基于渗流网络的极限定理
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摘要
本学位论文主要研究了Z~d上Bernoulli渗流开簇或网络的动态行为以及局部相依渗流,得到了中心极限定理,大数定律和大偏差定理等极限定理.全文的主要内容分为四章。
1.第一章中我们给出了有关渗流理论的基本知识,以及文中主要用到的几个不等式,这一章的大部分内容取自Grimmett(1989)(1999).
2.第二章研究了Z~d上Bernoulli边渗流开簇的随机着色模型:按照Z~d上的边渗流机制随机的选择一个子图,然后给每个开簇上的点随机的染色,要保证这种不同的开簇上的染色行为是互不相关的,而且同一开簇上的点被染的颜色是相同的。这个模型是H(a|¨)ggstr(o|¨)m(2001)研究的Dac(divide and color)模型的推广。我们注意到Garet(2001)中对于Dac模型研究了诸如大数定律和中心极限定理等极限理论,我们这里采用比Garet(2001)的方法更简单的方法和技巧,直接利用Penrose(2003)中关于正态估计的定理,分别就上临界和下临界情形、淬火分布和退火分布情形,证明了相应的中心极限定理和大数定律。
3.第三章中我们研究了在Z~d上Bernoulli点渗流网络上的马尔科夫链.不同于第二章,我们不能直接在渗流开簇上定义马尔科夫链,而是在无序的渗流图上定义马尔科夫过程。我们研究了渗流网络上的马尔科夫链大偏差理论,并给出了大偏差定理的速率函数的显式表达式。此外我们还利用Doburushin定理证明了中心极限定理。
4.第四章我们主要研究二维平面格点Z~2上的局部相依渗流,分别对格点盒子序列中最大开簇和原点0处的开簇,证明了相应的中心极限定理.此外我们还对Z~d上的局部相依渗流证明了无穷有向开簇的唯一性定理.
In this thesis,we mainly study the dynamic behaviors of random processes on Bernoulli bond or site percolation clusters or networks.As a generalization of the classic percolation process,locally dependent percolation process are studied.Limit theorems,such as central limit theorem,law of large number and large deviation for these random processes are presented.
1.In Chapter 1,we presented the basic theory for the classic percolation process on Z~d.Moreover,some inequalities are given.
2.In Chapter 2,we study the random coloring model on bond percolation clusters.The model is a generalization of Dac(divide and color)model given by H(a|¨)ggstr(o|¨)m(2001).The Dac model is easily described:choose a graph at random according to bond percolation,and then paint randomly and independently the different clusters,each cluster being monochromatic.We generalize the Dac model in the sense that the independent condition is weakened to uncorrelate.Apparently, this is not an essential generalization.However,the methods to the proofs of limit theorems in Garet(2001) cut no ice without the assumption that the colors painted on different clusters have independent and identical distributions.Therefore,we apply a new method to prove the law of large numbers and central limit theorems for our random coloring model under the subcritical and supercritical cases,quenched law and annealed law.
3.In Chapter 3,we study the Markov chain on supercritical site percolation process.Explicit expression of speed function for large deviation is obtained.And we apply Doburushin's theorem to prove a central limit theorem fbr the Markov chain model on supercritical site percolation process.
4.In Chapter 4,we mainly study the locally dependent percolation process on Z~2.For this model,we define the notion of the cluster.We present the central limit theorems for the size of biggest cluster and the size of the cluster at the origin in the lattice boxes sequences.Moreover,techniques of Burton-Keane,developed earlier for independent percolation on Z~d,is adapted to the setting of locally dependent percolation on Z~d for d≥2.The uniqueness of theorem of infinite directed cluster at the origin is proved.
1.第一章中我们给出了有关渗流理论的基本知识,以及文中主要用到的几个不等式,这一章的大部分内容取自Grimmett(1989)(1999).
