Uniqueness of the Infinite Cluster and Related Results in Percolation

1987 ◽  
pp. 13-20 ◽  
Author(s):  
M. Aizenman ◽  
H. Kesten ◽  
C. M. Newman
Keyword(s):  
Infinite Cluster

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

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 .

scholarly journals New sorbents for electrochromatography based on polymer-inorganic dielectric composites

2021 ◽  
Vol 2103 (1) ◽  
pp. 012124
Author(s):  
A Y Shmykov ◽  
S V Mjakin ◽  
N A Bubis ◽  
L M Kuztetzov ◽  
N A Esikova ◽  
...  
Keyword(s):  
Electron Microscopy ◽  
Filler Content ◽  
Structural Features ◽  
Polymer Binder ◽  
Dielectric Losses ◽  
Infinite Cluster ◽  
Confocal Scanning ◽  
Scanning Electron

Abstract Oligomeric diisocyanate based coatings with different contents of barium titanate (BaTiO3) submicron sized particles as a ferroelectric filler are synthesized on poly(dimethylsiloxane) (PDMS) supports. The study of thus obtained coatings using confocal scanning electron microscopy allowed the characterization of their morphology and features of BaTiO3 particles distribution in the polymer binder, including the determination of threshold filler contents corresponding to the formation of an infinite cluster, matrix-island and chain-like structures as well as the percolation. Dielectric permittivity and dielectric losses of the composites are measured and studied depending on BaTiO3 filler content and relating structural features.

Research On The Fractal and Percolation Characteristics of Coal-Based Porous Media For Filtration and Impregnation

2021 ◽  
Author(s):  
Qili Wang ◽  
Jiarui Sun ◽  
Yuehu Chen ◽  
Yuyan Qian ◽  
Shengcheng Fei ◽  
...  
Keyword(s):  
Porous Media ◽  
Fractal Structure ◽  
Mercury Intrusion Porosimetry ◽  
Fractal Theory ◽  
Fractal Dimensions ◽  
Infinite Cluster ◽  
Porous Graphite ◽  
Local Aggregation ◽  
Gradual Expansion ◽  
The Difference

Abstract In order to distinguish the difference in the heterogeneous fractal structure of porous graphite used for filtration and impregnation, the fractal dimensions obtained through the mercury intrusion porosimetry (MIP) along with the fractal theory were used to calculate the volumetric FD of the graphite samples. The FD expression of the tortuosity along with all parameters from MIP test was optimized to simplify the calculation. In addition, the percolation evolution process of mercury in the porous media was analyzed in combination with the experimental data. As indicated in the analysis, the FDs in the backbone formation regions of sample vary from 2.695 to 2.984, with 2.923 to 2.991 in the percolation regions and 1.224 to 1.544 in the tortuosity. According to the MIP test, the mercury distribution in porous graphite manifested a transitional process from local aggregation, gradual expansion, and infinite cluster connection to global connection.

scholarly journals Opening reionization: quantitative morphology of the epoch of reionization and its connection to the cosmic density field

2020 ◽  
Vol 498 (3) ◽  
pp. 4533-4549
Author(s):  
Philipp Busch ◽  
Marius B Eide ◽  
Benedetta Ciardi ◽  
Koki Kakiichi
Keyword(s):  
Morphological Characterization ◽  
Bubble Size Distribution ◽  
Density Field ◽  
Volume Element ◽  
Quantitative Morphology ◽  
Infinite Cluster ◽  
Spatially Resolved ◽  
Binary Fields ◽  
The Universe

ABSTRACT We introduce a versatile and spatially resolved morphological characterization of binary fields, rooted in the opening transform of mathematical morphology. We subsequently apply it to the thresholded ionization field in simulations of cosmic reionization and study the morphology of ionized regions. We find that an ionized volume element typically resides in an ionized region with radius ∼8 h−1 cMpc at the midpoint of reionization (z ≈ 7.5) and follow the bubble size distribution even beyond the overlap phase. We find that percolation of the fully ionized component sets in when 25 per cent of the universe is ionized and that the resulting infinite cluster incorporates all ionized regions above ∼8 h−1 cMpc. We also quantify the clustering of ionized regions of varying radius with respect to matter and on small scales detect the formation of superbubbles in the overlap phase. On large scales, we quantify the bias values of the centres of ionized and neutral regions of different sizes and not only show that the largest ones at the high-point of reionization can reach b ≈ 30, but also that early small ionized regions are positively correlated with matter and large neutral regions and late small ionized regions are heavily antibiased with respect to matter, down to b ≲ −20.

scholarly journals Random Walk on the Incipient Infinite Cluster for Oriented Percolation in High Dimensions

2008 ◽  
Vol 278 (2) ◽  
pp. 385-431 ◽  
Author(s):  
Martin T. Barlow ◽  
Antal A. Járai ◽  
Takashi Kumagai ◽  
Gordon Slade
Keyword(s):  
Random Walk ◽  
Oriented Percolation ◽  
High Dimensions ◽  
Infinite Cluster ◽  
Incipient Infinite Cluster

scholarly journals Geometry controlled anomalous diffusion in random fractal geometries: looking beyond the infinite cluster

2015 ◽  
Vol 17 (44) ◽  
pp. 30134-30147 ◽  
Author(s):  
Yousof Mardoukhi ◽  
Jae-Hyung Jeon ◽  
Ralf Metzler
Keyword(s):  
Random Walk ◽  
Percolation Threshold ◽  
Anomalous Diffusion ◽  
Infinite Cluster ◽  
Random Fractal

We study the strongly non-ergodic effects of a random walk on a percolation geometry below, at, and above the percolation threshold.

On the infinite cluster of Bernoulli bond percolation in Scherk's graph

2001 ◽  
Vol 38 (04) ◽  
pp. 828-840
Author(s):  
Dayue Chen
Keyword(s):  
Three Dimensional ◽  
The Other ◽  
Bond Percolation ◽  
Infinite Cluster ◽  
Dimensional Lattice ◽  
Supercritical Region ◽  
Other Hand

Scherk's graph is a subgraph of the three-dimensional lattice. It was shown by Markvorsen, McGuinness and Thomassen (1992) that Scherk's graph is transient. Consider the Bernoulli bond percolation in Scherk's graph. We prove that the infinite cluster is transient for p &gt; ½ and is recurrent for p &lt; ½. This implies the well-known result of Grimmett, Kesten and Zhang (1993) on the transience of the infinite cluster of the Bernoulli bond percolation in the three-dimensional lattice for p &gt; ½. On the other hand, Scherk's graph exhibits a new dichotomy in the supercritical region.

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