PERCOLATION: DISCRETE AND CONTINUUM MODELS
A thesis submitted to Kent State University in partial fulfillment of the requirements for the degree of Master of Science
by
Wei Bai
August 2014 Thesis written by
Wei Bai
B.S., University Science Malaysia, 2011
M.S., Kent State University, 2014
Approved by
Dr. Artem Zvavitch, Advisor
Dr. Andrew Tonge, Chair, Department of Mathetics Science
Dr. James L.Blank, Dean, College of Arts and Sciences
ii TABLE OF CONTENTS
LIST OF FIGURES ...... v
Acknowledgements ...... vi
Abstract ...... vii
1 Introduction ...... 1
2 Basic Mathematical Definitions ...... 3
2.1 Basic Terminology ...... 3
2.2 Trees ...... 4
2.3 Lattices ...... 5
2.4 Bond Percolation and Site Percolation ...... 6
3 Discrete Percolation ...... 7
3.1 Percolation on Binary Trees ...... 7
3.2 Percolation on k-ary Trees ...... 10
3.3 Percolation on Square Lattice ...... 12
3.4 Percolation on d-dimensional Lattices ...... 15
4 Continuum Percolation ...... 17
4.1 The Boolean-Poisson Model of Euclidean Balls with fixed Radii . . . . . 18
4.2 Generalization to Random Radii ...... 24 iii 4.3 Generalization to Convex Body ...... 26
BIBLIOGRAPHY ...... 28
iv LIST OF FIGURES
1 Fluid motion of ground water is one kind of percolation phenomenon. . . 2
2 Two generations of labeled binary tree ...... 5
3 Square Integer Lattice Z2 ...... 6
4 Bond Percolation ...... 6
5 Site Percolation ...... 6
6 Fixed points for fp(x)...... 9
7 The percolation function of binary tree ...... 10
8 Dual lattice (From [16]) ...... 13
9 The full line is a path of the Z2 and the dotted line is corresponding path
in the dual lattice. (From [16]) ...... 14
10 The thick line indicates a possible path of the plastic lily pads ...... 18
11 The cross section map in two-dimension ...... 20
12 The circles with fixed radii 1 centered at α and β not overlap. But their
√ d concentric circles with radii ln = 1 + n are overlapping...... 23
v Acknowledgements
I would like to address my sincere gratitude towards my advisor, Dr. Artem Zvav- itch, who guided me through the three years Applied Mathematics Master Program and helped me on the challenging path finishing my master program with Thesis option.
As my research advisor at the Department of Mathematical Science, he is very patient, encouraging and supportive.
I would like to express my appreciation to my Master Thesis Committee: Dr. Artem
Zvavitch, Dr. Benjamin Jaye, Dr. Dmitry Ryabogin, and Dr. Joe Diestel.
I also would like to give my deepest appreciation to my parents’ supporting and understanding.
vi Abstract
The purpose of this thesis is to review and study certain kinds of percolation models.
The thesis is divided into four chapters and dedicated to two classical types of percolation models.
We will give a brief introduction in the first chapter. We will continue the introduction into the second chapter, where we will give main mathematical definitions used in this thesis.
Then, in the third chapter, we will learn about the discrete percolation from four models, which are binary tree model, k-ary tree model and also give a short review for the percolation models on two dimensional and d-dimensional lattices.
In the final chapter, we focus on studying continuum percolation. We first look at the
Boolean-Poisson Model of Euclidean balls with fixed radii, and then extend to random radii. Finally, we will generalize the previous model into arbitrary convex bodies.
vii CHAPTER 1
Introduction
As early as the 1940s, Flory (1941) and Stockmeyer (1943) attempted to use perco-
lation theory to understand the gelation process, which is the process of small molecules
with random chemical bonds forming larger macromolecules. This is seen as the beginning
of percolation theory. In 1957, percolation was introduced into mathematics literature by
S. Broadbent and J. Hammersley, and has been studied intensively by mathematicians
and physicists since.(See [14])
Percolation can explain and describe a lot of natural phenomena, like cells osmotic
effect and water molecule in a coffee percolator. The most popular application is fluid
motion of ground water, which is also the earliest model studied by Broadbent and
Hammersley.
Suppose that the pores of the ground are connected through channels and that each
channel may be randomly open or closed to the passage of the flow, independently one
of the other (Figure 1). We assume a channel is open with probability p and closed with
probability 1 − p, where 0 < p < 1. Our main question is: does there exist a parameter p such that the probability of fluid to reach the center of the earth is greater than zero.
