Universality of critical exponents on transitive graphs of polynomial growth

Let Bernoulli percolation and lattice trees be defined on transitive graphs of polynomial growth, and let the volume-growth dimension denote the dimension governing the graph's polynomial volume growth. Universality conjecture. The critical exponents describing Bernoulli percolation and lattice trees on transitive graphs of polynomial growth are each determined by the volume-growth dimension of the graph. The paper reports strong numerical evidence for this claim from simulations on several non-Euclidean graphs, but provides no theoretical proof; it remains open whether the exponents depend only on volume-growth dimension rather than other large-scale geometric features.