The critical probability for the frog model is not a monotonic function of the graph

2004 ◽  
Vol 41 (1) ◽  
pp. 292-298 ◽  
Author(s):  
L. R. Fontes ◽  
F. P. Machado ◽  
A. Sarkar
Keyword(s):  
Percolation Model ◽  
Monotonic Function ◽  
Critical Probability ◽  
Frog Model

We show that the critical probability for the frog model on a graph is not a monotonic function of the graph. This answers a question of Alves, Machado and Popov. The nonmonotonicity is unexpected as the frog model is a percolation model.

The critical probability for the frog model is not a monotonic function of the graph

2004 ◽  
Vol 41 (01) ◽  
pp. 292-298 ◽  
Author(s):  
L. R. Fontes ◽  
F. P. Machado ◽  
A. Sarkar
Keyword(s):  
Percolation Model ◽  
Monotonic Function ◽  
Critical Probability ◽  
Frog Model

We show that the critical probability for the frog model on a graph is not a monotonic function of the graph. This answers a question of Alves, Machado and Popov. The nonmonotonicity is unexpected as the frog model is a percolation model.

scholarly journals Line-of-Sight Percolation

2009 ◽  
Vol 18 (1-2) ◽  
pp. 83-106 ◽  
Author(s):  
BÉLA BOLLOBÁS ◽  
SVANTE JANSON ◽  
OLIVER RIORDAN
Keyword(s):  
Random Walk ◽  
Higher Dimensions ◽  
Percolation Model ◽  
The Other ◽  
Branching Random Walk ◽  
Line Of Sight ◽  
Continuum Percolation ◽  
Critical Probability ◽  
Site Percolation ◽  
Vertex Set

Given ω ≥ 1, let $\Z^2_{(\omega)}$ be the graph with vertex set $\Z^2$ in which two vertices are joined if they agree in one coordinate and differ by at most ω in the other. (Thus $\Z^2_{(1)}$ is precisely $\Z^2$.) Let pc(ω) be the critical probability for site percolation on $\Z^2_{(\omega)}$. Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that limω→∞ωpc(ω)=log(3/2). We also prove analogues of this result for the n-by-n grid and in higher dimensions, the latter involving interesting connections to Gilbert's continuum percolation model. To prove our results, we explore the component of the origin in a certain non-standard way, and show that this exploration is well approximated by a certain branching random walk.

scholarly journals UNIVERSALITY CLASS FOR BOOTSTRAP PERCOLATION WITH m=3 ON THE CUBIC LATTICE

1999 ◽  
Vol 10 (05) ◽  
pp. 921-930 ◽  
Author(s):  
N. S. BRANCO ◽  
CRISTIANO J. SILVA
Keyword(s):  
Monte Carlo Simulation ◽  
Finite Size ◽  
Universality Class ◽  
Percolation Model ◽  
Bootstrap Percolation ◽  
Stable Configuration ◽  
Finite Size Scaling ◽  
Critical Probability ◽  
Cubic Lattice ◽  
Free Boundary Conditions

We study the m = 3 bootstrap percolation model on a cubic lattice, using Monte Carlo simulation and finite-size scaling techniques. In bootstrap percolation, sites on a lattice are considered occupied (present) or vacant (absent) with probability p or 1-p, respectively. Occupied sites with less than m occupied first-neighbors are then rendered unoccupied; this culling process is repeated until a stable configuration is reached. We evaluate the percolation critical probability, pc, and both scaling powers, yp and yh, and, contrary to previous calculations, our results indicate that the model belongs to the same universality class as usual percolation (i.e., m=0). The critical spanning probability, R(pc), is also numerically studied for systems with linear sizes ranging from L=32 up to L=480; the value we found, R(pc)=0.270±0.005, is the same as for usual percolation with free boundary conditions.

Substitution Method Critical Probability Bounds for the Square Lattice Site Percolation Model

1995 ◽  
Vol 4 (2) ◽  
pp. 181-188 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Upper Bound ◽  
Lattice Site ◽  
Percolation Model ◽  
Square Lattice ◽  
Critical Probability ◽  
Site Percolation ◽  
Substitution Method ◽  
Probability Bounds

The square lattice site percolation model critical probability is shown to be at most .679492, improving the best previous mathematically rigorous upper bound. This bound is derived by extending the substitution method to apply to site percolation models.

An Improved Upper Bound for the Critical Probability of the Frog Model on Homogeneous Trees

2005 ◽  
Vol 119 (1-2) ◽  
pp. 331-345 ◽  
Author(s):  
Élcio Lebensztayn ◽  
Fábio P. Machado ◽  
Serguei Popov
Keyword(s):  
Upper Bound ◽  
Critical Probability ◽  
Homogeneous Trees ◽  
Frog Model

Ergodic properties of a dynamical system arising from percolation theory

1995 ◽  
Vol 15 (4) ◽  
pp. 653-661 ◽  
Author(s):  
Cor Kraaikamp ◽  
Ronald Meester
Keyword(s):  
Dynamical System ◽  
Percolation Theory ◽  
Percolation Model ◽  
Null Vector ◽  
Critical Probability ◽  
Ergodic Properties ◽  
Dependent Percolation ◽  
Almost All

AbstractWe consider the following dynamical system: take a d-dimensional real vectorwith positive coordinates. Now keep the smallest coordinate and subtract this one from the others, and iterate this process. When the starting vector is x we denote by xn the result after n iterations. It is shown that for almost all x, limn→∞xn ≠ 0 (the null vector). This is shown to be equivalent to the conjectured finiteness of an algorithm which produces the critical probability in a certain dependent percolation model.

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