For a fixed p, we denote such probability by θ(p). It is called percolation function, which is the probability that there is an infinite open path starting from the origin, e.g., ground water to go to ”infinity” (the center of the earth).
1 Figure 1: Fluid motion of ground water is one kind of percolation phenomenon.
Generally speaking, percolation is a tool of studying connectivity problems of some randomly distributed objects on given spaces and models. In addition, percolation process is a branch of probability theory which exhibits what is known as critical phenomena, and which means the status of the system drastically changes with a natural parameter.
Basic mathematical definitions used in this thesis will be given in the next chapter.
Then, in chapter three, we are going to discuss discrete percolation on trees and lattices.
We focus on finding the critical parameter pc, the value of p at which the function θ(p) changes from being zero to being positive. We review the proof of critical probability
1 pc = 2 for binary tree model. As a generalization, we show the critical probability pc is
1 equal to k on k-ary tree model. Moreover, we show the existence of the critical probability pc on lattice models.
Finally, we will go to continuum percolation in chapter four. We will begin to look at the Boolean-Poisson Model of Euclidean balls with fixed radii and get the critical density
λc ∈ (0, ∞). Then, we are going to show that the critical density λc still belong to (0, ∞) for percolation with Euclidean balls of random radii. Next, we will consider a model when arbitrary convex bodies are instead of Euclidean balls.
2 CHAPTER 2
Basic Mathematical Definitions
2.1 Basic Terminology
We start our thesis by giving some fundamental mathematical definitions used in this thesis. Most of the following definitions with additional details may be found in [8] and
[12].
Definition 1. A graph is a set of objects involving points and links between them.
Consider a graph G = (V,E), the elements of V are called vertices and elements of E are called edges. The vertices assisting a edge to connect with other edges are called its endpoints.
Definition 2. A subgraph H = (VH ,EH ) of a graph G = (VG,EG) is a graph such that
VH ⊂ VG and EH ⊂ EG.
Definition 3. If two vertices x and y are jointed by an edge, we define them to be adjacent and call them neighbors, defined by x ∼ y.
Definition 4. A walk in a graph G is a finite (or infinite) sequence of vertices and edges,
W = {V0,E1,V1, ..., En,Vn},
where Vi−1 and Vi are the endpoints of the edge Ei, for i = 1, ..., n.
3 Definition 5. A path in a graph G is a walk such that no edge and vertex occur more
than once. Such as
P ath = {V0,E1,V1, ..., En,Vn}.
Definition 6. A walk with no repeating edges is called a circuit, if V0 = Vn.
Definition 7. A group is a set G together with a binary operation ∗: G × G → G. For
all a, b, c ∈ G, they are satisfying the following properties:
1.(Closure) a ∗ b ∗ c ∈ G
2.(Associativity) (a ∗ b) ∗ c = a ∗ (b ∗ c)
3.(Identity) There exists an element e ∈ G such that: a ∗ e = e ∗ a = a.
4.(Inverse)F oreacha ∈ G, there exist ana0 ∈ Gsuch that: a ∗ a0 = a0 ∗ a = e.
Definition 8. A subgroup of a group G is a group H such that H is the subset of G and
they have the same operation.
Definition 9. A body, denoted by K, is a compact set equal to the closure of its interior.
A convex body is a compact set with nonempty interior.
2.2 Trees
A tree is a connected graph with no circuits. A designated vertex of a tree is called a
root, and there is a unique path from the root to any other vertices in the tree. Following
common terminology, we call such a tree as directed tree or rooted tree T .
In Figure 2, we label 0 as the root of this rooted tree T . We say that the root 0 is at
level 0, vertices 1 and 2 are at level 1, and so forth. The level number n of a vertex Vn
in T is the length of the path from the root 0 to Vn. It is also called nth generation. For
4 any vertex V in a rooted tree T , except the root, the parent of V is its adjacent vertex
on the path to the root. The children of V are vertices for which V is the parent, i.e.,
in Figure 2, 1 is the parent of vertices 3 and 4 and at the same time vertices 3 and 4 are
the children of vertex 1.
Figure 2: Two generations of labeled binary tree
A binary tree, which is a rooted tree and in which each vertex has two children.
Similarly, a k-ary tree is a rooted tree that each vertex has k children.
