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.

Random walk – the stochastic process formed by successive summation of independent, identically distributed random variables – is one of the most basic and well-studied topics in probability theory. For random walks on the integer lattice ℤd, the main reference is the classic book by Spitzer (1976). This text considers only a subset of such walks, namely those corresponding to increment distributions with zero mean and finite variance. In this case, one can summarize the main result very quickly: the central limit theorem implies that under appropriate rescaling the limiting distribution is normal, and the functional central limit theorem implies that the distribution of the corresponding path-valued process (after standard rescaling of time and space) approaches that of Brownian motion.

Researchers who work with perturbations of random walks, or with particle systems and other models that use random walks as a basic ingredient, often need more precise information on random walk behavior than that provided by the central limit theorems. In particular, it is important to understand the size of the error resulting from the approximation of random walk by Brownian motion. For this reason, there is need for more detailed analysis. This book is an introduction to the random walk theory with an emphasis on the error estimates. Although “mean zero, finite variance” assumption is both necessary and sufficient for normal convergence, one typically needs to make stronger assumptions on the increments of the walk in order to obtain good bounds on the error terms.