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What are the limits of universality?

Halberstam, Noah and Hutchcroft, Tom (2021) What are the limits of universality? . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20210924-203748960

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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 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. [Phys. Lett. A 2018].


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/2106.13218arXivDiscussion Paper
ORCID:
AuthorORCID
Hutchcroft, Tom0000-0003-0061-593X
Additional Information:The work of TH was supported by ERC starting grant 804166 (SPRS). NH was supported by the doctoral training centre, Cambridge Mathematics of Information (CMI). We thank Romain Tessera for very helpful correspondence on the quasi-isometric classification of nilpotent groups.
Funders:
Funding AgencyGrant Number
European Research Council (ERC)804166
Cambridge Mathematics of Information (CMI)UNSPECIFIED
Record Number:CaltechAUTHORS:20210924-203748960
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210924-203748960
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111042
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:27 Sep 2021 16:11
Last Modified:27 Sep 2021 16:11

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