The Triangle Condition for Percolation

Aizenman and Newman introduced an unverified condition, the triangle condition, which has been shown to imply that a number of percolation critical exponents take their mean field values, and which is expected to hold above six dimensions for nearest neighbour percolation. We prove that the triangle condition is satisfied in sufficiently high dimensions for the nearest neighbour model, and above six dimensions for a class of "spread-out" models. The proof uses an expansion which is related to the lace expansion for self-avoiding walk. Percolation is a simple probabilistic model for which many interesting problems remain unsolved. The basic objects in percolation are random graphs on an infinite lattice. Typically there is a critical density for the graph edges (known as bonds) below which there is zero probability that a fixed point in the lattice is part of an infinite connected graph, but above which this probability is strictly positive. This abrupt change in behaviour has been used in chemistry and statistical physics to model phase transitions in a variety of physical systems, such as fluid flow through a porous medium (hence the name percolation), random resistor networks, and the gelation of branched polymers. For an introduction, see [15, 20]. To define the models we are interested in, we consider the of-dimensional integer lattice Z^ (whose elements are referred to as sites) and the set of pairs b = {x,y} of distinct sites (the bonds). To each bond b is associated an independent Bernoulli random variable nb9 with nb = 1 with probability p • Jb and nb — 0 with probability \ p Jh. The Jb are fixed and Z invariant, and p is a parameter. The nearest-neighbour model is defined by taking Jb = 1 for those b = {x,y} such that \\x y\\i = 1, and Jb = 0 otherwise. Expectation with respect to the joint distribution of the {nb} is denoted by (•••)/?• If nb = 1 we say b is occupied, and otherwise b is vacant. Given a bond configuration {nb}, two sites x and y are said to be connected if there is a path from x to y which consists of occupied bonds. The set of sites which are connected to x is called the connected cluster of x and is denoted C(x). The probability that x is connected to y is written Tp(x,y), (1) ip(x,y) — (I[x and y are connected])^ = {I[y e C(x)])p, where I denotes the indicator function. Denoting by \C(x)\ the number of sites in C(x), the expected cluster size or susceptibility is given by