2.3 Lattices
In graph theory, Z is the set of all integers and Zd is the set of all vectors x =
{x1, x2, ...xd} with integral coordinates. More precisely, we use xi to represent the ith coordinate of x ∈ Zd. Turned into a graph, Zd is called d-dimensional cubic lattice. By adding edges between any two neighbor points having Euclidean distance 1, we denote this lattice by Ld. Thus, we can write Ld = (Zd, Ed), where Zd is the set of vertices of
Ld and Ed is the set of all its edges (see Grimmett [7]). A lattice Ld in Rd is a discrete
subgroup (definition 7) of Rd, as they are satisfying the following properties:
1.(Discrete) There exist an ² > 0 such that any two distinct lattice points a 6= b ∈ Ld
5 are at distance at least ka − bk > ² > 0.
2.(Closure) Ld is closed under addition (or subtraction).
Figure 3: Square Integer Lattice Z2
2.4 Bond Percolation and Site Percolation
Models where edges are declared to be open or closed, are known as bond percolation, e.g., percolation on trees and on lattices. Alternatively, if we consider open or close situa- tion on vertices, those percolation are called site percolation, e.g., continuum percolation in this thesis. Figures below are two different typies of percolation.
Figure 4: Bond Percola- Figure 5: Site Percolation tion
Definition 10. We call a path open if all its vertices and edges are open.
6 CHAPTER 3
Discrete Percolation
3.1 Percolation on Binary Trees
Our discussion of discrete percolation is starting from percolation on an infinite binary
tree. We still denote the root of the tree T as 0, as in Figure 2.
Fix p ∈ [0, 1], and let each edge of T be independently open with probability p and
closed with probability 1 − p. Let Pn be the probability that there is an open path
from root 0 to a vertex at nth generation. Note that Pn ∈ [0, 1] is a non increasing and
bounded sequence, so the limit of Pn exists. Thus we have
θ(p) = lim Pn. n→∞
Theorem 1. The percolation function θ(p) is a non decreasing function of p.
Proof. Consider percolation on graph G = (V,E). Our idea is to glue percolation pro- cesses for different p into one process. Let X(e), for e ∈ E, is a collection of independent random variables, where each X(e) is uniformly distributed on [0, 1]. Fix a Pn ∈ [0, 1].
We imagine ηp as a random outcome of the process on G whose edges open with p, i.e.,
P (ηp(e) = 0) = P (X(e) ≥ p) = 1 − p and P (ηp(e) = 1) = P (X(e) < p) = p. More
precisely, define ηp (e) by 1 if X(e) < p η (e) = . p 0 if X(e) ≥ p
Note that for every edge e : ηp1 (e) ≤ ηp2 (e) whenever p1 ≤ p2, i.e., ηp1 ≤ ηp2 . 7 If there exists an infinite path starting from the origin on ηp1 , then so does ηp2 . Equiv-
alently, the probability that there is an infinite path for p1 is less than the probability
that there is an infinite path for p2. Then we get θ(p1) ≤ θ(p2).
Theorem 2. Consider a percolation process with parameter p on a binary rooted tree T .
If Pn is the probability of an open pass from the root to nth generation, then
2 Pn = 1 − (1 − pPn−1) .
Proof. First we consider the path goes through vertex 1 (See Figure 2). The probability
that an open path passes from root 0 through vertex 1 to a vertex Vn is pPn−1, where
p is the probability that the edge going from root 0 to vertex 1 is open and Pn−1 is
the probability that there is an open path starting from vertex 1 to vertex Vn. This is
because for an infinite rooted tree, the probability of a path starting from root 0 is the
same as starting from any vertex other than the root. Therefore, the probability that
there is no open path passing through vertex 1 from root 0 to a vertex Vn is 1 − pPn−1.
By self similarity, the probability that there is no open path passing through vertex 2 to a vertex Vn is also 1 − pPn−1. So the probability that there is no open path passing from
2 root 0 to a vertex Vn is (1 − pPn−1) and therefore
2 Pn = 1 − (1 − pPn−1) .
8 Next, we want to find the critical probability pc, at which the behavior of θ(p) dra- matically changing, i.e., changes from zero to positive. We define the function
2 fp(x) = 1 − (1 − px) .
Thus fp(Pn−1) = Pn. Taking the limit of both sides of this equation as fp is continuous, we get
lim Pn = lim fp(Pn−1) n→∞ n→∞ and
θ(p) = fp(θ(p)).
