Site percolation and random walks ond-dimensional Kagomé lattices

DOI10.1088/0305-4470/31/15/010zbMATH Open0930.60092arXivcond-mat/9801112OpenAlexW3102817562WikidataQ60228937 ScholiaQ60228937MaRDI QIDQ4254277

Steven C. van der Marck

Publication date: 15 February 2000

Published in: Journal of Physics A: Mathematical and General (Search for Journal in Brave)

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).

Full work available at URL: https://arxiv.org/abs/cond-mat/9801112

zbMATH Keywords

bond percolation Bethe approximation random walker

Mathematics Subject Classification ID

Interacting random processes; statistical mechanics type models; percolation theory (60K35) Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics (82B41) Percolation (82B43) Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics (82B20)