What are the limits of universality?

TL;DR

This paper investigates the limits of universality in critical phenomena beyond Euclidean lattices, showing that critical exponents depend mainly on volume-growth dimension in certain graphs but not on fractal dimensions, challenging previous conjectures.

Contribution

It provides numerical evidence that universality extends to transitive graphs with polynomial volume growth but fails on fractals, clarifying the scope of universality in statistical mechanics models.

Findings

Critical exponents depend only on volume-growth dimension in certain graphs.

Universality does not hold on fractals with identical fractal dimensions.

Percolation exponents are consistent across different geometries with the same volume-growth dimension.

Abstract

It is a central prediction of renormalisation group theory that the critical behaviours of many statistical mechanics models on Euclidean lattices depend only on the dimension and not on the specific choice of lattice. We investigate the extent to which this universality continues to hold beyond the Euclidean setting, taking as case studies Bernoulli bond percolation and lattice trees. We present strong numerical evidence that the critical exponents governing these models on transitive graphs of polynomial volume growth depend only on the volume-growth dimension of the graph and not on any other large-scale features of the geometry. For example, our results strongly suggest that percolation, which has upper-critical dimension six, has the same critical exponents on the four-dimensional hypercubic lattice $\mathbb{Z}^4$ and the Heisenberg group despite the distinct large-scale geometries of these two lattices preventing the relevant percolation models from sharing a common scaling limit. On the other hand, we also show that no such universality should be expected to hold on fractals, even if one allows the exponents to depend on a large number of standard fractal dimensions. Indeed, we give natural examples of two fractals which share Hausdorff, spectral, topological, and topological Hausdorff dimensions but exhibit distinct numerical values of the percolation Fisher exponent $\tau$. This gives strong evidence against a conjecture of Balankin et al. [Phys. Lett. A 2018].