General Percolation Model, The Critical Point Of Uniqueness

Percolation model was first proposed by Broadbent S.R. and Hammersly J.M. in 1957.It has been to study in depth in the past 60 years .This statistical physics model of the probability expanded field of study ,and also provided a rigorous mathematical model based on this .In the first chapter,Let G be a infinite graph, we consider bond percolation on G. Let p_c(G) be the critical probability of the model, i.e., p_c(G) := sup{p : P_p(|C|=∞) = 0}, where C be the open cluster containing the origin. We prove in thispaper that, for any p < p_c(G), there exists someφ(p) > 0 such that (?)P_p(v(?)(?)B(v,n))≤exp{-φ(p)n} for all n≥0. As a corollary, we get the coincidence ofcritical points of the model: p_c(G) = p_T(G), where p_T(G) := sup{p : E_P(|C|) <∞}.In the second chapter,we introduced the related formula and inequality . Then we used the Russo formula, BK inequality and so on to prove the the results .