Lorentz model

In a famous sequence of papers of 1906, Hendrik Lorentz introduced a version of the following problem. Large (heavy) particles are distributed about ℝd. A small (light) particle is fired through ℝd, with a trajectory comprising straight-line segments between the points of interaction with the heavy particles. When the small particle hits a heavy particle, the small particle is reflected at its surface, and the large particle remains motionless. See Figure 12.1 for an illustration.

We may think of the heavy particles as objects bounded by reflecting surfaces, and the light particle as a photon. The problem is to say something non-trivial about how the trajectory of the photon depends on the ‘environment’ of heavy particles. Conditional on the environment, the photon pursues a deterministic path about which the natural questions include:

1. Is the path unbounded?

2. How distant is the photon from its starting point after time t?

For simplicity, we assume henceforth that the large particles are identical to one another, and that the small particle has negligible volume.

Erdős–Rényi graphs

Let V = {1, 2, …, n}, and let (Xi,j : 1 ≤ i < j ≤ n) be independent Bernoulli random variables with parameter p. For each pair i < j, we place an edge 〈i, j〉 between vertices i and j if and only if Xi,j = 1. The resulting random graph is named after Erdős and Rényi, and it is commonly denoted Gn,p. The density p of edges may vary with n, for example, p = λ/n with λ ∈ (0, ∞), and one commonly considers the structure of Gn,p in the limit as n → ∞.

The original motivation for studying Gn,p was to understand the properties of ‘typical’ graphs. This is in contrast to the study of ‘extremal’ graphs, although it may be noted that random graphs have on occasion manifested properties more extreme than graphs obtained by more constructive means.

Random graphs have proved an important tool in the study of the ‘typical’ runtime of algorithms.

Introductory remarks

There are many beautiful problems of physical type that may be modelled as Markov processes on the compact state space = {0, 1}V for some countable set V. Amongst the most studied to date by probabilists are the contact, voter, and exclusion models, and the stochastic Ising model.

Subcritical phase

In language borrowed from the theory of branching processes, a percolation process is termed subcritical if p < pc, and supercritical if p > pc.

In the subcritical phase, all open clusters are (almost surely) finite. The chance of a long-range connection is small, and it approaches zero as the distance between the endpoints diverges. The process is considered to be ‘disordered’, and the probabilities of long-range connectivities tend to zero exponentially in the distance. Exponential decay may be proved by elementary means for sufficiently small p, as in the proof of Theorem 3.2, for example. It is quite another matter to prove exponential decay for all p < pc, and this was achieved for percolation by Aizenman and Barsky and Menshikov around 1986.

Within the menagerie of objects studied in contemporary probability theory, there are a number of related animals that have attracted great interest amongst probabilists and physicists in recent years. The inspiration for many of these objects comes from physics, but the mathematical subject has taken on a life of its own, and many beautiful constructions have emerged. The overall target of these notes is to identify some of these topics, and to develop their basic theory at a level suitable for mathematics graduates.

If the two principal characters in these notes are random walk and percolation, they are only part of the rich theory of uniform spanning trees, self-avoiding walks, random networks, models for ferromagnetism and the spread of disease, and motion in random environments. This is an area that has attracted many fine scientists, by virtue, perhaps, of its special mixture of modelling and problem-solving. There remain many open problems. It is the experience of the author that these may be explained successfully to a graduate audience open to inspiration and provocation.

The material described here may be used for personal study, and as the bases of lecture courses of between 24 and 48 hours duration. Little is assumed about the mathematical background of the audience beyond some basic probability theory, but students should be willing to get their hands dirty if they are to profit.