Triangle Point Grid And Hexagonal Point Lattice Side Seepage Estimates

This paper studies the band percolation of the models of the triangle lattice and the hexagonal lattice. Before we study the models of the triangle lattice and the hexagonal lattice,We will introduce period.Let G = (Vc,Ec) be a connected graph,we write VG for the set of vertices of G and EGfor the set of its edges,u e Vc is a vertex,and v is the coordinate vector. We say v is the period for graph G if u + vVG.The model of the triangle lattice:Divide Z2into equilateral triangles by means of the horizontal linesand lines under an angle of 60 or 120 with the firstcoordinate-axis through the points (k,0) k Z,the vertices of the graph are the vertices of the equilateral triangles and two such vertices are adjacent iff they are vertices of one and the same triangle. Even though the vector (1,0) is a period,the vector (0,1) is not. However,if wechange the vertical scale by a factor,we will obtain a periodic graphin Z2. Its vertices are located at all points of the form (i1,i2) oreach vertex has six neighbors. The six neighborsof the vertex v areThe set of vertices The set of edges EThe model of the hexagonal latticeThe usual way to imbed the hexagonal lattice is such that its face are regular hexagons .the vertices are the points.The origin is in the center of one of the hexagons and the periods are(,0) and (0,3). If we change the vertical scale by a factor,we willobtain a periodic graph in Z2.This paper studies the band percolation of the models of the triangle lattice and the hexagonal lattice with foundational means. This paper had proved that P (LR(n))=1/2 .LR(n) is that there is an open path in rectangle [0,n] x [0,n] joining some vertex on its left side to some vertex on its right side when p = pc in the models of the triangle lattice and the hexagonal lattice. This paper had also proved that P(LR(n))-1 n-. LR(n) is that there is an open path in rectangle [0,n]x[0,n] joining some vertex on its left side to some vertex on its right side when p > pc in the models of the triangle lattice and the hexagonal lattice,oTheorem In the models of the hexagonal lattice,let LR(n) be is an open path in rectangle [0,n]x[0,n] joining some vertex on its left side to some vertex on its right side. If (p) >0,thenTheorem In the models of the hexagonal lattice,let LR(n) be is an open path in rectangle [0,n+l]x[0,n] joining some vertex on its left side to some vertex on its right side. IfThis paper also studies the band percolation of the models of the triangle lattice and the hexagonal lattice when p = pc and p > pc .such as the estimation of power of ,x and xf and their upper and lower bound.Theorem In the models of the hexagonal lattice,Ep is the expect withp . when p = pc=1-2sin,then there exist positive finiteconstants Ai,ai i=1 2 3,such that the following statements are valid :Theorem In the models of the hexagonal lattice,Ep is the expect with p . let x(p) = Epc xf(p) = Ep(c;c<),then there exist positive finite constants Ai,ai;(i=4,5,6,7),such that the following statements are valid :HereIn the end this paper gives the general property and proposes about the band percolation of the models of the triangle lattice and the hexagonal lattice.This paper consists of four parts.Chapter one Percolation and The Main Inequality in Percolation,Chapter two Main Results of This Paper,Chapter three Proof of The Main ResultsChapter four The general property and proposition about the bandpercolation of the models of the triangle lattice,the rectangle lattice and the hexagonal lattice.