Figure 6: Fixed points for fp(x)
So the limit must be a fixed point of fp. Now we are interested in finding fixed points
1 for fp(x) on [0, 1]. By calculating fp(x) = x, we have two cases. If p < 2 , then x0 = 0 is
1 2p−1 the only fixed point and θ(p) = 0. And if p ≥ 2 , fixed points are x0 = 0 and x1 = p2 , indicated in Figure 6. Notice that Pn ≤ Pn−1, thus
fp(Pn−1) ≤ Pn−1, 9 and also notice that fp(x) ≥ x when x ∈ [x0, x1]. Thus Pn−1 cannot belong to the interval
[x0, x1]. x1 becomes the only possible limit for Pn. So the fixed point cannot be zero and
2p − 1 θ(p) = lim Pn = x1 = . n→∞ p2
Finally, the percolation function is 1 0 if p < 2 θ(p) = . 2p−1 1 p2 if p ≥ 2
1 Thus, the critical probability pc = 2 and the percolation function of a binary tree can be seen on Figure 7.
Figure 7: The percolation function of binary tree
3.2 Percolation on k-ary Trees
Now we extend the percolation process from a binary tree to a k-ary tree. Similar to the binary tree, the probability that there is no open path from root 0 to a vertex Vn
k is (1 − pPn−1) and the probability that there is an open path from root 0 to the nth
10 generations is
k Pn = 1 − (1 − pPn−1) .
Again, consider
k fp(x) = 1 − (1 − px)
as x = Pn−1 and Pn = fp(Pn−1).
We use the same logic as in the previous section. Our goal is to study fixed points of
k fp(x) = 1 − (1 − px) , for x ∈ [0, 1]. We observe that x0 = 0 is always a fixed point of
fp(x). Consider
F (x) = 1 − (1 − px)k − x.
We need to find the roots of F (x) = 0. We know that F (0) = 0. To see if there are
any other roots, consider
F 0(x) = kp(1 − px)k−1 − 1.
Thus, the critical point of function F must satisfy the following equality
µ ¶ 1 1 k−1 px = 1 − . c (kp)
³ ´ 1 1 1 k−1 We see that if p < k , then pxc = 1 − (kp) < 0 and xc < 0, i.e., F (x) has no
0 critical points on [0, 1]. More over, F (0) = kp − 1 < 0 and the sign for interval (xc, ∞)
is negative. Then it follows that F (x) is decreasing on (xc, ∞), so x0 = 0 is the only
1 solution for F (x) = 0 on interval x ∈ [0, 1]. Thus θ(p) = lim Pn = 0, for p < , which n→∞ k means the probability for an infinite open pass is zero of the k-ary tree.
1 Now consider the second case, if p ≥ k , then it gives the critical point xc > 0 and
0 F (0) = kp − 1 > 0. Then F is increasing on (0, xc) and decreasing on (xc, ∞). Our 11 goal is to understand if xc ∈ [0, 1]. Assume towards the contradiction, if xc > 1, then
k F (1) > F (0) > 0. But, in fact, F (1) = −(1 − p) and F (0) = 0, so we get xc is less than 1. Combining with F (0) = 0 and F (1) < 0, it gives us that there must exist a solution x1 ∈ (xc, 1) for F (x1) = 0. We use the same logic as in case k = 2 to claim that
Pn−1 ≥ x1. So we proved : θ(p) = lim Pn = x0. Therefore, we conclude that the critical n→∞
1 probability pc = k .
3.3 Percolation on Square Lattice
2 1 For percolation on square lattice Z , the critical probability is pc = 2 , which is given
by Harris and Kesten and known as The Harris-Kesten Theorem. In 1960s, Harris stated
1 1 1 and proved that θ( 2 ) = 0, which implied that pc ≥ 2 . Then pc ≤ 2 was proven by kesten in 1982. A good brief proof of The Harris-Kesten Theorem is introduced by B. Bollobas and O. Riordan ([2]). We will not present those proofs in our thesis. Instead, we will show a weaker result. We just show the existence of critical probability pc.
1 2 In the following two theorems, we will show that pc ∈ ( 3 , 3 ).
Let an open cluster C(x) contains x is the set of y that x and y are connected by
an open path. Obversely, C(0), abbreviated by C, is a open cluster at the origin. For a
fixed p ∈ (0, 1), let Pp = Π µp where µp is Bernoulli measure on (0, 1), given by e∈Ed
µp(edge is closed) = 1 − p, and µp(edge is open) = p.
Thus, we have θ(p) = Pp(|C| = ∞).
