Frontiers in Physics (May 2019)
Euler Number and Percolation Threshold on a Square Lattice With Diagonal Connection Probability and Revisiting the Island-Mainland Transition
Abstract
In our study we report on some of the novel properties of a square lattice filled with white sites, randomly occupied by black sites (with probability p). We consider connections up to the second nearest neighbors, according to the following rule. Edge-sharing sites, i.e., nearest neighbors of similar type are always considered to belong to the same cluster. A pair of black corner-sharing sites, i.e., second nearest neighbors may form a “cross-connection” with a pair of white corner-sharing sites. In this case assigning connected status to both pairs simultaneously, makes the system quasi-three dimensional, with intertwined black and white clusters. The two-dimensional character of the system is preserved by considering the black diagonal pair to be connected with a probability q, in which case the crossing white pair of sites are deemed disjoint. If the black pair is disjoint, the white pair is considered connected. In this scenario we investigate (i) the variation of the Euler number χ(p) [= NB(p) − NW(p)] vs. p graph for varying q, (ii) variation of the site percolation threshold with q, and (iii) size distribution of the black clusters for varying p, when q = 0.5. Here NB is the number of black clusters and NW is the number of white clusters, at a certain probability p. We also discuss the earlier proposed “Island-Mainland” transition [1] and show mathematically that the proposed transition is not in fact a critical phase transition and does not survive finite size scaling. It is also explained mathematically why clusters of size 1 are always the most numerous.
Keywords
