A random walk approach to high-dimensional critical phenomena

Abstract: One of the main goals of statistical mechanics is to understand critical phenomena of lattice models. This can be achieved by computing critical exponents, which govern the universal behaviour of the model at and near its critical point. This task is generally difficult due to the intricate interplay between model-specific features and the underlying graph geometry.

A striking observation was made in the 20th century: above the upper critical dimension d_c, the geometry becomes irrelevant and critical exponents adopt their mean-field values (as on Cayley trees or complete graphs). Classical approaches—renormalization group, differential inequalities, and the lace expansion—are powerful but model-specific and technically demanding. We present a new, unified, probabilistic, and relatively simple proof of mean-field critical behaviour for high-dimensional models containing a small parameter. Applications include spin systems and self-avoiding walks in dimensions d>4, percolation in dimensions d>6, and lattice trees in dimensions d>8.

Joint work with Hugo Duminil-Copin, Aman Markar, and Gordon Slade.

mathematical physics

Audience: researchers in the discipline

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