  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].
