scholarly journals Critical point and percolation probability in a long range site percolation model on Zd

2011 ◽  
Vol 121 (9) ◽  
pp. 2043-2048 ◽  
Author(s):  
Bernardo N.B. de Lima ◽  
Rémy Sanchis ◽  
Roger W.C. Silva
Keyword(s):  
Critical Point ◽  
Long Range ◽  
Percolation Model ◽  
Site Percolation ◽  
Percolation Probability

scholarly journals Percolation of Words on Zd with Long-Range Connections

2011 ◽  
Vol 48 (4) ◽  
pp. 1152-1162 ◽  
Author(s):  
B. N. B. de Lima ◽  
R. Sanchis ◽  
R. W. C. Silva
Keyword(s):  
Long Range ◽  
Percolation Model ◽  
Binary Sequences ◽  
Site Percolation ◽  
Almost All ◽  
Independent Site

Consider an independent site percolation model on Zd, with parameter p ∈ (0, 1), where all long-range connections in the axis directions are allowed. In this work we show that, given any parameter p, there exists an integer K(p) such that all binary sequences (words) ξ ∈ {0, 1}N can be seen simultaneously, almost surely, even if all connections with length larger than K(p) are suppressed. We also show some results concerning how K(p) should scale with p as p goes to 0. Related results are also obtained for the question of whether or not almost all words are seen.

scholarly journals Percolation of Words on Z d with Long-Range Connections

2011 ◽  
Vol 48 (04) ◽  
pp. 1152-1162
Author(s):  
B. N. B. de Lima ◽  
R. Sanchis ◽  
R. W. C. Silva
Keyword(s):  
Long Range ◽  
Percolation Model ◽  
Binary Sequences ◽  
Site Percolation ◽  
Almost All ◽  
Independent Site

Consider an independent site percolation model onZd, with parameterp∈ (0, 1), where all long-range connections in the axis directions are allowed. In this work we show that, given any parameterp, there exists an integerK(p) such that all binary sequences (words) ξ ∈ {0, 1}Ncan be seen simultaneously, almost surely, even if all connections with length larger thanK(p) are suppressed. We also show some results concerning howK(p) should scale withpaspgoes to 0. Related results are also obtained for the question of whether or not almost all words are seen.

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.

Scaling properties of a percolation model with long-range correlations

1996 ◽  
Vol 54 (4) ◽  
pp. 3870-3880 ◽  
Author(s):  
Muhammad Sahimi ◽  
Sumit Mukhopadhyay
Keyword(s):  
Long Range ◽  
Percolation Model ◽  
Scaling Properties ◽  
Long Range Correlations

Critical behavior of percolation and Markov fields on branching planes

1993 ◽  
Vol 30 (3) ◽  
pp. 538-547 ◽  
Author(s):  
C. Chris Wu
Keyword(s):  
Continuous Function ◽  
Critical Behavior ◽  
Mean Field ◽  
Percolation Model ◽  
Homogeneous Tree ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Percolation Probability ◽  
Triangle Condition ◽  
Markov Fields

For an independent percolation model on, whereis a homogeneous tree andis a one-dimensional lattice, it is shown, by verifying that the triangle condition is satisfied, that the percolation probabilityθ(p) is a continuous function ofpat the critical pointpc, and the critical exponents,γ,δ, and Δ exist and take their mean-field values. Some analogous results for Markov fields onare also obtained.

EUCLIDEAN GIBBS STATES OF QUANTUM LATTICE SYSTEMS

2002 ◽  
Vol 14 (12) ◽  
pp. 1335-1401 ◽  
Author(s):  
S. ALBEVERIO ◽  
YU. KONDRATIEV ◽  
YU. KOZITSKY ◽  
M. RÖCKNER
Keyword(s):  
Critical Point ◽  
Long Range ◽  
Gibbs State ◽  
Classical Model ◽  
Gibbs Measures ◽  
Gibbs States ◽  
Multiplication Operators ◽  
Approximation Technique ◽  
Lattice Systems ◽  
Euclidean Gibbs States

An approach to the description of the Gibbs states of lattice models of interacting quantum anharmonic oscillators, based on integration in infinite dimensional spaces, is described in a systematic way. Its main feature is the representation of the local Gibbs states by means of certain probability measures (local Euclidean Gibbs measures). This makes it possible to employ the machinery of conditional probability distributions, known in classical statistical physics, and to define the Gibbs state of the whole system as a solution of the equilibrium (Dobrushin–Lanford–Ruelle) equation. With the help of this representation the Gibbs states are extended to a certain class of unbounded multiplication operators, which includes the order parameter and the fluctuation operators describing the long range ordering and the critical point respectively. It is shown that the local Gibbs states converge, when the mass of the particle tends to infinity, to the states of the corresponding classical model. A lattice approximation technique, which allows one to prove for the local Gibbs states analogs of known correlation inequalities, is developed. As a result, certain new inequalities are derived. By means of them, a number of statements describing physical properties of the model are proved. Among them are: the existence of the long-range order for low temperatures and large values of the particle mass; the suppression of the critical point behavior for small values of the mass and for all temperatures; the uniqueness of the Euclidean Gibbs states for all temperatures and for the values of the mass less than a certain threshold value, dependent on the temperature.

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