1 Theorem 3. If p < 3 , θ(p) = 0.
Proof. Let Fn be an event that there is an open path starting from origin to vertex n 12 in Z2. The probability for such an open path is pn and there are at most 4(3)(n−1) such paths. Since paths do not turn back in Z2, there are 4 choices for the first step and 3 choices for each of the following steps. Then we have,
4 P (F ) = pn4(3)(n−1) = (3p)n. p n 3
1 Note that when p < 3 as n → ∞, Pp(Fn) = Pp(|C| = ∞) = 0. It implies θ(p) = 0 if
1 p < 3 .
Figure 8: Dual lattice (From [16])
Then we introduce dual lattice (Figure 8) formed from the square lattice Z2: place
one vertex in the center of each square and connect these vertices with an edge. Notice
that for each edge in the square lattice, in the corresponding dual lattice, there is exactly
one edge crossing it. In fact, dual lattice is the square lattice translated by the vector
1 1 ( 2 , 2 ).
Consider an edge in dual lattice to be open if and only if the corresponding edge in
Z2 is open. In other words, it is open when edge in Z2 crossing it is closed, i.e., Figure
9. The distribution of the set of open edge in dual lattice is also determined by Pp. 13 Figure 9: The full line is a path of the Z2 and the dotted line is corresponding path in the dual lattice. (From [16])
2 Theorem 4. θ(p) > 0 if p > 3 .
If there does not exist an open cluster in dual lattice, then |C| < ∞ while if |C| < ∞, there would not be an infinite open cluster. This is a theorem given by Whitney in graph theory.([16])
Lemma 1. |C| < ∞ if and only if there exists a simple cycle in dual lattice surrounding
0 consisting of all closed edges.
Proof. Let Gn be an event that there is a simple circle in dual lattice surrounding 0 having length n. Note that at least 4 vertices can form a circle, then we can have the
14 following equation. ∞ Pp(|C| < ∞) = Pp( ∪ Gn) n=4 X∞ ≤ Pn(Gn) n=4 X∞ (3.1) ≤ n4(3)n−1(1 − p)n n=4 4X∞ = n(3(1 − p))n 3 n=4 since any of the n vertices can be seen as the beginning of a circle and there are at most
4(3)(n−1) such circles.
Based on equation 3.1, we get
θ(p) = Pp(|C| = ∞)
= 1 − Pp(|C| < ∞) . 4X∞ = 1 − n(3(1 − p))n 3 n=4
2 Therefore, if p > 3 , then θ(p) → 1 almost surely.
1 2 2 Thus, we prove pc ∈ ( 3 , 3 ) in Z .
3.4 Percolation on d-dimensional Lattices
As before, let θd(p) be the probability there is an infinite path from origin on d- dimensional lattice. The critical probability is known as:
pc(d) = sup{p : θd(p) = 0} = inf{p : θd(p) ≥ 0}.
1 2 Obviously, when d = 1, the critical probability is 1. As we just showed, pc ∈ ( 3 , 3 ) when d = 2. We notice that pc(d) will be decreasing when d is increasing. This is because in higher dimensional lattices, there exist more possible paths to go to infinity. Thus, the 15 critical probability pc(d) will become smaller and smaller as dimension increasing. Thus,
2 1 from Theorem 4 we get pc(d) < 3 and pc(d) ≤ 2 when d ≥ 2.
Next lemma shows that pc(d) is greater than 0.
1 Lemma 2. If p < 2d−1 , θd(p) = 0.
Proof. The proof process is similar with the proof of theorem 3. But now, Fn happens in
Zd. The number of corresponding path is at most 2d(2d − 1)n−1, since there are at most
2d choices for the first step and (2d − 1) choices for the remaining n − 1 steps. Similarly, we have 2d P (F ) ≤ pn2d(2d − 1)n−1 = (p(2d − 1))n. p n 2d − 1
1 This implies when p < 2d−1 , Pp(Fn) → 0 as n → ∞.
Thus, in d-dimensional lattice, there exist a critical probability pc(d) and pc(d) ∈
1 1 ( 2d−1 , 2 ).
