interpretation of the formal RSB solution will be proposed, and some new concepts and quantities will be introduced. The crucial concept that is needed to understand physics behind the RSB structures is that of the pure states.

The pure states

Consider again a simple example of the ferromagnetic system. Here, spontaneous symmetry breaking takes place below the critical temperature Tc, and at each site the non-zero spin magnetizations 〈σi〉 = ±m appear. As we have already discussed in Section 2.2, in the thermodynamic limit the two ground states with the global magnetizations 〈σi〉 = +m and 〈σi〉 = –m are separated by an infinite energy barrier. Therefore, once the system has happened to be in one of these states, it will never be able (during any finite time) to jump into the other one. In this sense, the observable state is not the Gibbs one (which is obtained by summing over all the states), but one of these two states with non-zero global magnetizations. To distinguish them from the Gibbs state they could be called the ‘pure states’. More formally, the pure states could also be defined by the property that all the connected correlation functions in these states, such as 〈σiσj〉c ≡ 〈σiσj〉 – 〈σi〉〈σj〉, tend towards zero at large distances.

In the previous chapter we obtained a special type of spin-glass ground-state solution.

This part of the book explores the subject of anomalous diffusion in fractals and disordered media. Our goal here is twofold: to introduce various approaches to the problem – exact, as well as approximate; and to become intimately acquainted with the phenomenon of anomalous diffusion, its characteristics, and its causes.

Chapter 5 discusses diffusion in the Sierpinski gasket. Anomalous diffusion is demonstrated through simulations and exact enumerations, and through an exact renormalization of the first-passage time. The important relation between diffusion and conductivity (the Einstein relation), introduced in Section 3.4, is used to rederive the anomalous diffusion exponent in yet another way. The chapter closes with a discussion of probability-density-distribution functions and of fractons and spectral dimensions.

In Chapter 6 we present a summary of diffusion in percolation clusters. The question of diffusion in the incipient infinite cluster versus diffusion in all of the clusters is analyzed through scaling and simulations. The chapter describes also the attempt of the Alexander–Orbach conjecture to connect between static and dynamic exponents, diffusion in chemical space, and the multifractality of conductivity and diffusion in percolation clusters.

Chapter 7 discusses diffusion in loopless fractals: Eden trees, combs, etc. In this important case some exact results may be derived, including general relations between static and dynamic exponents, which shed some light on the more general situation in which loops are relevant.

Harris criterion

In studies of the phase-transition phenomena, the systems considered before were assumed to be perfectly homogeneous. In real physical systems, however, some defects or impurities are always present. Therefore, it is natural to consider what effect impurities might have on the phase-transition phenomena. As we have seen in the previous chapter, the thermodynamics of the second-order phase transition is dominated by large-scale fluctuations. The dominant scale, or the correlation length, Rc ∼ |T/TC – l|–v grows as T approaches the critical temperature Tc, where it becomes infinite. The large-scale fluctuations lead to singularities in the thermodynamical functions as |τ| ≡ |T/Tc – 1| → 0. These singularities are the main subject of the theory.

If the concentration of impurities is small, their effect on the critical behavior remains negligible so long as Rc is not too large, i.e. for T not too close to Tc. In this regime the critical behavior will be essentially the same as in the perfect system. However, as |τ| → 0 (T → Tc) and Rc becomes larger than the average distance between impurities, their influence can become crucial.

As Tc is approached the following change of length scale takes place. First, the correlation length of the fluctuations becomes much larger than the lattice spacing, and the system ‘forgets’ about the lattice. The only relevant scale that remains in the system in this regime is the correlation length Rc(τ).

A stochastic particles system is truly fully characterized only when the infinite hierarchy of multiple-point density correlation functions - the probability of finding any given number of particles at some specified locations, simultaneously -is known. The IPDF method is capable of handling this complicated question, and, in fact, in several cases the complete exact solution may be thus obtained. Such studies reveal a peculiar property of “shielding”, particular to reversible coalescence, whereby a particle at the edge of the system seems to shield the rest of the particles from the imposed boundary conditions.

We begin with an analysis of inhomogeneous systems, when translational symmetry is broken, at the simple level of point densities (i.e., the particle concentration). An interesting application is to the study of Fisher waves, and the effect of internal fluctuations on this well-known mean-field model for invasion of an unstable phase by a stable phase.

Inhomogeneous initial conditions

Until now we have discussed only translationally symmetric systems. The method of interparticle distribution functions can be generalized to inhomogeneous situations (Doering et al, 1991). To this end, En(t) need simply be replaced by En,m(t) – the probability that the sites n, n + 1,…, m are empty at time t.

Random walks normally obey Gaussian statistics, and their average square displacement increases linearly with time; 〈r2〉 ∼ t. In many physical systems, however, it is found that diffusion follows an anomalous pattern: the mean-square displacement is 〈r2〉 ∼ t2/dw, where dw ≠ 2. Here we discuss several models of anomalous diffusion, including CTRWs (with algebraically long waiting times), Lévy flights and Lévy walks, and a variation of Mandelbrot's fractional-Brownianmotion (FBM) model. These models serve as useful, tractable approximations to the more difficult problem of anomalous diffusion in disordered media, which is discussed in subsequent chapters.

Random walks as fractal objects

The trail left by a random walker is a complicated random object. Remarkably, under close scrutiny it is found that the trail is self-similar and can be thought of as a fractal (Exercise 1). The ubiquity of diffusion in Nature makes it one of the most fundamental mechanisms giving rise to random fractals.

The fractal dimension of a random walk is called the walk dimension and is denoted by dw. If we think of the sites visited by a walker as “mass”, then the mass of the walk is proportional to time.