Definitions and Heuristics

In an Erdös–Rényi random graph, vertices have degrees that have asymptotically a Poisson distribution. However, as discussed in Section 1.4, in social and communication networks, the distribution of degrees is much different from the Poisson and in many cases has a power law form, that is, the fraction of vertices of degree k, pk ~ Ck-β as k → ∞. Molloy and Reed (1995) were the first to construct graphs with specified degree distributions. We will use the approach of Newman, Strogatz, and Watts (2001, 2002) to define the model.

Let d1,…dn be independent and have P(di = k) = pk. Since we want di to be the degree of vertex i, we condition on En = {d1 + … + dn is even}. If the probability P(E1) ∊ (0, 1) then P(En) → ½ as n → ∞ so the conditioning will have little effect on the finite-dimensional distributions. If d1 is always even then P(En) = 1 for all n, while if d1 is always odd, P(E2n) = 1 and P(E2n+1) = 0 for all n.

To build the graph we think of di half-edges attached to i and then pair the half-edges at random. The picture gives an example with eight vertices.

Introduction to the Introduction

The theory of random graphs began in the late 1950s in several papers by Erdös and Rényi. However, the introduction at the end of the twentieth century of the small world model of Watts and Strogatz (1998) and the preferential attachment model of Barabási and Albert (1999) have led to an explosion of research. Querying the Science Citation Index in early July 2005 produced 1154 citations for Watts and Strogatz (1998) and 964 for Barabási and Albert (1999). Survey articles of Albert and Barabási (2002), Dorogovstev and Mendes (2002), and Newman (2003) each have hundreds of references. A book edited by Newman, Barabási, and Watts (2006) contains some of the most important papers. Books by Watts (2003) and Barabási (2002) give popular accounts of the new science of networks, which explains “how everything is connected to everything else and what it means for science, business, and everyday life.”

While this literature is extensive, many of the papers are outside the mathematical literature, which makes writing this book a challenge and an opportunity. A number of articles have appeared in Nature and Science. These journals with their impressive impact factors are, at least in the case of random graphs, the home of 10 second sound bite science.

Barabási–Albert Model

As we have noted, many real-world graphs have power law degree distributions. Barabási and Albert (1999) introduced a simple model that produces such graphs. They start with a graph with

Bollobás, Riordan, Spencer, and Tusnády (2001) complain: “The description of the random graph process quoted above is rather imprecise. First as the degrees are initially zero, it is not clear how the process is supposed to get started. More seriously, the expected number of edges linking a new vertex to earlier vertices is ΣiΠ(ki) = 1, rather than m. Also when choosing in one go a set S of m earlier vertices as neighbors of v, the distribution is not specified by giving the marginal probability that each vertex lies in S.”

As we will see below there are several ways to make the process precise and all of them lead to the same asymptotic behavior.

We now turn to the main topic of this book, disordered systems. We split this into two parts, treating in turn lattice models and mean-field models. From the physical point of view, the former should be more realistic and hence more relevant, so it is natural that we present the general mathematical framework in this context. However, the number of concrete problems one can to this day solve rigorously is quite limited, so that the examples we will treat can mostly be considered as random perturbations of the Ising model. In the second part we will be able to look, in a simplified setting, at more complex, genuinely random models, that rather surprisingly will turn out to produce fascinating mathematics, but also lead to applications that are beyond the standard scope of physics.

Introduction

We have seen that statistical mechanics is a theory that treats the dynamical degrees of freedom of a large system as random variables, distributed according to the Gibbs measure.

We now return to the class of Gaussian mean-field models whose covariance is a function of the Hamming distance, respectively the overlap RN. They were the original mean-field models for spin-glasses, introduced by D. Sherrington and S. Kirkpatrick in, and are considered to be the most natural, or physically relevant ones. Certainly, a large part of the excitement and interest, both in physics and in mathematics, has come from the fact that Parisi suggested a very complex solution, more precisely a formula for the free energy of these models, that has in many respects been seen as rather mysterious and unexpected. Moreover, this formula for the free energy suggested a structure of the Gibbs measures, which is to some extent similar to what we have encountered already in the CREMs. Much of the mystery of Parisis solution certainly resulted from the original method of its derivation, which involved objects such as zero-dimensional matrices, and that appeared to transcend the realm of standard mathematics.

Fortunately, these mysteries have been greatly clarified, mainly due to an ingenious discovery of F. Guerra in 2002, and subsequent work by Aizenman, Sims, and Starr, and M. Talagrand, that allows a clear and mathematically rigorous formulation of the Parisi solution.

