The Critical Behavior of Random Systems

Self-avoiding walks, lattice trees and lattice animals, and percolation are among the simplest models exhibiting the general features of critical phenomena. A basic problem is to prove the existence of critical exponents governing their behavior near the critical point. This problem gains importance from interrelations between these models and models of ferromagnetism such as the Ising model, and from their role in the theory of polymer molecules. In low spatial dimensions, the principal mathematical questions about critical exponents for these models remain unsolved. This is true in spite of significant progress on such questions in the physics, chemistry, and numerical literature. Proofs of existence of critical exponents are currently restricted to high spatial dimensions, above the so-called upper critical dimensions. These proofs are based on joint work with Hara, and for percolation, also on results of [4], [6]. The proofs make use of an expansion method first introduced by Brydges and Spencer [9] and known as the lace expansion. The results are summarized below. More extensive summaries, with further references, can be found in [22], [26]. The results described below form a natural sequel to those for ferromagnetic spin systems [1], [13] presented in Aizenman's ICM 1983 lecture [2], and provide a resolution of an issue raised there for percolation.