Bond percolation between k separated points on a square lattice

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 ${\overline{p}}_{ck}$ 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 $L\to \infty$. We study ${\overline{p}}_{ck}$ for bond percolation on the square lattice and find that ${\overline{p}}_{ck}$ are above the normal percolation threshold $p_c=1/2$ and represent specific supercritical states. The ${\overline{p}}_{ck}$ can be related to integrals over powers of the function $P_{\infty }\left(p\right)$ equal to the probability a point is connected to the infinite cluster; we find numerically from both direct simulations and from measurements of $P_{\infty }\left(p\right)$ on $L\times L$ systems that for $L\to \infty , {\overline{p}}_{c1}=0.51755\left(5\right), {\overline{p}}_{c2}=0.53219\left(5\right), {\overline{p}}_{c3}=0.54456\left(5\right)$, and ${\overline{p}}_{c4}=0.55527\left(5\right)$. The percolation thresholds ${\overline{p}}_{ck}$ remain the same, even when the $k$ points are randomly selected within the lattice. We show that the finite-size corrections scale as $L^{-1/{\nu }_k}$ where ${\nu }_k=\nu /(k\beta +1)$, with $\beta =5/36$ and $\nu =4/3$ being the ordinary percolation critical exponents, so that ${\nu }_1=48/41, {\nu }_2=24/23, {\nu }_3=16/17, {\nu }_4=6/7$, etc. We also study three-point correlations in the system and show how for $p>p_c$, the correlation ratio goes to 1 (no net correlation) as $L\to \infty$, while at $p_c$ it reaches the known value of $1.022$.