Site percolation and random walks on d-dimensional Kagome lattices

TL;DR

This study investigates site percolation thresholds and random walk properties on d-dimensional Kagome lattices, revealing unique scaling behaviors that differ from hypercubic lattices and challenging the Bethe approximation in high dimensions.

Contribution

It provides numerical estimates of percolation thresholds and analyzes their scaling, highlighting differences from hypercubic lattices and deriving new theoretical insights.

Findings

Percolation thresholds scale as 2/q, not 1/(q-1)

Random walk return probability scales as 2/q

Results challenge the Bethe approximation in high dimensions

Abstract

The site percolation problem is studied on d-dimensional generalisations of the Kagome' lattice. These lattices are isotropic and have the same coordination number q as the hyper-cubic lattices in d dimensions, namely q=2d. The site percolation thresholds are calculated numerically for d= 3, 4, 5, and 6. The scaling of these thresholds as a function of dimension d, or alternatively q, is different than for hypercubic lattices: p_c ~ 2/q instead of p_c ~ 1/(q-1). The latter is the Bethe approximation, which is usually assumed to hold for all lattices in high dimensions. A series expansion is calculated, in order to understand the different behaviour of the Kagome' lattice. The return probability of a random walker on these lattices is also shown to scale as 2/q. For bond percolation on d-dimensional diamond lattices these results imply p_c ~ 1/(q-1).