2.第二章研究了Z~d上Bernoulli边渗流开簇的随机着色模型:按照Z~d上的边渗流机制随机的选择一个子图,然后给每个开簇上的点随机的染色,要保证这种不同的开簇上的染色行为是互不相关的,而且同一开簇上的点被染的颜色是相同的。这个模型是H(a|¨)ggstr(o|¨)m(2001)研究的Dac(divide and color)模型的推广。我们注意到Garet(2001)中对于Dac模型研究了诸如大数定律和中心极限定理等极限理论,我们这里采用比Garet(2001)的方法更简单的方法和技巧,直接利用Penrose(2003)中关于正态估计的定理,分别就上临界和下临界情形、淬火分布和退火分布情形,证明了相应的中心极限定理和大数定律。
3.第三章中我们研究了在Z~d上Bernoulli点渗流网络上的马尔科夫链.不同于第二章,我们不能直接在渗流开簇上定义马尔科夫链,而是在无序的渗流图上定义马尔科夫过程。我们研究了渗流网络上的马尔科夫链大偏差理论,并给出了大偏差定理的速率函数的显式表达式。此外我们还利用Doburushin定理证明了中心极限定理。
4.第四章我们主要研究二维平面格点Z~2上的局部相依渗流,分别对格点盒子序列中最大开簇和原点0处的开簇,证明了相应的中心极限定理.此外我们还对Z~d上的局部相依渗流证明了无穷有向开簇的唯一性定理.
In this thesis,we mainly study the dynamic behaviors of random processes on Bernoulli bond or site percolation clusters or networks.As a generalization of the classic percolation process,locally dependent percolation process are studied.Limit theorems,such as central limit theorem,law of large number and large deviation for these random processes are presented.
1.In Chapter 1,we presented the basic theory for the classic percolation process on Z~d.Moreover,some inequalities are given.
2.In Chapter 2,we study the random coloring model on bond percolation clusters.The model is a generalization of Dac(divide and color)model given by H(a|¨)ggstr(o|¨)m(2001).The Dac model is easily described:choose a graph at random according to bond percolation,and then paint randomly and independently the different clusters,each cluster being monochromatic.We generalize the Dac model in the sense that the independent condition is weakened to uncorrelate.Apparently, this is not an essential generalization.However,the methods to the proofs of limit theorems in Garet(2001) cut no ice without the assumption that the colors painted on different clusters have independent and identical distributions.Therefore,we apply a new method to prove the law of large numbers and central limit theorems for our random coloring model under the subcritical and supercritical cases,quenched law and annealed law.
3.In Chapter 3,we study the Markov chain on supercritical site percolation process.Explicit expression of speed function for large deviation is obtained.And we apply Doburushin's theorem to prove a central limit theorem fbr the Markov chain model on supercritical site percolation process.
4.In Chapter 4,we mainly study the locally dependent percolation process on Z~2.For this model,we define the notion of the cluster.We present the central limit theorems for the size of biggest cluster and the size of the cluster at the origin in the lattice boxes sequences.Moreover,techniques of Burton-Keane,developed earlier for independent percolation on Z~d,is adapted to the setting of locally dependent percolation on Z~d for d≥2.The uniqueness of theorem of infinite directed cluster at the origin is proved.
引文
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[3]Balogh,J.and Bollolás,B.(2003) Sharp thresholds in bootstrap percolation.Phys.A 326 305-312.
[4]Balogh J.and Bollolás,B.(2005) Bootstrap percolation on the hypercube.Probab.Theory Relat.Fields 134 624-648.
[5]Balogh,J.,Peres,Y.and Pete,G.(2006) Bootstrap percolation on infinite trees and non-amenable groups.Combinatorics,Probability and Computing 15 715-730.
[6]Barbato D.(2005) FKG inequality for Brownian motion and stochastic differential equations.Electron.Commun.Probab.10 7-16.
[7]Barlow,M.T.(2004) Random walks on supercritical percolation clusters.Ann.Probab.32 3024-3084.
[8]Belhadji,L.and Lanchier,N.(2006) Individual versus cluster recoveries within a spatially structured population,Ann.Appl.Probab.16 403-422.
