Explosive site percolation and finite-size hysteresis

Abstract

We report the critical point for site percolation for the "explosive" type for 2D square lattices using Monte Carlo simulations and compare it to the classical well known percolation. We use similar algorithms as have been recently reported for bond percolation and networks. We calculate the "explosive" site percolation threshold as pc = 0.695 and we find evidence that "explosive" site percolation surprisingly may belong to a different universality class than bond percolation on lattices, providing that the transitions (a) are continuous and (b) obey the conventional finite size scaling forms. We do not attempt to determine the order of the explosive transition. Finally, we study and compare the direct and reverse processes, showing that while the reverse process is different from the direct process for finite size systems, the two cases become equivalent in the thermodynamic limit of large L. I.INTRODUCTIONRandom percolation (RP) [1][2][3][4][5][6][7] is a well studied model for phase transitions in statistical physics and condensed matter physics. It consists in randomly occupying sites (site percolation) or bonds (bond percolation) on a given lattice with probability p. Neighboring occupied sites merge to form clusters. As p increases, new clusters are formed or coalesce, until at a critical value, p c , a giant component emerges, transversing the system, the wellknown infinite percolating cluster. Classical percolation and all of its variants (continuum percolation, site-bond percolation, bootstrap percolation, invasion percolation, etc [1]), are known to be continuous transitions, exhibiting a divergent correlation length ξ at p c . Thus, there was a stupendous surprise that recently, Achlioptas et al [8], proposed a new method for the occupation of sites which produces "explosive" transitions: when filling sequentially an empty lattice with occupied sites, instead of randomly occupying a site or bond, we choose two candidates and investigate which one of them leads to the smaller clustering. The one that does this is kept as a new occupied site on the lattice while the second one is discarded. This procedure considerably slows down the emergence of the giant component, which is now formed abruptly, thus the term "explosive". Although it is not clear if "explosive" percolation is a first-order transition [9][10][11] or continuous transition with a very small β exponent [12], it is certainly a very sharp transition with a lot of interesting and unusual properties and has stimulated considerable interest. Following these first announcements new work emerged [12,13] that doubted the result of the Achlioptas problem being a first-order phase transition, thus making this problem a hot contentious issue. Recently, applications of the "explosive" percolation transition for "real" systems have been proposed [14,15]. Up to now the systems investigated include networks and lattice bond percolation.In this paper we extend these investigations to include now site explosive percolation and its finite-size hysteresis pro...

Explosive site percolation with a product rule

“…For the systematic analysis, we show the dependence of area, A(L), enclosed by P LC (L) for the increasing and decreasing processes on L. If the system undergoes a continuous transition, then A(L) should vanish in the limit L → ∞. However, our data clearly shows that A(L) increases as L increases or, at least, seems to saturate to a nonzero value unlike the sum rule [19] in which A(L) → 0 as L → ∞. This shows that 2DSAP undergoes a discontinuous transition.…”

“…We therefore cannot exclude the possibility that the transition nature of the AP in the meanfield limit can be different from that in lower-dimensional systems, like the Potts model [18]. Moreover, based on the measurement of hysteresis [19], a sum rule for the 2D site percolation possibly makes the transition continuous in the thermodynamic limit. This indicates that under the AP-like processes the bond percolation and site percolation may have different transition natures in the 2D lattice.…”

“…In contrast, the results in Refs. [8,17,19] show the possibility that under AP the bond percolation with a product rule and the site percolation with a sum rule do not belong to the same universality class. Therefore, it is theoretically important and interesting to investigate whether in a low-dimensional system the product rule and the sum rule belong to the same universality class or not.…”

“…(1) is replaced by summation, then the rule is called as a sum rule. Recent study for site percolation with sum rule shows that the transition is continuous when the linear size of the lattice, L, goes to infinity [19]. Since we use a 2D square lattice n(m) in Eq.…”

“…The hysteresis measurement for the explosive percolation has also been emphasized in Refs. [11,19]. The hysteresis is a history-dependent property of a system and usually observed in the discontinuous phase transition because of the metastable state.…”