Physicists often use various lattices to apply their favorite models in them. For instance, the most favorite lattice is perhaps the square lattice. There are 14 Bravais space lattice where every cell has exactly the same number of nearest, next nearest, nearest of next nearest etc. neighbors and hence they are called regular lattice. Often physicists and mathematicians study phenomena which require disordered lattice where each cell do not have exactly the same number of neighbors rather the number of neighbors can vary wildly. For instance, if one wants to study the spread of disease, viruses, rumors etc. then the last thing one would look for is the square lattice. In such cases a disordered lattice is necessary. One way of constructing a disordered lattice is by doing the following.
Starting with a square, say of unit area, and dividing randomly at each step only one block, after picking it preferentially with respect to ares, into four smaller blocks creates weighted planar stochastic lattice (WPSL). Essentially it is a disordered planar lattice as its block size and their coordination number are random.
Description
In applied mathematics, a weighted planar stochastic lattice (WPSL) is a structure that has properties in common with those of lattices and those of graphs. In general, space-filling planar cellular structures can be useful in a wide variety of seemingly disparate physical and biological systems. Examples include grain in polycrystalline structures, cell texture and tissues in biology, acicular texture in martensite growth, tessellated pavement on ocean shores, soap froths and agricultural land division according to ownership etc. The question of how these structures appear and the understanding of their topological and geometrical properties have always been an interesting proposition among scientists in general and physicists in particular. Several models prescribe how to generate cellular structures. Often these structures can mimic directly the structures found in nature and they are able to capture the essential properties that we find in natural structures. In general, cellular structures appear through random tessellation, tiling, or subdivision of a plane into contiguous and non-overlapping cells. For instance, Voronoi diagram and Apollonian packing are formed by partitioning or tiling of a plane into contiguous and non-overlapping convex polygons and disks respectively.
Regular planar lattices like square lattices, triangular lattices, honeycomb lattices, etc., are the simplest example of the cellular structure in which every cell has exactly the same size and the same coordination number. The planar Voronoi diagram, on the other hand, has neither a fixed cell size nor a fixed coordination number. Its coordination number distribution is rather Poissonian in nature. That is, the distribution is peaked about the mean where it is almost impossible to find cells which have significantly higher or fewer coordination number than the mean. Recently, Hassan et al proposed a lattice, namely the weighted planar stochastic lattice. For instance, unlike a network or a graph, it has properties of lattices as its sites are spatially embedded. On the other hand, unlike lattices, its dual (obtained by considering the center of each block of the lattice as a node and the common border between blocks as links) display the property of networks as its degree distribution follows a power law. Besides, unlike regular lattices, the sizes of its cells are not equal; rather, the distribution of the area size of its blocks obeys dynamic scaling, whose coordination number distribution follows a power-law.