Critical exponents of the explosive percolation transition

Costa, Rui; Dorogovtsev, S. N.; Goltsev, A. V.; Mendes, J. F. F.

doi:10.1103/physreve.89.042148

Cited by 23 publications

(32 citation statements)

References 39 publications

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“…Solving the problem analytically, we have shown that the explosive percolation transitions are actually continuous [14]. These transitions have a so small critical exponent of the percolation cluster size, that in simulations of finite systems they can be easily perceived as discontinuous [14,15]. This conclusion was supported by subsequent works of physicists [16][17][18][19] and mathematicians [20].…”

Section: Introductionmentioning

confidence: 64%

“…Forτ > 2 + 1/(2m− 1) the transition occurs at t c > 0, with the critical exponents and scaling functions calculated in [15,21]. Ifτ < 2 + 1/(2m − 1) the size of the percolation cluster follows the power-law S ∼ = Bt β , with β and B given by Eqs.…”

Section: (57)mentioning

confidence: 77%

“…(14) to the integrals of Eq. (15). In general, we estimate the difference of the respective integral and sum with arbitrary exponents ψ and φ…”

Section: Scaling Of P (S T)mentioning

confidence: 99%

“…We consider the following explosive percolation model [14,15,21]. The evolution starts from a given distribution of clusters.…”

Section: Effect Of Initial Conditions On Explosive Percolation (Mmentioning

confidence: 99%

“…We obtained the full set of relevant critical exponents and scaling functions in the typical situation, in which the evolution starts from isolated nodes or clusters with sufficiently rapidly decaying size distribution. In our papers [14,15,21] we employ the following model of explosive percolation, at which the number of nodes N is fixed and links are added one by one. At each step we choose two sets of m random nodes, from each set we select the node that is in the smallest of m clusters, and then we add a new link between these two nodes.…”

Section: Introductionmentioning

confidence: 99%

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Solution of the explosive percolation quest. II. Infinite-order transition produced by the initial distributions of clusters

et al. 2015

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We describe the effect of power-law initial distributions of clusters on ordinary percolation and its generalizations, specifically, models of explosive percolation processes based on local optimization. These aggregation processes were shown to exhibit continuous phase transitions if the evolution starts from a set of disconnected nodes. Since the critical exponents of the order parameter in explosive percolation transitions turned out to be very small, these transitions were first believed to be discontinuous. In this article we analyze the evolution starting from clusters of nodes whose sizes are distributed according to a power law. We show that these initial distributions change dramatically the position and order of the phase transitions in these problems. We find a particular initial power-law distribution producing a peculiar effect on explosive percolation, namely before the emergence of the percolation cluster, the system is in a "critical phase" with an infinite generalized susceptibility. This critical phase is absent in ordinary percolation models with any power-law initial conditions. The transition from the critical phase is an infinite order phase transition, which resembles the scenario of the Berezinskii-Kosterlitz-Thouless phase transition. We obtain the critical singularity of susceptibility at this peculiar infinite-order transition in explosive percolation. It turns out that susceptibility in this situation does not obey the Curie-Weiss law.

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Section: Introductionmentioning

confidence: 64%

Section: (57)mentioning

confidence: 77%

“…(14) to the integrals of Eq. (15). In general, we estimate the difference of the respective integral and sum with arbitrary exponents ψ and φ…”

Section: Scaling Of P (S T)mentioning

confidence: 99%

“…We consider the following explosive percolation model [14,15,21]. The evolution starts from a given distribution of clusters.…”

Section: Effect Of Initial Conditions On Explosive Percolation (Mmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Solution of the explosive percolation quest. II. Infinite-order transition produced by the initial distributions of clusters

et al. 2015

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Percolation transitions in growing networks under achlioptas processes: Analytic solutions

Son

Kahng

2021

Chaos, Solitons & Fractals

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On multi-scale percolation behaviour of the effective conductivity for the lattice model

Olchawa

Wiśniowski

Frączek

et al. 2015

Physica A: Statistical Mechanics and its Applications

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 We explore the multi-scale percolation behaviour in a few lattice models.  A change in the size of the basic cluster alters the value of the percolation threshold.  In the extended three-phase model, double-threshold routes appear on the conductivity surface. A B S T R A C TMacroscopic properties of heterogeneous media are frequently modelled by regular lattice models, which are based on a relatively small basic cluster of lattice sites. Here, we extend one of such models to any cluster's size k  k. We also explore its modified form. The focus is on the percolation behaviour of the effective conductivity of random twoand three-phase systems. We consider only the influence of geometrical features of local configurations at different length scales k. At scales accessible numerically, we find that an increase in the size of the basic cluster leads to characteristic displacements of the percolation threshold. We argue that the behaviour is typical of materials, whose conductivity is dominated by a few linear, percolation-like, conducting paths. Such a system can be effectively treated as one-dimensional medium. We also develop a simplified model that permits of an analysis at any scale. It is worth mentioning that the latter approach keeps the same thresholds predicted by the former one. We also briefly discuss a three-phase system, where the double-thresholds paths appear on model surfaces.

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Critical exponents of the explosive percolation transition

Cited by 23 publications

References 39 publications

Solution of the explosive percolation quest. II. Infinite-order transition produced by the initial distributions of clusters

Solution of the explosive percolation quest. II. Infinite-order transition produced by the initial distributions of clusters

Percolation transitions in growing networks under achlioptas processes: Analytic solutions

On multi-scale percolation behaviour of the effective conductivity for the lattice model

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