Exact site-percolation probability on the square lattice

2021 ◽  
Author(s):  
Stephan Mertens
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Percolation Threshold ◽  
Percolation Cluster ◽  
Square Lattice ◽  
Site Percolation ◽  
Exact Probability ◽  
Percolation Probability ◽  
A Site ◽  
Time And Space Complexity

Abstract We present an algorithm to compute the exact probability $R_{n}(p)$ for a site percolation cluster to span an $n\times n$ square lattice at occupancy $p$. The algorithm has time and space complexity $O(\lambda^n)$ with $\lambda \approx 2.6$. It allows us to compute $R_{n}(p)$ up to $n=24$. We use the data to compute estimates for the percolation threshold $p_c$ that are several orders of magnitude more precise than estimates based on Monte-Carlo simulations.

Monte Carlo simulations of site percolation on the directed square lattice

1984 ◽  
Vol 101 (4) ◽  
pp. 221-223 ◽  
Author(s):  
K. De'Bell ◽  
T. Lookman ◽  
D.L. Hunter
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Square Lattice ◽  
Site Percolation

scholarly journals REEXAMINATION OF SEVEN-DIMENSIONAL SITE PERCOLATION THRESHOLD

2000 ◽  
Vol 11 (01) ◽  
pp. 205-209 ◽  
Author(s):  
DIETRICH STAUFFER ◽  
ROBERT M. ZIFF
Keyword(s):  
Monte Carlo ◽  
Renormalization Group ◽  
Monte Carlo Simulations ◽  
Percolation Threshold ◽  
Critical Dimension ◽  
Corrections To Scaling ◽  
Site Percolation ◽  
Upper Critical Dimension ◽  
Size Dependent ◽  
Previous Series

Monte Carlo simulations alone could not clarify the corrections to scaling for the size-dependent pc(L) above the upper critical dimension. Including the previous series estimate for the bulk threshold pc (∞) gives preference for the complicated corrections predicted by renormalization group and against the simple 1/L extrapolation. Additional Monte-Carlo simulations using the Leath method corroborate the series result for pc.

scholarly journals Probability of Incipient Spanning Clusters in Critical Square Bond Percolation

1997 ◽  
Vol 08 (03) ◽  
pp. 473-481 ◽  
Author(s):  
Lev N. Shchur ◽  
Sergey S. Kosyakov
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Percolation Threshold ◽  
Square Lattice ◽  
Simultaneous Occurrence ◽  
Bond Percolation ◽  
Free Boundaries ◽  
Single Event ◽  
A Value ◽  
Infinite Lattices

The probability of simultaneous occurrence of at least k spanning clusters has been studied by Monte Carlo simulations on the 2D square lattice with free boundaries at the bond percolation threshold pc =1/2. It is found that the probability of k and more Incipient Spanning Clusters (ISC) have the values P(k>1) ≈ 0.00658(3) and P(k>2) ≈ 0.00000148(21) provided that the limit of these probabilities for infinite lattices exists. The probability P(k>3) of more than three ISC could be estimated to be of the order of 10-11 and is beyond the possibility to compute such a value by nowadays computers. So, it is impossible to check in simulations the Aizenman law for the probabilities when k≫1. We have detected a single sample with four ISC in a total number of about 1010 samples investigated. The probability of this single event is 1/10 for that number of samples. The influence of boundary conditions is discussed in the last section.

MARKET VOLATILITY AND THE DISTRIBUTION OF MEAN CLUSTER SIZES FOR PERCOLATION

2000 ◽  
Vol 11 (03) ◽  
pp. 519-524 ◽  
Author(s):  
DIETRICH STAUFFER
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Percolation Threshold ◽  
Stock Price ◽  
Cluster Size ◽  
Market Volatility ◽  
Site Percolation ◽  
Price Fluctuations ◽  
The Mean

The tail of the distribution for the "mean cluster size" (susceptibility) at the site percolation threshold is found by Monte Carlo simulations to be exponential. Similar distributions might be expected for the market volatility, if stock price fluctuations are described by Cont–Bouchaud percolation.

scholarly journals Pinning of interfaces in a random elastic medium and logarithmic lattice embeddings in percolation

2015 ◽  
Vol 145 (3) ◽  
pp. 481-512 ◽  
Author(s):  
Patrick W. Dondl ◽  
Michael Scheutzow ◽  
Sebastian Throm
Keyword(s):  
Percolation Threshold ◽  
Elastic Medium ◽  
Driving Force ◽  
Percolation Cluster ◽  
Nearest Neighbour ◽  
Site Percolation ◽  
Kinetic Relation ◽  
Force Velocity ◽  
Linear Force ◽  
Neighbour Site

For a model of a driven interface in an elastic medium with random obstacles we prove the existence of a stationary positive supersolution at non-vanishing driving force. This shows the emergence of a rate-independent hysteresis through the interaction of the interface with the obstacles despite a linear (force = velocity) microscopic kinetic relation. We also prove a percolation result, namely, the possibility to embed the graph of an only logarithmically growing function in a next-nearest neighbour site percolation cluster at a non-trivial percolation threshold.

Qualitative Equivalence Between Electrical Percolation Threshold and Effective Thermal Conductivity in Polymer/Carbon Nanocomposites

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
Majid Baniassadi ◽  
Akbar Ghazavizadeh ◽  
Yves Rémond ◽  
Said Ahzi ◽  
David Ruch ◽  
...  
Keyword(s):  
Thermal Conductivity ◽  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Percolation Threshold ◽  
Effective Thermal Conductivity ◽  
Electrical Percolation ◽  
Aspect Ratios ◽  
Wide Range ◽  
Statistical Continuum ◽  
Electrical Percolation Threshold

In this study, a qualitative equivalence between the electrical percolation threshold and the effective thermal conductivity of composites filled with cylindrical nanofillers has been recognized. The two properties are qualitatively compared on a wide range of aspect ratios, from thin nanoplatelets to long nanotubes. Statistical continuum theory of strong-contrast is utilized to estimate the thermal conductivity of this type of heterogeneous medium, while the percolation threshold is simultaneously evaluated using the Monte Carlo simulations. Statistical two-point probability distribution functions are used as microstructure descriptors for implementing the statistical continuum approach. Monte Carlo simulations are carried out for calculating the two-point correlation functions of computer generated microstructures. Finally, the similarities between the effective conductivity properties and percolation threshold are discussed.

Transition under noise in the Sznajd model on square lattice

2016 ◽  
Vol 27 (03) ◽  
pp. 1650026 ◽  
Author(s):  
F. W. S. Lima
Keyword(s):  
Phase Transition ◽  
Monte Carlo ◽  
Social Behavior ◽  
Monte Carlo Simulations ◽  
Order Phase Transition ◽  
Opinion Dynamics ◽  
Two Dimensions ◽  
Square Lattice ◽  
Behavior Model ◽  
Second Order Phase Transition

In order to describe the formation of a consensus in human opinion dynamics, in this paper, we study the Sznajd model with probabilistic noise in two dimensions. The time evolution of this system is performed via Monte Carlo simulations. This social behavior model with noise presents a well defined second-order phase transition. For small enough noise q < 0.33 most agents end up sharing the same opinion.

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