The existence of the free energy

A key observation that has led to the recent breakthroughs in spin-glass theory, made by F.Guerra and F.-L. Toninelli, was that Lemma 10.2.1, or rather the interpolation idea used in its proof, is a powerful tool in this context.

Starting with the Newtonian revolution, the eighteenth and nineteenth centuries saw with the development of analytical mechanics an unprecedented tool for the analysis and prediction of natural phenomena. The power and precision of Hamiltonian perturbation theory allowed even the details of the motion observed in the solar system to be explained quantitatively. In practical terms, analytical mechanics made the construction of highly effective machines possible. Unsurprisingly, these successes led to the widespread belief that, ultimately, mechanics could explain the functioning of the entire universe. On the basis of this confidence, new areas of physics, outside the realm of the immediate applicability of Newtonian mechanics, became the target of the new science of theoretical (analytical) physics. One of the most important of these new fields was the theory of heat, or thermodynamics. One of the main principles of Newtonian mechanics was that of the conservation of energy. Now, such a principle could not hold entirely, due to the ubiquitous loss of energy through friction.

In the previous chapters we have seen that with considerable work it is possible to study some simple aspects of random perturbations of the Ising model. On the other hand, models that are genuinely random and promise to bring some new features to light are at the moment not seriously accessible to rigorous analysis. In such situations it it natural to turn to simplified models, and a natural reflex in statistical mechanics is to turn to the mean-field approximation. In the case of disordered systems, this has turned out, quite surprisingly, to open a Pandora's box.

The common feature of mean-field models is that the spatial structure of the lattice ℤd is abandoned in favour of a simpler setting, where sites are indexed by the natural numbers and all spins are supposed to interact with each other, irrespective of their distance. The prototype of all such models is the Curie–Weiss model, that we studied in Section 3.5.

We now turn to the question of what should be the natural class of disordered mean-field models to study? This question requires some thought, as there are at least two natural classes that propose themselves.

With the Hopfield model we have seen that the range of applications of statistical mechanics goes beyond standard physics into biology and neuroscience. In recent years, many such extensions have been noted. A particularly lively area is that of combinatorial optimization, notably with applications to computer science. While I cannot cover this subject adequately, I will in this last chapter concentrate on possibly the simplest example, the number partitioning problem. The connection to statistical mechanics has been pointed out by S. Mertens, from whom I learned about this interesting problem.

Number partitioning as a spin-glass problem

The number partitioning problem is a classical optimization problem: Given N numbers n1, n2, …, nN, find a way of distributing them into two groups such that their sums in each group are as similar as possible. One can easily imagine that this problem occurs all the time in real life, albeit with additional complication: Imagine you want to stuff two moving boxes with books of different weights. You clearly have an interest in making both boxes more or less the same weight, so that neither of them is too heavy. In computing, you want to distribute a certain number of jobs on, say, two processors, in such a way that all of your processes are executed in the fastest way, etcClearly, the two examples indicate that the problem has a natural generalization to partitionings into more than two groups. What is needed in practice is an algorithm that, when presented with the N numbers, finds the optimal partitioning as quickly as possible.

Statistical mechanics is the branch of physics that attempts to understand the laws of the behaviour of systems that are composed of very many individual components, such as gases, liquids, or crystalline solids. The statistical mechanics of disordered systems is a particularly difficult, but also particularly exciting, branch of the general subject, that is devoted to the same problem in situations when the interactions between these components are very irregular and inhomogeneous, and can only be described in terms of their statistical properties. From the mathematical point of view, statistical mechanics is, in the spirit of Dobrushin, a ‘branch of probability theory’, and the present book adopts this point of view, while trying not to neglect the fact that it is, after all, also a branch of physics.

This book grew out of lecture notes I compiled in 2001 for a Concentrated Advanced Course at the University of Copenhagen in the the MaPhySto programme and that appeared in the MaPhySto Lecture Notes series in the same year. In 2004 I taughta two-semester course on Statistical Mechanics at the Technical University of Berlin with in the curriculum of mathematical physics for advanced undergraduate students, both from thephysics and the mathematics departments. It occurred to me that the material I was going to cover in this course could indeed provide a suitable scope for a book, in particular as the mathematical understanding of the field was going through a period of stunning progress, and that an introductory textbook, written from a mathematical perspective, was maybe more sought after than ever.