Dynamical sensitivity of the infinite cluster in critical percolation

AbstractWe introduce a new percolation model on planar lattices. First, impurities (“holes”) are removed independently from the lattice. On the remaining part, we then consider site percolation with some parameter p close to the critical value $$p_c$$ p c . The mentioned impurities are not only microscopic, but allowed to be mesoscopic (“heavy-tailed”, in some sense). For technical reasons (the proofs of our results use quite precise bounds on critical exponents in Bernoulli percolation), our study focuses on the triangular lattice. We determine explicitly the range of parameters in the distribution of impurities for which the connectivity properties of percolation remain of the same order as without impurities, for distances below a certain characteristic length. This generalizes a celebrated result by Kesten for classical near-critical percolation (which can be viewed as critical percolation with single-site impurities). New challenges arise from the potentially large impurities. This generalization, which is also of independent interest, turns out to be crucial to study models of forest fires (or epidemics). In these models, all vertices are initially vacant, and then become occupied at rate 1. If an occupied vertex is hit by lightning, which occurs at a very small rate $$\zeta $$ ζ , its entire occupied cluster burns immediately, so that all its vertices become vacant. Our results for percolation with impurities are instrumental in analyzing the behavior of these forest fire models near and beyond the critical time (i.e. the time after which, in a forest without fires, an infinite cluster of trees emerges). In particular, we prove (so far, for the case when burnt trees do not recover) the existence of a sequence of “exceptional scales” (functions of $$\zeta $$ ζ ). For forests on boxes with such side lengths, the impact of fires does not vanish in the limit as $$\zeta \searrow 0$$ ζ ↘ 0 . This surprising behavior, related to the non-monotonicity of these processes, was not predicted in the physics literature.

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Power-law bounds for critical long-range percolation below the upper-critical dimension

Probability Theory and Related Fields ◽

10.1007/s00440-021-01043-7 ◽

2021 ◽

Author(s):

Tom Hutchcroft

Keyword(s):

Long Range ◽

Power Law ◽

Upper Bound ◽

Mean Field ◽

Maximum Size ◽

Finite Graph ◽

Critical Dimension ◽

Point Function ◽

Upper Critical Dimension ◽

Infinite Cluster

AbstractWe study long-range Bernoulli percolation on $${\mathbb {Z}}^d$$ Z d in which each two vertices x and y are connected by an edge with probability $$1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })$$ 1 - exp ( - β ‖ x - y ‖ - d - α ) . It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if $$0<\alpha <d$$ 0 < α < d then there is no infinite cluster at the critical parameter $$\beta _c$$ β c . We give a new, quantitative proof of this theorem establishing the power-law upper bound $$\begin{aligned} {\mathbf {P}}_{\beta _c}\bigl (|K|\ge n\bigr ) \le C n^{-(d-\alpha )/(2d+\alpha )} \end{aligned}$$ P β c ( | K | ≥ n ) ≤ C n - ( d - α ) / ( 2 d + α ) for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality $$(2-\eta )(\delta +1)\le d(\delta -1)$$ ( 2 - η ) ( δ + 1 ) ≤ d ( δ - 1 ) relating the cluster-volume exponent $$\delta $$ δ and two-point function exponent $$\eta $$ η .

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Estimates of critical percolation probabilities for a set of two-dimensional lattices

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100050325 ◽

1972 ◽

Vol 71 (1) ◽

pp. 97-106 ◽

Cited By ~ 16

Author(s):

D. G. Neal

Keyword(s):

Monte Carlo ◽

Two Dimensions ◽

Two Dimensional ◽

Critical Percolation ◽

Site Percolation

AbstractThis paper describes new detailed Monte Carlo investigations into bond and site percolation problems on the set of eleven regular and semi-regular (Archimedean) lattices in two dimensions.

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A PERCOLATION MODEL OF DIAGENESIS

International Journal of Modern Physics C ◽

10.1142/s0129183102003176 ◽

2002 ◽

Vol 13 (03) ◽

pp. 319-331 ◽

Cited By ~ 7

Author(s):

S. S. MANNA ◽

T. DATTA ◽

R. KARMAKAR ◽

S. TARAFDAR

Keyword(s):

Numerical Simulation ◽

Critical Behavior ◽

Sedimentary Rocks ◽

Two Dimensions ◽

Percolation Model ◽

Type Model ◽

Stable Configuration ◽

Critical Percolation ◽

Restructuring Process ◽

Simulation Results

The restructuring process of diagenesis in the sedimentary rocks is studied using a percolation type model. The cementation and dissolution processes are modeled by the culling of occupied sites in rarefied and growth of vacant sites in dense environments. Starting from sub-critical states of ordinary percolation the system evolves under the diagenetic rules to critical percolation configurations. Our numerical simulation results in two dimensions indicate that the stable configuration has the same critical behavior as the ordinary percolation.

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