The Nature Of The Two-dimensional Double Oriented Percolation Critical Probability Function Qc (p)

This paper is based on the Grimmett's conjecture about the random oriented percolation; In this paper.I summarized the basic knowledge about the random oriented percolation. When the dimension d=2 ,1 construct the equipollence model of the two dimension random oriented percolation:two dimensional double oriented percolation.This paper gives the proof that the critical percolation function qc(p) of two dimensional double oriented percolation is monotone.symmetrical and continuous about p.Let ,where . E3+(Ed₎ is the set of the oriented edges.Without thinking the edge's direction.E+(Ed₎ - Ed,&nd the edge in Ec$_Ed is followed the coordinate axis poaitive(negative) direction.Ld = Ld+. U Ld - (Zll,Ei U Ed),the vetex set of J? is Zd,hr all two adjacent vertices in Zd(that is the Euclid distance between them equals 1).there are two oriented edges between them.one is in .El,the other is in Ed.Now we think of the oriented percolation with the parameter p on the subgraph L+ of Ld,the percolation probability of it is defined as P^; accordingly,the percolation probability of the oriented percolation with the parameter q on the subgraph Ld of Ld is defined as P-q;then the percolation probability of double oriented percolation is defined as Pptq ?P^ x P^.The mark e+ (respectively.e"")denotes an edge in E^E'L), the mark "0" denotes ''closed'',the mark "!'' denotes "opened".Now all kinds probability measures are described as following: where {X(e+) : e+ Ed+] , {Y(e⁾ : e- E-d} are two i.i.d random variables sequences.Denoting the percolation probability function asThe Critical Probability Function of two dimensional double oriented percolation is denned as following:(Definition) For p G [0, l],the Critical Probability Function of two dimensional double oriented percolation is given byWe verify the following theorems.(Theorem) For qc(p),-we have : (1)(2)(Monotony) qc(p) is a strictly decrease function on (Symmetry) qc(p) is a symmetry function on |0. pc] about p = q. (Continuum) qc(p)is a continuous function on [0,pc].As the application of above theorems , we prove : (Theorem) qc(1/2) = 1/2...</sub>