Lower bounds for the critical probability in percolation models with oriented bonds

1980 ◽  
Vol 17 (04) ◽  
pp. 979-986 ◽  
Author(s):  
Lawrence Gray ◽  
John C. Wierman ◽  
R. T. Smythe
Keyword(s):  
Lower Bounds ◽  
Oriented Percolation ◽  
Critical Probability ◽  
Critical Percolation ◽  
Simple Method

In completely or partially oriented percolation models, a conceptually simple method, using barriers to enclose all open paths from the origin, improves the best previous lower bounds for the critical percolation probabilities.

Lower bounds for the critical probability in percolation models with oriented bonds

1980 ◽  
Vol 17 (4) ◽  
pp. 979-986 ◽  
Author(s):  
Lawrence Gray ◽  
John C. Wierman ◽  
R. T. Smythe
Keyword(s):  
Lower Bounds ◽  
Oriented Percolation ◽  
Critical Probability ◽  
Critical Percolation ◽  
Simple Method

In completely or partially oriented percolation models, a conceptually simple method, using barriers to enclose all open paths from the origin, improves the best previous lower bounds for the critical percolation probabilities.

Approximate upper bounds for the critical probability of oriented percolation in two dimensions based on rapidly mixing Markov chains

1997 ◽  
Vol 34 (04) ◽  
pp. 859-867
Author(s):  
Béla Bollabás ◽  
Alan Stacey
Keyword(s):  
Monte Carlo ◽  
Markov Chains ◽  
Monte Carlo Simulations ◽  
Statistical Tests ◽  
Upper Bounds ◽  
Two Dimensions ◽  
Oriented Percolation ◽  
Critical Probability ◽  
Confidence Levels

We develop a technique for establishing statistical tests with precise confidence levels for upper bounds on the critical probability in oriented percolation. We use it to give pc < 0.647 with a 99.999967% confidence. As Monte Carlo simulations suggest that pc ≈ 0.6445, this bound is fairly tight.

scholarly journals CRITICAL PERCOLATION OF FREE PRODUCT OF GROUPS

2008 ◽  
Vol 18 (04) ◽  
pp. 683-704 ◽  
Author(s):  
IVA KOZÁKOVÁ
Keyword(s):  
Cayley Graph ◽  
Free Product ◽  
Finite Groups ◽  
Cluster Size ◽  
Critical Probability ◽  
Critical Percolation ◽  
Residually Finite Groups ◽  
Quotient Groups ◽  
Free Product Of Groups ◽  
Product Of Groups

In this article we study percolation on the Cayley graph of a free product of groups. The critical probability pc of a free product G1 * G2 * ⋯ * Gn of groups is found as a solution of an equation involving only the expected subcritical cluster size of factor groups G1, G2, …, Gn. For finite groups this equation is polynomial and can be explicitly written down. The expected subcritical cluster size of the free product is also found in terms of the subcritical cluster sizes of the factors. In particular, we prove that pc for the Cayley graph of the modular group PSL2(ℤ) (with the standard generators) is 0.5199…, the unique root of the polynomial 2p5 - 6p4 + 2p3 + 4p2 - 1 in the interval (0, 1). In the case when groups Gi can be "well approximated" by a sequence of quotient groups, we show that the critical probabilities of the free product of these approximations converge to the critical probability of G1 * G2 * ⋯ * Gn and the speed of convergence is exponential. Thus for residually finite groups, for example, one can restrict oneself to the case when each free factor is finite. We show that the critical point, introduced by Schonmann, p exp of the free product is just the minimum of p exp for the factors.

scholarly journals Generalisations of Turán's main theorems on lower bounds for sums of powers

1970 ◽  
Vol 2 (1) ◽  
pp. 15-37 ◽  
Author(s):  
A.J. van der Poorten
Keyword(s):  
Lower Bounds ◽  
Exponential Sums ◽  
Simple Method ◽  
Polynomial Coefficients ◽  
Sums Of Powers ◽  
Constant Coefficients ◽  
New Type

In his book, Eine neue Methode in der Analysis und deren Andwendungen, P. Turán proved a number of new theorems given lower bounds for sums of powers. Since it was only his intention to demonstrate a new type of result, his bounds are by no means best possible nor are his proofs easily susceptible of improvement.We generalise Turán's so-called Main Theorems to exponential sums with polynomial coefficients by a simple method involving only the evaluation and estimation of certain determinants. This approach gives in each case a result known to be asymptotically correct in the various exponents, and when specialised to the case of constant coefficients it provides in each case best-known results.Our method moreover applies in more general circumstances and provided only that the determinants which arise can be conveniently estimated serves to provide lower bounds for other than exponential sums.

scholarly journals Lower Bounds of Periods of Periodic Solutions for a Class of Differential Equations with Variable Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-3
Author(s):  
Xin-Ge Liu ◽  
Mei-Lan Tang
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Lower Bounds ◽  
Functional Differential Equations ◽  
Hölder Inequality ◽  
Simple Method ◽  
Variable Delays ◽  
Nonautonomous Differential Equations ◽  
Functional Differential ◽  
Wirtinger’S Inequality

Based on generalized Wirtinger's inequality, periods of periodic solutions of the nonautonomous differential equations with variable delays are investigated. Based on Hölder inequality, lower bounds of periods of periodic solutions for a class of functional differential equations with variable delays are obtained by a simple method.

Upper bounds for the critical probability of oriented percolation in two dimensions

1993 ◽  
Vol 440 (1908) ◽  
pp. 201-220 ◽  
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Two Dimensions ◽  
Oriented Percolation ◽  
Bond Percolation ◽  
Critical Probability ◽  
Site Percolation

We give a method for obtaining upper bounds on the critical probability in oriented bond percolation in two dimensions. This method enables us to prove that the critical probability is at most 0.6863, greatly improving the best published upper bound, 0.84. We also prove that our method can be used to give arbitrarily good upper bounds. We also use a slight variant of our method to obtain an upper bound of 0.72599 for the critical probability in oriented site percolation.

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