Analysis on the Sierpinski Carpet

The ‘analysis on fractals’ and ‘analysis on metric spaces’ communities have tended to work independently. Metric spaces such as the Sierpinski carpet fail to satisfy some of the properties which are generally assumed for metric spaces. This survey discusses analysis on the Sierpinski carpet, with particular emphasis on the properties of the heat kernel. 1. Background and history Percolation was introduced by Broadbent and Hammersley [BH] in 1957. The easiest version to describe is bond percolation. Take any graph G = (V,E) and a probability p ∈ (0, 1). For each edge e keep it with probability p and delete it with probability 1− p, independently of all other edges. This is one of the core models of statistical physics, and has many applications in other areas – one is contact networks for infectious diseases. For books on percolation see [BR, Gri]. Write Gp = (V,Ep) for the (random) graph obtained by the percolation process. The connected component of x is called the cluster containing x and is denoted C(x). Percolation on Z with d ≥ 2 has a phase transition. Set θ(p) = Pp(|C(0)| = ∞), pc = pc(d) = inf{p : θ(p) > 0}. Theorem 1.1. [BH] For the lattice Z, pc ∈ (0, 1). When p is small Gp consists of a number of (mainly small) finite clusters. For large p the cluster C(0) may or may not be infinite, but a zero-one law combined with the stationary ergodic nature of the percolation process gives that infinite clusters exist with probability 1. A less elementary argument gives that if p > pc then there is exactly one infinite cluster. (As always in such contexts, this statement has to be qualified by ‘with probability 1’.) In spite of its 50 year history, many open problems remain. The most fundamental of these is what happens at pc in low dimensions: Open Problem 1. Is θ(pc) = 0 if 3 ≤ d ≤ 18? It is known that θ(pc) = 0 if d = 2 or d ≥ 19 – see [Ke1, HS2]. Physicists are interested in ‘transport’ problems of percolation clusters, that is, in the behaviour of solutions to the wave or heat equation. A percolation cluster is c ©0000 (copyright holder)