Dual lattice

The dual lattice is obtained by drawing the bisectors of lines comrecting neighbouring lattice points. Examples of lattices in two dimensions and their duals are shown in figure A2.3.28. A square lattice is self-dual. 

Fig. 12.8 Schematic showing tiie equivalence between n-dimensional cells (outlined by dark lines) and the dual lattice formed by contracting them to nodes (solid circles) which retaining their connectivity (outlined by light lines).

Using this simplicial decomposition, we may now define a dual lattice constructively as follows [tdleeSSb]. Let i be an arbitrary site in , We say that a point p G D belongs to i if i is the nearest site to p in . The dual to site i is the volume iJi = p p belongs to f. We call Wj an n-dimensional cell.f The dual lattice of the random lattice consists of the cells into which s volume D is divided using the above construction. 

Figure 12.10 shows a small section of a two-dimensional lattice and its dual lij is the length of the link between sites i and j, while lij is the border length of their dual cells. More generally, in n-dimensions, bj is the volume of the (n - 1)-dimensional surface in the dual lattice. In terms of these variables, Lee defines the weights Xij to be equal to... 

Fig. 12.10 An example of a two dimensional random lattice (dark lines and solid circles) and its dual lattice (lighter lines and open circles).

In a number of papers [16-23,25], the discrete variant of the PCAO-model is considered the chain is modeled by a random walk on the lattice with spacing a and the topological constraints are placed on the dual lattice with period c. 

Dual lattice and dual transformation, (a) Square lattice is self-dual S = S. (b) The dual lattice of a honeycomb lattice is a triangular lattice H = T and vice versa T = H. 

Some of the critical values of percolation can be derived using a simple method based on the duality of the lattices. In general, the new lattice A constructed from A by connecting the centers of lattice cells is called the dual lattice of A. For instance, the dual lattice of the square lattice is a square lattice (S = S). It is self-dual. A honeycomb lattice and a triangular lattice are dual to each other (T = H, H =T) (Figure 8.12). 

Because bonds on the lattice A and bonds on its dual lattice A cross each other, percolation problems on them are related by... 

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