What are the limits of universality?
It is a central prediction of renormalization 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 6, has the same critical exponents on Z⁴ 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 τ. This gives strong evidence against a conjecture of Balankin et al. (2018 Phys. Lett. A382, 12–19 (doi:10.1016/j.physleta.2017.10.035)).
Additional Information© 2022 The Author(s). Published by the Royal Society. Manuscript received 28/11/2021; Manuscript accepted 22/02/2022; Published online 30/03/2022; Published in print 30/03/2022. We thank Romain Tessera for very helpful correspondence on the quasi-isometric classification of nilpotent groups, and thank Tyler Helmuth for enjoyable discussions on models of large upper-critical dimension. We thank Aleks Reinhardt for helpful advice on stylistic matters. The Cambridge Faculty of Mathematics HPC system, fawcett, was used for all Monte Carlo simulations. The work of T.H. was carried out while he was a Senior Research Associate at the University of Cambridge and was supported by ERC starting grant no. 804166 (SPRS). N.H. was supported by the doctoral training centre, Cambridge Mathematics of Information (CMI). We declare we have no competing interests.
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