Two Results On The First Passage Percolation

First-passage percolation is one branch of percolation process. It is a time-dependent model.It was introduced by Hammersley and Welsh in 1965.It mainly studies the property of vertex set to which from one certain point during some given time,under conditions that the network of the path has been given.but the travel time of the path is random.We write Z = {0,+1,+2,...} for the set of all integers.andfor the set of all vertices v = (x1,X2,xd) with integral coordinates. For any vertices x = (x1,x2,...,xd),y=(y1,y2,...,yd) Zd,the distance from x to y isdefined by say that x and y are adjacent,inthis case,we write x - y and we represent the edge e from x to y as e= (x,y). Write Ed for the set of these edges.then we may get d-dimension cubic lattice. We denote this lattice by Ld = (Zd,Ed)A path of Ld is an alternating sequenceof distinct vertices in which each edge e connects the two vertices vk-1 and vk. A path is said to be self-avoiding if all the vertices and thus all the edges are distinct. A circuit is a path for which v0=vn. we denote this path by r.and we denote by r the number of edges in r and call it route length.To each edge of Ed,we assume the values 0 and 1 only,independent of all other edges,with probabilities p and 1 - p,respectively.Generally speaking,edges with value 0 are called open,those with value 1 are closed.Thus,we may get Bernoulli bond percolation model with parameter p in Ld.In Bernoulli bond percolation model with parameter P in I,2,we call a path open if all of its edges are open and closed if all of its edges are closed,we shall write u-v if there is an open path joining vertex u to vertex v. we write C(x) for the open cluster containing vertex x,defined by C(x) = {y :x -y. y Zd}. we represent the open cluster C(O) at the origin by the singlr letterC.we denote by C(x) the number of vertices in C(x). If C(x) = oc. we call it infinite open cluster.The percolation probability (p) is the probability that the origin belongs to an infinite open cluster,we defineIt is fundamental to percolation theory that there exists a critical value 0 < pc = pc(d) < 1,if d>2 such thatpc(d) is called critical probability and is defined formally byIt has been proved that if d = 2,pc(2) = 1/2 [10].For Ld - (Zd,Ed):assign to each edge of Ed an independent.non-negative random variable u(e)( which we think of as being the time required for fluid to flow along e) with distribution function F. This is our first-passage percolation model.We will mainly describe first passage percolation model in plane square lattice L2 = (Z2,E2) below.Let random variable w(e) with distribution function F(x) = P(w(e) < x). For w(e) is non-negative,we knowFor any path we define the passage time t(r) byFor Z2,the passage time between vertices u and v is given byhere r is a path from u to v.Suppose that m < n,we definehere r is a path from point (m,0) to point (n. 0).here r is a path from point (m,0) to line x = n.It has been proved in [1] that when Eu(e) < (this condition may be weakened.see [l]),there exists constant (only dependent of distribution F),such thathere u is called time constant .moreover we havehere 1/2 is percolation critical probability of Bernoulli bond percolation model in planr square lattice,for general d dimension first-passage percolation,change 1/2 into relevant critical probability.we can also get relevant results,see[lj.A path r from (m,0) to (n,0) . if it satisfies t(r) = amn,then we call r is route of amn.Similiarly.we can define route of bmn. For L2,existence of routes of aon,bon has been proved[2]. in higher dimensional lattices,only at critical state,existence of routes has not been completely solved yet[5]. Generally speaking .routes of a0n or b0n need not be unique.Writehere,r is route of mn, = a b,Nn is the length of the shortest route of 9mn.In first passage percolation,an important guess is that for 9 - a. b.when F(0) 7 1/2,the limit lim exists.(when F(0) = 1/2,people conject the above limit is ). in the seventi...