### Bond percolation between k separated points on a square lattice

S. S. Manna,
Published: 26 June 2020

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 $L×L$ systems that for , and ${\overline{p}}_{c4}=0.555\phantom{\rule{0.16em}{0ex}}27\left(5\right)$. 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 ${\nu }_{k}=\nu /\left(k\beta +1\right)$, with $\beta =5/36$ and $\nu =4/3$ being the ordinary percolation critical exponents, so that , 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 $1.022$.
Keywords: lattice / percolation / k points / size / square / function / thresholds

#### Scifeed alert for new publications

Never miss any articles matching your research from any publisher
• Get alerts for new papers matching your research
• Find out the new papers from selected authors
• Updated daily for 49'000+ journals and 6000+ publishers
• Define your Scifeed now