Limit Theorems Based On Percolation Networks

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.