Newman-Ziff algorithm for the bootstrap percolation: Application to the Archimedean lattices

Abstract

We propose very efficient algorithms for the bootstrap percolation and the diffusion percolation models by extending the Newman-Ziff algorithm of the classical percolation [M. E. J. Newman and R. M. Ziff, Phys. Rev. Lett. 85 (2000) 4104]. Using these algorithms and the finite-size-scaling, we calculated with high precision the percolation threshold and critical exponents in the eleven two-dimensional Archimedean lattices. We present the condition for the continuous percolation transition in the bootstrap percolation and the diffusion percolation, and show that they have the same critical exponents as the classical percolation within error bars in two dimensions. We conclude that the bootstrap percolation and the diffusion percolation almost certainly belong to the same universality class as the classical percolation.

Universality class of the percolation in two-dimensional lattices with distortion

“…W 1 (p, L) is the wrapping probability in any one direction but not in the other direction. Critical exponents ν and β can be estimated from the scaling relations [28,29,26] Max dW (p, L) dp ∼ L 1/ν , dW (p, L) dp…”

“…) A new site of the distorted lattice is chosen uniformly at random within the sampling domain. Two occupied nearest neighbors are set to be connected if and only if their Euclidean distance is smaller than the connection threshold d. In this work, the Newman-Ziff algorithm, which is much more efficient than brute-force methods when the percolation threshold is not known in advance [24,25,26], is used. In this algorithm, a set of average values Q(n) is obtained first, where Q(n) is any physical quantity such as the size of the largest cluster for a fixed number of occupied sites n. A value of Q(p) as a function of occupation probability p can be calculated by the binomial transformation:…”

“…Both models were concluded to belong to the same universality class as the ordinary percolation model based on the values of their critical exponents. However, the conclusion of Mitra et al is questionable because all the values of critical exponent (β/ν = 0.1143 (20), 0.1158 (26), and 0.1158 (30) for three kinds of distortion) show systematic deviation from β/ν = 5/48 ≈ 0.1042 [1] of the ordinary two-dimensional percolation by four or five times of the error bars. Therefore, the conclusion should have been that the model with distortion belongs to a different kind of the universality class from their results.…”

Universal behavior of site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in 2 and 3 dimensions

“…Further variations involve correlated percolation [13,14], like for example drilling percolation [15,16]. In bootstrap percolation [17][18][19][20], sites and/or bonds are first occupied and then successively culled from a system if a site does not have at least k neighbors. Another important model of percolation, which is in a different universality class, is directed percolation [21][22][23][24], where connectivity along a bond depends upon the direction of the flow.…”

Phase transition in the diffusion and bootstrap percolation models on regular random and Erdős-Rényi networks

“…The DP and BP have been studied intensively on lattices [6,7,8,9,4,10,11,12,13,14,15,16,17]. In the ∆-regular Archimedean lattices and three-dimensional lattices (simple cubic, body-centered cubic, and face-centered cubic lattices), it was shown that the DP and BP have first-order percolation transitions with the percolation thresholds p c = 1 and p c = 0, respectively, if m > m c and k < (∆ + 1 − m c ), where m c = (∆ + 1)/2 (the only exception is the bounce lattice, which has m c = (∆ + 1)/2 + 1) [6,18,4,14,16,17]; otherwise, the DP and BP have second-order percolation transitions at finite p c (0 < p c < 1) with the same critical exponents as the CP [16,17] (exceptionally, the BP transition with m = m c = 6 in the face-centered cubic lattice is first-order [17]).…”