Universality class of site and bond percolation on multifractal scale-free planar stochastic lattice

Universality class of explosive percolation in Barabási-Albert networks

In this work, we study explosive percolation (EP) in Barabási-Albert (BA) network, in which nodes are born with degree k = m, for both product rule (PR) and sum rule (SR) of the Achlioptas process. For m = 1 we find that the critical point t_c = 1 which is the maximum possible value of the relative link density t; Hence we cannot have access to the other phase like percolation in one dimension. However, for m > 1 we find that t_c decreases with increasing m and the critical exponents ν, α, β and γ for m > 1 are found to be independent not only of the value of m but also of PR and SR. It implies that they all belong to the same universality class like EP in the Erdös-Rényi network. Besides, the critical exponents obey the Rushbrooke inequality α + 2β + γ ≥ 2 but always close to equality.

    PACS numbers: 61.43.Hv, 64.60.Ht, 68.03.Fg, 82.70.Dd.

Product-sum universality and Rushbrooke inequality in explosive percolation

We study explosive percolation (EP) on an Erdös-Rényi network for product rule (PR) and sum rule (SR). Initially, it was claimed that EP describes discontinuous phase transition; now it is well accepted as a probabilistic model for thermal continuous phase transition (CPT). However, no model for CPT is complete unless we know how to relate its observable quantities with those of thermal CPT. To this end, we define entropy and specific heat, redefine susceptibility, and show that they behave exactly like their thermal counterparts. We obtain the critical exponents ν,α,β, and γ numerically and find that both PR and SR belong to the same universality class and obey Rushbrooke inequality.

Entropy, specific heat, susceptibility, and Rushbrooke inequality in percolation

We investigate percolation, a probabilistic model for continuous phase transition, on square and weighted planar stochastic lattices. In its thermal counterpart, entropy is minimally low where order parameter (OP) is maximally high and vice versa. In addition, specific heat, OP, and susceptibility exhibit power law when approaching the critical point and the corresponding critical exponents α,β,γ respectably obey the Rushbrooke inequality (RI) α+2β+γ≥2. Their analogs in percolation, however, remain elusive. We define entropy and specific heat and redefine susceptibility for percolation and show that they behave exactly in the same way as their thermal counterpart. We also show that RI holds for both the lattices albeit they belong to different universality classes.

Explosive percolation on a scale-free multifractal weighted planar stochastic lattice

In this article, we investigate explosive bond percolation (EBP) with the product rule, formally known as the Achlioptas process, on a scale-free multifractal weighted planar stochastic lattice. One of the key features of the EBP transition is the delay, compared to the corresponding random bond percolation (RBP), in the onset of the spanning cluster. However, when it happens, it happens so dramatically that initially it was believed, although ultimately proved wrong, that explosive percolation (EP) exhibits a first-order transition. In the case of EP, much effort has been devoted to resolving the issue of its order of transition and almost no effort has been devoted to finding the critical point, critical exponents, etc., to classify it into universality classes. This is in sharp contrast to the situation for classical random percolation. We do not even know all the exponents of EP for a regular planar lattice or for an Erdös-Renyi network. We first find the critical point p_{c} numerically and then obtain all the critical exponents, β, γ, and ν, as well as the Fisher exponent τ and the fractal dimension d_{f} of the spanning cluster. We also compare our results for EBP with those for RBP and find that all the exponents of EBP obey the same scaling relations as do those for RBP. Our findings suggest that EBP is not special except for the fact that the exponent β is unusually small compared to that for RBP.