Numerical evidence of the critical percolation probability Pc=1 for site problems on Sierpinski gaskets

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 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 τ . This gives strong evidence against a conjecture of Balankin et al. (2018 Phys. Lett. A 382 , 12–19 ( doi:10.1016/j.physleta.2017.10.035 )).

Fractal-like tree networks reducing the thermal conductivity

The effective thermal conductivity of composites with embedded self-similar H-shaped fractal-like tree networks is studied. It is found that the effective thermal conductivity of the composites with these networks is related to the structures of the networks and the ratio of the component thermal conductivities: the longer the branches, the lower the thermal conductivity; the smaller the ratio (beta) of successive branch diameters, the lower the thermal conductivity; the denser the network, the lower the thermal conductivity. It is also found that the thermal conductivity of the H-shaped fractal-like tree networks does not obey Murray's law. The present results show that a network embedded in a composite plays an important role, and the thermal conductivity of the network itself may be less than that of the original material by several orders of magnitude. Fractal-like tree networks can significantly reduce the thermal conductivity compared to an equivalent single cylinder.