[9]Benjamini,I.and Schramm,O.(1996) Percolation beyond Z~d,many questions and a few answers.Elect.Comm.in Probab.1 71-82.
[10]Berger,N.,Gantert,N.and Peres,Y.(2003) The speed of biased random walk on percolation clusters.Probab.Theory Related Fields 126 221-242.
[11]Berg,J.van den,Grimmett,G.R.and Schinazi,R.(1998) Dependent random graphs and spatial epidemics.Ann.Appl.Probab.8 317-336.
[12]Berger,N.(2002) Transience,recurrence and critical behavior for long-range percolation.Commun.Math.Phys.226 531-558.
[13]Berger,N.and Biskup,M.(2007) Quenched invariance principle for simple random walk on percolation clusters.Probab.Theory Related Fields 137 83-120.
[14]Blanchard,P.and Gandolfo,D.(2001) Percolation:Concepts,Tools and Application to Real World Phenomena.The Sciences of complexity.
[15]Bollobás,B.and Riordan,O.(2006) Percolation.Cambridge University Press.
[16]Braga,G.A.,Sanchis,R.and Schieber,T.A.(2005) Critical percolation on a Bethe lattice revisited.SIAM Review 47 349-365.
[17]Broadbent,S.R.and Hammersley,J.M.(1957) Percolation processes,Ⅰ and Ⅱ.Proc.Cambridge Philos.Soc.53 629-645.
[18]Burton R.M.and Eeane M.(1989) Density and uniqueness in percolation.Commun.Math.Phys.121 501-505.
[19]Chen,D.(2001) On the infinite cluster of the Bernoulli bond percolation in the Scherk's graph.J.Appl.Probab.38 828-840.
[20]Chem D.and Peres,Y.(2003) The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters:an Overview.Discrete Mathematice and Theoretical Computer ScienceAC 39-44.
[21]Chem D.,Peres,Y.and Pete,G.(2004) Anchored expansion,percolation and speed.Ann.Probab.32 2978-2995.
[22]Chen,D.and Zhang,F.(2007) On the monotonicity of the speed of random walks on a percolation cluster of trees.Acta.Mathematica Sinica.23 1949-1954.
[23]Chen,R.(1997) Critical points of three-dimensional bootstrap percolation-like cellular automata.Chin.J.Appl.Probab.Stat.13 399-406.
[24]Cox,J.T.and Durrett,R.(1988) Limit theorems for the spread of epidemics and forest fires.Stochastic Processes Appl.30 171-191.
[25]De Lima,B.N.B.and Sidoravicius,V.(2008) On the truncated long-range percolation on Z~2.J.Appl.Probab.45 287-291.
[26]Dietz,Z.and Sethuraman,S.(2005) Large deviations for a class of nonhomogeneous Markov chains.Ann.Appl.Probab.15 421-486.
[27]Durrett,R.(1984) Oriented percolation in two dimensions.Ann.Probab.12999-1040.
[28]Durrett,R.(1991) Probability:Theory and Examples.2nd edition,Duxbury Press.
[29]Durrett,R.and Schonmann,R.(1988) The contact process on a finite set Ⅱ.Ann.Probab.16 1570-1583.
[30]Durrett,R.,Schonmann,R.and Tanaka,N.I.(1989) Correlation lengths for oriented percolation.J.Stat.Phys.55 965-979.
[31]Durrett,R.and Tanaka,N.I.(1989) Scaling inequalities for oriented percolation,J.Stat.Phys.981-995.
[32]Essam,J.W.(1980) Percolation theory.Rep.Prog.Phys.43 S33-912.
[33]Felinto,D.and Brady Moreira,F.G.(2001) Correlated invasion percolation.Phys.A 293 307-314.
[34]Friedli,S.,De Lima,B.N.B.and Sidoravicius,V.(2004) On long range percolation with heavy tails.Elect.Comm.Prob.9 175-177.
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