Bond percolation between k separated points on a square lattice

Physical Review. E
S S Manna, Robert M Ziff

Abstract

We consider a percolation process in which k points separated by a distance proportional the system size L simultaneously connect together (k>1), or a single point at the center of a system connects to the boundary (k=1), through adjacent connected points of a single cluster. These processes yield new thresholds p[over ¯]_{ck} defined as the average value of p at which the desired connections first occur. These thresholds not sharp, as the distribution of values of p_{ck} for individual samples remains broad in the limit of L→∞. We study p[over ¯]_{ck} for bond percolation on the square lattice and find that p[over ¯]_{ck} are above the normal percolation threshold p_{c}=1/2 and represent specific supercritical states. The p[over ¯]_{ck} can be related to integrals over powers of the function P_{∞}(p) equal to the probability a point is connected to the infinite cluster; we find numerically from both direct simulations and from measurements of P_{∞}(p) on L×L systems that for L→∞, p[over ¯]_{c1}=0.51755(5), p[over ¯]_{c2}=0.53219(5), p[over ¯]_{c3}=0.54456(5), and p[over ¯]_{c4}=0.55527(5). The percolation thresholds p[over ¯]_{ck} remain the same, even when the k points are randomly selected within the lattice. We show that th...Continue Reading

References

Nov 1, 2000·Physical Review Letters·M E Newman, R M Ziff
Jul 20, 2001·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·M E Newman, R M Ziff
Sep 21, 2002·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·R M Ziff, M E J Newman
Jul 15, 2016·Physical Review. E·Sumanta Kundu, S S Manna
Mar 20, 2018·Physical Review. E·Wei HuangYoujin Deng
May 15, 2018·Physical Review Letters·Cesar I N Sampaio FilhoAndré A Moreira
May 16, 2018·Physical Review. E·L S RamirezA J Ramirez-Pastor
Jun 17, 2018·Physical Review. E·M M H Sabbir, M K Hassan
Sep 27, 2018·Physical Review. E·Stephan Mertens, Cristopher Moore
Feb 20, 2019·Physical Review. E·Sayantan MitraAnkur Sensharma
Oct 3, 2019·Physical Review. E·Ivan KryvenGinestra Bianconi

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