Critical sponge dimensions in percolation theory

1981 ◽  
Vol 13 (2) ◽  
pp. 314-324 ◽  
Author(s):  
G. R. Grimmett
Keyword(s):  
Positive Constant ◽  
Percolation Theory ◽  
Square Lattice ◽  
Bond Percolation ◽  
Critical Probability ◽  
Percolation Process ◽  
Open Path ◽  
Image Position

In the bond percolation process on the square lattice, with let S(k) be the probability that some open path joins the longer sides of a sponge with dimensions k by a log k. There exists a positive constant α = αp such that Consequently, the subset of the square lattice {(x, y):0 ≦ y ≦ f(x)} which lies between the curve y = f(x) and the x-axis has the same critical probability as the square lattice itself if and only if f(x)/log x → ∞ as x → ∞.

Critical sponge dimensions in percolation theory

1981 ◽  
Vol 13 (02) ◽  
pp. 314-324 ◽  
Author(s):  
G. R. Grimmett
Keyword(s):  
Positive Constant ◽  
Percolation Theory ◽  
Square Lattice ◽  
Bond Percolation ◽  
Critical Probability ◽  
Percolation Process ◽  
Open Path ◽  
Image Position

In the bond percolation process on the square lattice, with let S(k) be the probability that some open path joins the longer sides of a sponge with dimensions k by a log k. There exists a positive constant α = αp such that Consequently, the subset of the square lattice {(x, y):0 ≦ y ≦ f(x)} which lies between the curve y = f(x) and the x-axis has the same critical probability as the square lattice itself if and only if f(x)/log x → ∞ as x → ∞.

On the time constant and path length of first-passage percolation

1980 ◽  
Vol 12 (04) ◽  
pp. 848-863 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Distribution Function ◽  
Time Constant ◽  
Path Length ◽  
Passage Time ◽  
Square Lattice ◽  
Bond Percolation ◽  
First Passage Percolation ◽  
Critical Probability ◽  
First Passage ◽  
Individual Bond

Let U be the distribution function of the passage time of an individual bond of the square lattice, and let pT be the critical probability above which the expected size of the open component of the origin (in the usual bond percolation) is infinite. It is shown that if (∗)U(0–) = 0, U(0) < pT , then there exist constants 0 < a, C 1 < ∞ such that a self-avoiding path of at least n steps starting at the origin and with passage time ≦ an} ≦ 2 exp (–C 1 n). From this it follows that under (∗) the time constant μ (U) of first-passage percolation is strictly positive and that for each c > 0 lim sup (1/n)Nn (c) <∞, where Nn (c) is the maximal number of steps in the paths starting at the origin with passage time at most cn.

scholarly journals Disproof of the conjectured subexponentiality of certain functions in percolation theory

1984 ◽  
Vol 16 (03) ◽  
pp. 690-691
Author(s):  
J. Van den berg
Keyword(s):  
Percolation Theory ◽  
Bond Percolation ◽  
Open Path

Consider bond-percolation on a graph G with sites S(G). We disprove the conjecture of Hammersley (1957) that the function n → sup s ϵ S(G) E [the number of sites s′ at distance n from s which can be reached from s by an open path which, except for s′, only passes through sites at distance smaller than n from s] is always subexponential.

scholarly journals Disproof of the conjectured subexponentiality of certain functions in percolation theory

1984 ◽  
Vol 16 (3) ◽  
pp. 690-691
Author(s):  
J. Van den berg
Keyword(s):  
Percolation Theory ◽  
Bond Percolation ◽  
Open Path

Consider bond-percolation on a graph G with sites S(G). We disprove the conjecture of Hammersley (1957) that the function n → sups ϵ S(G)E [the number of sites s′ at distance n from s which can be reached from s by an open path which, except for s′, only passes through sites at distance smaller than n from s] is always subexponential.

On the time constant and path length of first-passage percolation

1980 ◽  
Vol 12 (4) ◽  
pp. 848-863 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Distribution Function ◽  
Time Constant ◽  
Path Length ◽  
Passage Time ◽  
Square Lattice ◽  
Bond Percolation ◽  
First Passage Percolation ◽  
Critical Probability ◽  
First Passage ◽  
Individual Bond

Let U be the distribution function of the passage time of an individual bond of the square lattice, and let pT be the critical probability above which the expected size of the open component of the origin (in the usual bond percolation) is infinite. It is shown that if (∗)U(0–) = 0, U(0) < pT, then there exist constants 0 < a, C1 < ∞ such that a self-avoiding path of at least n steps starting at the origin and with passage time ≦ an} ≦ 2 exp (–C1n).From this it follows that under (∗) the time constant μ (U) of first-passage percolation is strictly positive and that for each c > 0 lim sup (1/n)Nn(c) <∞, where Nn(c) is the maximal number of steps in the paths starting at the origin with passage time at most cn.

scholarly journals The Distribution of Heights of Binary Trees and Other Simple Trees

1993 ◽  
Vol 2 (2) ◽  
pp. 145-156 ◽  
Author(s):  
Philippe Flajolet ◽  
Zhicheng Gao ◽  
Andrew Odlyzko ◽  
Bruce Richmond
Keyword(s):  
Asymptotic Formula ◽  
Positive Constant ◽  
Binary Trees ◽  
Asymptotic Results ◽  
Image Position

The number, , of rooted plane binary trees of height ≤ h with n internal nodes is shown to satisfyuniformly for δ−1(log n)−1/2 ≤ β ≤ δ(log n)1/2, where and δ is a positive constant. An asymptotic formula for is derived for h = cn, where 0 < c < 1. Bounds for are also derived for large and small heights. The methods apply to any simple family of trees, and the general asymptotic results are stated.

Functions regular in the unit circle

1956 ◽  
Vol 52 (1) ◽  
pp. 49-60 ◽  
Author(s):  
C. N. Linden ◽  
M. L. Cartwright
Keyword(s):  
Unit Circle ◽  
Positive Constant ◽  
Upper Bounds ◽  
Image Position

Letbe a function regular for | z | < 1. With the hypotheses f(0) = 0 andfor some positive constant α, Cartwright(1) has deduced upper bounds for |f(z) | in the unit circle. Three cases have arisen and according as (1) holds with α < 1, α = 1 or α > 1, the bounds on each circle | z | = r are given respectively byK(α) being a constant which depends only on the corresponding value of α which occurs in (1). We shall always use the symbols K and A to represent constants dependent on certain parameters such as α, not necessarily having the same value at each occurrence.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

On spàtial general epidemics and bond percolation processes

1984 ◽  
Vol 21 (4) ◽  
pp. 911-914 ◽  
Author(s):  
Kari Kuulasmaa ◽  
Stan Zachary
Keyword(s):  
Lower Bound ◽  
Bond Percolation ◽  
Percolation Process ◽  
Percolation Probability ◽  
Percolation Processes

We show that a lower bound for the probability that a spatial general epidemic never becomes extinct is given by the percolation probability of an associated bond percolation process.

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