Euler Number and Percolation Threshold on a Square Lattice With Diagonal Connection Probability and Revisiting the Island-Mainland Transition

Abstract

We report some novel properties of a square lattice filled with white sites, randomly occupied by black sites (with probability p). We consider connections up to second nearest neighbours, according to the following rule. Edgesharing sites, i.e. nearest neighbours of similar type are always considered to belong to the same cluster. A pair of black corner-sharing sites, i.e. second nearest neighbours 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) [= N B (p) − N W (p)] versus 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 N B is the number of black clusters and N W is the number of white clusters, at a certain probability p. We also discuss the earlier proposed 'Island-Mainland' transition (Khatun, T., Dutta, T. & Tarafdar, S. Eur. Phys. J. B (2017) 90: 213) 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. arXiv:1904.05748v1 [cond-mat.stat-mech]

A Multi-Scale Network with Percolation Model to Describe the Spreading of Forest Fires

“…For the case where only two classes of vegetation are accounted, i.e., occupied/empty cells of the cellular automaton, the results obtained for the percolation threshold in a square lattice with Moore neighbourhood, p c ≈ 0.407, are consistent with the literature [29][30][31]. The case for a neighbourhood of warm trees for each burning cell with different action radius is considered, changing the threshold value to p c ≈ 0.725, and p c ≈ 0.375, for first and second neighbours, respectively.…”

Aggregation patterns of copper sulphate salt via droplet drying: mediation by surface hydrophobicity and salt concentration

“…A 'cluster' definition may extend upto a 'r-neighbourhood' about the centre of a grid [52]. In a square or triangular lattice, their can be a certain ambiguity regarding connectivity of clusters through vertices, which is usually resolved by assigning a certain probability to the vertex neighbours of being connected [36,53]. Once all the clusters are counted and numbered, the Euler characteristic is calculated following Eq.…”

Evaluation of Effective Heat Transport and Temperature Gradients in Thermally Anisotropic Particulate Arrangements

“…For example, the percolation threshold of a honeycomb lattice is 0.69, whereas, for a random closed packing, it is 0.27 . Percolation thresholds for different lattice types, with and without the influence of the local environment, have been calculated. ,, The percolation threshold is further affected by the intrinsic properties of the single constituents. , Several studies investigate the percolation behavior using components with elliptical particles, i.e., anisotropic shapes. ,, In addition, rod-like systems are the focus of current research. − …”

“…33 Percolation thresholds for different lattice types, with and without the influence of the local environment, have been calculated. 37,40,41 The percolation threshold is further affected by the intrinsic properties of the single constituents. 42,43 Several studies investigate the percolation behavior using components with elliptical particles, 44 i.e., anisotropic shapes.…”