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The K sub tree seepage percolation critical exponents

Author: LiuGengYang
Tutor: ZhangZuoCi
School: Capital Normal University
Course: Basic mathematics
Keywords: K-ary Tree Percolation Critical Exponents
CLC: O357.3
Type: Master's thesis
Year: 2002
Downloads: 29
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Abstract


The main context of this paper is to generalize some rigurous results of ’critical exponents’ from binary tree to k-nary tree.Percolation on the trees is defined as follows. We write T = (Z,E) for the bond percolation on the tree .we write Z for the set of vertices of T. arid E for the set of its edges. Let p satisfy 0 < p < 1. We declare each edge of T to be open with probability p and dosed otherwise,independently of all other edges. We call the resulting process (bond) percolation on TA path of T is an alternating sequeiice:{v0;e1,v1,e2,v2,.....,en,vn} of distinctvertices vi and edges ei = (vi-1,vi). Such a path has length n. We call a path open if all of its edges are open. If an open path has infinite length,we call it infinite open path.Consider the random subgraph of T containing the vertex set Z and the open edges only. The connected components of this graph are called open cluter. usually,we write it in the short term C. Some people say that ’percolation occurs’when there exists an infinite open cluster.Here are the relative concepts1. A principal concept is the percolation probability. (p) being the probability that a given vertex belongs to an infinite open cluster.2. It is fundamental to percolation theory that there exits a critical value pc = pc(d) of p,and pc(d) is called the critical probability and is defined by3. We take product measure with density p aswhere ue is Bernoulli measure on {0.1}. and its corresponding expectation as Ep.4. We write x(p) = EP(C) for the mean number of vertices in the open cluster at the origin.The main results of this paper are as follows:1. For the percolation on K-ary tree,Critical Probability2. For the percolation on K-ary tree.3. For the percolation on K-ary tree.4. For the subcritical percolation on K-ary tree,when (p < pc)5. For the subcritical percolation on K-ary tree,when (p < pc), for any positive integer l,

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CLC: > Mathematical sciences and chemical > Mechanics > Fluid Mechanics > Viscous fluid > Seepage
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