16 CHAPTER 4
Continuum Percolation
Instead of working with discrete models, we are now focusing on studying continuum percolation. Imagine a swimming pool in a amusement park, we are interested in if children can pass the swimming pool by following the randomly thrown plastic lily pads while two pads are connected if and only if they overlap to each other. Now let us give a mathematical definition of this model. Let X = {x1, x2, x3...} be the points of a Poisson process, the process of throwing plastic lily pads (whose centers belong to X) into a swimming pool is a continuous random process. This process is called Boolean-Poisson
Model, or the Poisson blob model ([7]).
d Definition 11. Let X = {x1, x2, x3...} be a poisson point process on R , for any bounded region A ⊂ Rd, we define X(A) = #(X ∩ A) and we have
(λµ(A))k P (X(A) = k) = e−λµ(A) , λ k! where µ(A) denotes the d-dimensional volume of A and k = 1, 2, 3...
This process is called Poisson Process with density λ > 0.
Note that the expected number of points from X in a set A is
X∞ (λµ(A))k E (X(A) = k) = e−λµ(A) k = λµ(A). (4.1) λ k! k=0
If we consider an infinite large pool and ask for the probability of existence of infinite cluster shown in Figure 10. It is also interesting in asking how the radii changing and 17 shapes changing of lily pads effect probabilities of such paths’ existence. Now we start
our model from Euclidean balls with fixed radii.
Figure 10: The thick line indicates a possible path of the plastic lily pads
4.1 The Boolean-Poisson Model of Euclidean Balls with fixed Radii
Let xi, i ≥ 0 be the points of Poisson process with density λ > 0 and each point
d xi ∈ R is the center of a closed Euclidean ball with fixed radius.
From our experience with the binary tree model, we can expect that there exists a
critical density λc such that when λ > λc there exist an infinite open cluster and when
λ < λc there is no such cluster.
Notation Pλ indicates the probability corresponding to a Poisson process with density
λ. Similarly, we define θ(λ) as percolation function, the probability that there is an infinite
cluster in the continuum model,
θ(λ) = Pλ(|C| = ∞).
18 Theorem 5. There exists λc ∈ (0, ∞), such that = 0 if λ < λc θ(λ) . > 0 if λ > λc
The idea (See [7]) of this proof is by applying the method of studying probability of
discrete model to continuous model.
Proof. We start with relating the continuous process to an ordinary site percolation
d −1 d process. Let n be a positive integer and let Zn = n Z . More precisely,
n x x x o Zd = ( 1 , 2 , ..., d ): x ∈ Zd . n n n n
d d In this way, we partition R into cubes Bn(x) whose centers are at the points of Zn.
The cubes Bn(x) are defined as:
d 1 1 d Bn(x) = Π [xi − , xi + ) for x ∈ Zn. i=1 2n 2n
We use half open and half close interval to avoid overlapping edges, i.e., Bn(xi) ∩
d Bn(xj) = φ, for i 6= j in Zn.
The way we choosing balls follows the rule: the centers of balls should locate in cubes
Bn(x). We notice that two balls are connected, if their centers are at distance less than
2. Thus two cubes have chance to be connected if ∃α ∈ Bn(xi) and ∃β ∈ Bn(xj) with
d |α − β| ≤ 2, for i 6= j. Then we turn Zn into a lattice Ln whose vertices are the centers of cubes Bn(x), indicated in Figure 11. We shall consider site percolation on such lattice
Ln.
We declare a vertex x of Ln to be open if one or more points inside Bn(x) are selected
from the Poisson process, and closed otherwise. As cubes are not overlapping with each 19 Figure 11: The cross section map in two-dimension
other, so different vertices are independently open or closed. Let Pn(λ) be the probability that given vertex is open, and by definition of Poisson process we get
−λn−d Pn (λ) = 1 − Pn (no point selected from Poisson process is inside Bn(x) ) = 1 − e .
Notice that if there is an infinite cluster of balls, then there will be an infinite cluster in Ln.
Let pc (Ln) be the critical probability of an infinite cluster on Ln. We know it is easier to get an open vertex on Ln than to get an infinite cluster on Ln. So we have,
Pn (λ) ≥ pc (Ln) .
By computing
−λn−d Pn (λ) = 1 − e ≥ pc (Ln) , we get
d λ ≥ −n log (1 − pc (Ln)) .
20 For an infinite cluster, λ > λc as λc = inf{λ, θ(λ) > 0}, and we have
d λc ≥ −n log (1 − pc (Ln)) > 0, as 0 < pc (Ln) < 1.
Therefore, we get λc > 0. Next, we will provide an upper bound of λc.
In order to study upper bound of λc, we introduce a new parameter
λ λT = sup{λ : E |C| < ∞} ∈ (0, ∞),
where Eλ |C| is the average cluster size, for a fixed λ.
