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PERCOLATION: DISCRETE and CONTINUUM MODELS a Thesis Submitted to Kent State University in Partial Fulfillment of the Requirement

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PERCOLATION: DISCRETE and CONTINUUM MODELS a Thesis Submitted to Kent State University in Partial Fulfillment of the Requirement PERCOLATION: DISCRETE AND CONTINUUM MODELS A thesis submitted to Kent State University in partial ful¯llment 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 De¯nitions . 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 ¯xed 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 ¯xed radii 1 centered at ® and ¯ not overlap. But their p 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 ¯nishing 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 ¯rst chapter. We will continue the introduction into the second chapter, where we will give main mathematical de¯nitions 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 ¯nal chapter, we focus on studying continuum percolation. We ¯rst look at the Boolean-Poisson Model of Euclidean balls with ¯xed 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 e®ect and water molecule in a co®ee 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 ¯xed p, we denote such probability by θ(p). It is called percolation function, which is the probability that there is an in¯nite open path starting from the origin, e.g., ground water to go to "in¯nity" (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 de¯nitions 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 ¯nding 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 ¯xed radii and get the critical density ¸c 2 (0; 1). Then, we are going to show that the critical density ¸c still belong to (0; 1) 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 De¯nitions 2.1 Basic Terminology We start our thesis by giving some fundamental mathematical de¯nitions used in this thesis. Most of the following de¯nitions with additional details may be found in [8] and [12]. De¯nition 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. De¯nition 2. A subgraph H = (VH ;EH ) of a graph G = (VG;EG) is a graph such that VH ½ VG and EH ½ EG. De¯nition 3. If two vertices x and y are jointed by an edge, we de¯ne them to be adjacent and call them neighbors, de¯ned by x » y. De¯nition 4. A walk in a graph G is a ¯nite (or in¯nite) sequence of vertices and edges, W = fV0;E1;V1; :::; En;Vng; where Vi¡1 and Vi are the endpoints of the edge Ei, for i = 1; :::; n. 3 De¯nition 5. A path in a graph G is a walk such that no edge and vertex occur more than once. Such as P ath = fV0;E1;V1; :::; En;Vng: De¯nition 6. A walk with no repeating edges is called a circuit, if V0 = Vn. De¯nition 7. A group is a set G together with a binary operation ¤: G £ G ! G. For all a; b; c 2 G, they are satisfying the following properties: 1:(Closure) a ¤ b ¤ c 2 G 2:(Associativity) (a ¤ b) ¤ c = a ¤ (b ¤ c) 3:(Identity) There exists an element e 2 G such that: a ¤ e = e ¤ a = a: 4:(Inverse)F oreacha 2 G; there exist ana0 2 Gsuch that: a ¤ a0 = a0 ¤ a = e: De¯nition 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. De¯nition 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.
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