Since the breakthrough works of Hara and Slade in the early 90’s, there has been a well-developed theory of mean-field criticality for statistical physics models in high-dimensional Euclidean space. This means that critical models on these spaces are described by the same critical exponents as they are on, say, the 3-regular tree. While this is intuitively due to the “expansiveness” of high-dimensional space, the proofs are rather specific to the Euclidean setting. In 1996, Benjamini and Schramm proposed a program of understanding percolation and other models on arbitrary transitive graphs through their coarse geometric features, such as their isoperimetry. Relatively little progress has been made however, even under the presence of very strong geometric assumptions such as nonamenability.
In the first half of the talk, I will discuss the main problems and conjectures in the field. In the second half, I will outline my recent work on special cases of these conjectures in which the graph has certain special symmetry properties (namely, a nonunimodular transitive subgroup of automorphisms). In particular, I hope to be able to show a complete proof that self-avoiding walk on the product T x Z of a 3-regular tree with the integers (an example of historical interest) has mean-field critical exponents.