Jamming and percolation of parallel squares in single-cluster growth model

TL;DR

This paper investigates how the size and mixture of squares affect jamming and percolation thresholds in a cluster growth model, revealing that larger squares and mixtures can significantly alter percolation behavior.

Contribution

It introduces a single-cluster growth model for parallel squares and quantifies how size and mixture influence jamming and percolation thresholds.

Findings

Jamming concentration approaches 0.638 for large squares.

Percolation threshold increases with square size.

Mixtures of squares can suppress percolation of larger active squares.

Abstract

This work studies the jamming and percolation of parallel squares in a single-cluster growth model. The Leath-Alexandrowicz method was used to grow a cluster from an active seed site. The sites of a square lattice were occupied by addition of the equal size $k \times k$ squares (E-problem) or a mixture of $k \times k$ and $m \times m$ ($m \leqslant k$) squares (M-problem). The larger $k \times k$ squares were assumed to be active (conductive) and the smaller $m \times m$ squares were assumed to be blocked (non-conductive). For equal size $k \times k$ squares (E-problem) the value of $p_j = 0.638 \pm 0.001$ was obtained for the jamming concentration in the limit of $k\rightarrow\infty$. This value was noticeably larger than that previously reported for a random sequential adsorption model, $p_j = 0.564 \pm 0.002$. It was observed that the value of percolation threshold $p_{\mathrm{c}}$ (i.e., the ratio of the area of active $k \times k$ squares and the total area of $k \times k$ squares in the percolation point) increased with an increase of $k$. For mixture of $k \times k$ and $m \times m$ squares (M-problem), the value of $p_{\mathrm{c}}$ noticeably increased with an increase of $k$ at a fixed value of $m$ and approached 1 at $k\geqslant 10m$. This reflects that percolation of larger active squares in M-problem can be effectively suppressed in the presence of smaller blocked squares.