Some Exact Critical Percolation Probabilities for Bond and Site Problems in Two Dimensions

Sykes, M F; Essam, J W

doi:10.1103/physrevlett.10.3

Cited by 200 publications

(99 citation statements)

References 11 publications

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“…Shortly afterwards [12], he proved that p c = 1/2 also holds for site percolation on the triangular lattice T . Later, Wierman [36] used his 'substitution' method to give rigorous proofs of the values p b c (T ) = 2 sin(π/18) and p b c (H) = 1−2 sin(π/18) for bond percolation on the triangular and hexagonal lattices respectively; these values had been obtained heuristically much earlier by Sykes and Essam [33,34]. There are two further values that may be easily derived from these: the (3,6,3,6) or Kagomé lattice K is the line graph of the hexagonal lattice, so p s c (K) = p b c (H) = 1 − 2 sin(π/18).…”

Section: Introductionmentioning

confidence: 99%

Rigorous confidence intervals for critical probabilities

Riordan

Walters

2007

Phys. Rev. E

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We use the method of Balister, Bollobás and Walters [3] to give rigorous 99.9999% confidence intervals for the critical probabilities for site and bond percolation on the 11 Archimedean lattices. In our computer calculations, the emphasis is on simplicity and ease of verification, rather than obtaining the best possible results. Nevertheless, we obtain intervals of width at most 0.0005 in all cases.

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Section: Introductionmentioning

confidence: 99%

Rigorous confidence intervals for critical probabilities

Riordan

Walters

2007

Phys. Rev. E

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“…Yet to date, no analytical expression has been found for the numerical value of p c . The square site lattice lacks the symmetry that has allowed exact solutions on other topologies [2,4,5,6,7,8,9]. So long as the problem remains intractable, statistical estimates from Monte Carlo studies can, at least, offer approximate values.…”

Section: Introductionmentioning

confidence: 99%

Pseudo-random-number generators and the square site percolation threshold

Lee

2008

Phys. Rev. E

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“…That's why in the model has been included also the bond percolation theory [10]. The oxide film that forms between stainless steel and lead can be represented by a 2D network of randomly placed conducting elements (vertices) and it can be modelled using (bond) directed percolation theory (DP).…”

Section: Percolation Theorymentioning

confidence: 99%

“…by adding one column of horizontal and vertical segments. Then, transfer matrix A L+1 in the next step can be calculated using recursive relation [10]: [12]. The conductance of the total network is defined as the first element of inverse transfer matrix [13].…”

Section: Simulation Proceduresmentioning

confidence: 99%

Modelling and detection of failure in medical electrodes

2015

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In this paper we have studied the failures in medical electrodes such as electroencephalogram electrodes (EEG) being used for collecting brain signals. As those electrodes have to guarantee high level of reliability it is important to explore and predict the possible occurrence of failures in there structure. The electrode tip (needle) made of stainless steel is covered with thin oxide film acting as a dielectric and determing the total electrode resistance. In fact, studying the fluctuations of that resistance gives the insight into defects of the whole structure. The electrical properties of the oxide layer are characterized by charge hopping mechanism and the total resistance could be modeled by implementing random resistance network (RRN) methodology.The applied computational algorithm is based on Monte Carlo simulation procedure with direct and iteration methods. The obtained simulation results show non-gaussian Bramwell-Holdsworth-Pinton (BHP) distribution of the total resistance fluctuations, and they verified by the experiments. Key words: Cold solder, oxide layer, failure detection, biased percolation, RRN simulationModeliranje i detekcija kvara u medicinskim elektrodama. U ovom je radu razvijen model za simulaciju predviđanja defekata u posebnim elektrodama koje se koriste pri mjerenju i prikupljanju signala mozga. Ključni dio elektrode je njenčelični vrh prekriven tankim oksidnim slojem koji ujedno djeluje kao dielektrik. Upravo fluktuacije vrijednosti otpora dielektrika odnosno njihovo rasipanje od normalne do BHP raspodjele, korištene su kao osnova predloženog modela. Pritom je primjenjena metodologija mreže nasumičnih otpora (RRN), algoritam temeljen na Monte-Carlo simulacijama kao i teorija usmjerene perkolacije. Definiran je i poseban parametar kao indeks intervala valjanosti za svaku elektrodu. Simulacije su potvrđene mjerenjem u laboratorijskim uvjetima na komercijalnim EEG elektrodama.

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Some Exact Critical Percolation Probabilities for Bond and Site Problems in Two Dimensions

Cited by 200 publications

References 11 publications

Rigorous confidence intervals for critical probabilities

Rigorous confidence intervals for critical probabilities

Pseudo-random-number generators and the square site percolation threshold

Modelling and detection of failure in medical electrodes

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