Series study of percolation moments in general dimension

Abstract

Series expansions for general moments of the bond-percolation cluster-size distribution on hypercubic lattices to 15th order in the concentration have been obtained. This is one more than the previously published series for the mean cluster size in three dimensions and four terms more for higher moments and higher dimensions. Critical exponents, amplitude ratios, and thresholds have been calculated from these and other series by a variety of independent analysis techniques. A comprehensive summary of extant estimates for exponents, some universal amplitude ratios, and thresholds for percolation in all dimensions is given, and our results are shown to be in excellent agreement with the ε expansion and some of the most accurate simulation estimates. We obtain threshold values of 0.2488±0.0002 and 0.180 25±0.000 15 for the three-dimensional bond problem on the simple-cubic and body-centered-cubic lattices, respectively, and 0.160 05±0.000 15 and 0.118 19±0.000 04, for the hypercubic bond problem in four and five dimensions, respectively. Our direct exponent estimates are γ=1.805±0.02, 1.435±0.015, and 1.185±0.005, and β=0.405±0.025, 0.639±0.020, and 0.835±0.005 in three, four, and five dimensions, respectively. Disciplines Physics CommentsAt the time of publication, author A. Brooks Harris was also affiliated with Tel Aviv University. Currently, he is a faculty member in the Physics Department at the University of Pennsylvania.

4-dimensional layer phase as a gauge field localization: Extensive study of the 5-dimensional anisotropic U(1) gauge model on the lattice

“…In [19], [20], it is argued that percolation in three dimensions can be viewed as a gauge theory and it can capsulate most of the features of confinement and the glueball spectrum. The values of the exponents γ and ν given at the end of the Section 4.2 are in a good agreement with the values of the corresponding exponenents of the 5-D percolation model which are: γ 5Dperc = 1.18 and ν 5Dperc = 0.57 (see [21]). Although this fact alone can not justify any further argumentation on a possible universality class issues, however it might be useful in providing a new point of view for the confinement mechanism along the extra dimension.…”

Critical properties of the three- and four-dimensional gauge glass

“…For the percolation systems η(d) has a weak minimum around d = 3 before becoming positive at d = 2. 57 The present results show that for the GG there is a deep minimum in η(d) near d = 4. This is in stark contrast to the situation in the canonical Ising, XY , or Heisenberg systems with no disorder where η(d) hardly changes at all when the degrees of freedom of the spin are modified.…”

Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions

“…These results are more precise than some of the published values for p c = 0.16005 ± 0.00015 [17], 0.1407 ± 0.0003 [18], and 0.11819 ± 0.00004 [17] for 4D bond, 5D site, and 5D bond percolation, respectively and for τ = 2.41 for 5D percolation; for 4D site percolation, Ballesteros et al [19] found the comparably precise value p c = 0.196901 ± 0.000005 (just slightly higher than ours) and τ = 2.3127 ± 0.0007. All simulation parameters and our results are summarized in Table I.…”