Euler Number and Percolation Threshold on a Square Lattice with Diagonal Connection Probability and Revisiting the Island-Mainland Transition
| Autor: | Sujata Tarafdar, Tajkera Khatun, Tapati Dutta, Sanchayan Dutta, Sugata Sen |
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| Rok vydání: | 2019 |
| Předmět: |
Materials Science (miscellaneous)
Diagonal Biophysics General Physics and Astronomy FOS: Physical sciences Disjoint sets scaling theory 01 natural sciences percolation theory Combinatorics Critical phase Percolation theory 0103 physical sciences Cluster (physics) Euler-number Physical and Theoretical Chemistry 010306 general physics Scaling Condensed Matter - Statistical Mechanics Mathematical Physics Mathematics second-nearest-neighbor Statistical Mechanics (cond-mat.stat-mech) square-lattice Percolation threshold Square lattice lcsh:QC1-999 lcsh:Physics |
| Zdroj: | Frontiers in Physics, Vol 7 (2019) |
| DOI: | 10.48550/arxiv.1904.05748 |
| Popis: | 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. Edge-sharing 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 $��(p) \ [=N_B(p)-N_W(p)]$ 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. |
| Databáze: | OpenAIRE |
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