A random walk approach to high-dimensional critical phenomena
The study of critical phenomena in lattice statistical mechanical models such as percolation and spin systems has a long history in both physics and mathematics. A central problem is to prove existence and calculate the values of the critical exponents that govern the universal behaviour of the model at and near its critical point. Detailed proofs are typically available only for models in dimension two, or above an upper critical dimension where mean-field behaviour is observed. We present a new, unified, generic, probabilistic, and relatively very simple proof of mean-field critical behaviour for high-dimensional models containing a small parameter. The results apply to spin systems and self-avoiding walk in dimensions above 4, percolation in dimensions above 6, and lattice trees in dimensions above 8. Minimal model-specific data is required for these applications. This is joint work with Hugo Duminil-Copin, Aman Markar and Romain Panis.