Clearly,
λT ≤ λc.
λ since λc = inf{λ : E |C| = ∞}. This is because for any λ > λc, Pλ(|C| = ∞) > 0, then
Eλ |C| = ∞.
And our object is to show
λT = λc.
In order to show it, we note that each point X from the Poisson process lies in some
unique Bn(x). And we define a function x = πn(X): mapping points of Poisson process into vertices of Ln. Let Dn be the set of open vertices of Ln connected with origin or
d some vertex in the set {x ∈ Zn : x ∼ 0} of neighbors of the origin, where we define x ∼ 0 as adjacency relation if and only if ∃α ∈ Bn(x) and ∃β ∈ Bn(0) with |α − β| ≤ 2.
Suppose |Dn| = m. We know that each vertex of Dn is the center of a cube Bn(x) and by
definition of Dn, these vertices are open. Then at least one point from the Poisson process is contained in the cube Bn(x). Let E(N|N ≥ 1) be the average number of such points
21 −1 in such cubes. Then we have πn (Dn) is the set of all points from Poisson process which belong to cubes Bn(x), where x ∈ Dn. Based on the formula of conditional expectation, by given equation (4.1), we get
∞ λ −1 P λ −1 λ E |πn (Dn)| = E (πn (Dn)| |Dn| = m)P (|Dn| = m) m=0 ∞ P λ = m(E(N|N ≥ 1))P (|Dn| = m) m=0 ∞ P λ = E(N|N ≥ 1) mP (|Dn| = m) m=0
λ = E(N|N ≥ 1))E (|Dn| = m) ∞ λ P P (N=k∩N≥1) = E (|Dn| = m) P (N≥1) k k=1 ∞ λ P P (N=k) = E (|Dn| = m) P (N≥1) k k=1 ∞ λ P P (N=k) = E (|Dn| = m) 1−P (N=0) k k=1 ∞ −d λ P e−λn (λn−d)k = E (|Dn| = m) −d k 1−e−λn k! k=1 ∞ 1 λ P −λn−d (λn−d)k = −d E (|Dn| = m) e k 1−e−λn k! k=1
λn−d λ = −d E (|Dn| = m). 1−e−λn
Let Sn be the open cluster of Ln containing the origin. We note that getting the
d number of neighbors (e.g. x ∼ 0) of the origin of Ln is at most (4n) . And therefore
d |Dn| ≤ |C| + (4n) |Sn|
λ λ d d λ E |Dn| ≤ E (|Sn| + (4n) |Sn|) = (1 + (4n) )E (|Sn|).
−1 And as C ⊆ πn (Dn), then ¯ ¯ λ λ ¯ −1 ¯ E |C| ≤ E πn (Dn) λn−d = Eλ |D | (4.2) 1 − e−λn−d n λn−d(1 + (4n)d) ≤ Eλ |S | . 1 − e−λn−d n 22 λ From the definition of λT , for any λ > λT , we have E |C| = ∞. Thus from (4.2), we
λ get E |Sn| = ∞. So for this λ, we have an open cluster in Sn. Thus Pn(λ) ≥ Pc(Ln), and
d λT ≥ −n log (1 − pc (Ln)) .
Figure 12: The circles with fixed radii√ 1 centered at α and β not overlap. But their d concentric circles with radii ln = 1 + n are overlapping.
Now we are ready to provide an upper bound for λc. We will work with lattice Ln, and we define vertices to be adjacency differently. As in Figure 12, we now consider two
√ d circles with radii are not overlapping, but their concentric circles with radii ln = 1+ n do √ d overlap, where n is the diagonal of the cube Bn(x). Similarly, we say the two concentric
circles are connected if ∃α ∈ Bn(xi) and ∃β ∈ Bn(xj) with |α − β| ≤ 2ln. And then the
cluster of balls starting from the origin is infinite whenever there is an infinite cluster of
Ln. Therefore
d d Pn (λ) ≥ pc (Ln) implies λln > λc and λ ≥ −n log (1 − pc (Ln)) ,
and then we notice
d −d d d −n log (1 − pc (Ln)) ≥ ln λc and λc ≤ −lnn log (1 − pc (Ln)) . 23 Then we get the upper bound of λc. Therefore, we find the critical density λc :
d d 0 < λc ≤ −lnn log (1 − pc (Ln)) .
We have proved that
£ d ¤ λc = λT = lim −n log (1 − pc (Ln)) , n→∞
and λc < ∞ as n → ∞.
4.2 Generalization to Random Radii
In this section, we consider a generalization to the case of balls with discrete random radii as Rad = {r1, r2, . . . rn}. We get that the conclusion is the same as in the case of
fixed radii. Each time we select a center α, we also randomly pick a ball with radius r(α)
∈ Rad corresponding to a distribution:
P (r (α) = ri) = Pi > 0
Pn and Pi = 1, where ri ∈ Rad. i=1 c c Our goal is to study critical density λRad for this model. When λRad is defined as:
c = 0 if λ < λRad θ(λ) , c > 0 if λ > λRad where θ(λ) is a percolation function.
Now let rmin = min {r1, r2, . . . rn} and rmax = max {r1, r2, . . . rn} . Then we describe three models. Model one is all balls with fixed radii rmin. And model two is that balls have random radii r(α) ∈ Rad, which is the studying model. And in the third model, all 24 balls have fixed radii rmax. And let Pmin(λ),PRad(λ) and Pmax(λ) are the probabilities
that there is an open path for the three models we just state respectively. We notice that
the larger the radii are, the more chance there exist an open cluster for a model. So we
have
Pmin(λ) ≤ PRad(λ) ≤ Pmax(λ).
c Then for critical density λRad, we have
c c c λmax ≤ λRad ≤ λmin. (4.3)
c c From the proof of Theorem 5, we get λmax ∈ (0, ∞) and λmin ∈ (0, ∞) . Therefore,
c from inequality (4.3), we get the critical density λRad ∈ (0, ∞) .
Now let us consider a general case of random radii: for each center α, we select radius
r(α) distributed with law P (r(α) < x) = P (x). Again, we assume that r(α) and r(β) are
independent and identically distributed random variables if α 6= β.
c Corollary 1. The above continuous model has a critical probability λRad ∈ (0, ∞).
Let Yp be a Poisson process with density λ and in this process each point is picked up with probability p. Then we can get that
25 P∞ P (Y (A) = k) = P (Y (A) = k|X(A) = i) ∗ P (X(A) = i) i=0
P∞ i i = pk(1 − p)i−ke−λµ(A) (λµ(A)) i! i=0 k
∞ −λµ(A) k k P i! (1−p)i−k(λµ(A))i−k = e p (λµ(A)) (i−k)!k! i! i=k ∞ e−λµ(A)pk(λµ(A))k P 1 j j = k! j! ((1 − p) λµ(A)) let j = i − k j=0
−pλµ(A) (pλµ(A))k = e k!
∞ P 1 j j (1−p)λµ(A) since j! ((1 − p) λµ(A)) = e . j=0
So our new process Yp is a Poisson process with parameter pλ.
Consider a percolation with Poisson process Yr of random radii. For fixed r0, let
P (r > r0) = p > 0 with p ∈ (0, 1). Then consider a Poisson process Ycut following the rule: do not put a ball at a point if r < r0 and put one if r > r0. Then
P (Yp) ≤ P (Ycut) ≤ P (Yr),
so
λc ≤ (λp)c < ∞.
4.3 Generalization to Convex Body
In this section, we would like to consider another generalization of continuous perco-
lation model. We will again select centers α with respect to Poisson Process. But now
instead of putting balls, we will put a copy of fixed convex symmetric body K. Our goal
is again to show that the critical density λc belongs to (0, ∞) .
Let us start with a few standard definitions and notations about convex bodies. 26 A body K ⊂ Rd is called star-shaped if every straight line through the origin intersects the boundary of K at exactly two points different from the origin ([18]). The Minkowski function of a convex body K is defined as
||x||K = min {λ > 0 : x ∈ λK} ,
which is a continuous function on Rd.
A set K ⊂ Rd is called convex, if [x, y] ⊂ K, where [x, y] denotes the line segment
with endpoints x and y and for all x, y ∈ K. A set in Rd is called a convex body if it is
convex and compact with nonempty interior.
Then we can also get the same theorem as the balls with fixed radii.
Theorem 6. There exists λc, 0 < λc < ∞, such that = 0 if λ < λc θ(λ) . > 0 if λ > λc
The proof is same as the proof of Theorem 3, but replacing adjacency relation, xi ∼ xj,
if ∃α ∈ Bn(xi) and ∃β ∈ Bn(xj) with |α − β| ≤ 2 by using ||x − y||K ≤ 2.
27 BIBLIOGRAPHY
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