Percolation and jamming of random sequential adsorption samples of large linear $k$-mers on a square lattice

TL;DR

This study investigates percolation thresholds and jamming coverage in large linear k-mers on a square lattice using efficient simulations, revealing new behaviors and generalizing existing theorems for periodic boundary conditions.

Contribution

Developed a fast, memory-efficient parallel algorithm for large k-mers, identified new percolation threshold regimes, and extended a theorem to periodic boundary conditions.

Findings

Percolation thresholds up to k=2^17 were obtained.

A new large-k regime of percolation threshold behavior was identified.

At jamming, all clusters are percolating, and percolation occurs before jamming.

Abstract

The behavior of the percolation threshold and the jamming coverage for isotropic random sequential adsorption samples has been studied by means of numerical simulations. A parallel algorithm that is very efficient in terms of its speed and memory usage has been developed and applied to the model involving large linear $k$-mers on a square lattice with periodic boundary conditions. We have obtained the percolation thresholds and jamming concentrations for lengths of $k$-mers up to $2^{17}$. New large $k$ regime of the percolation threshold behavior has been identified. The structure of the percolating and jamming states has been investigated. The theorem of G.~Kondrat, Z.~Koza, and P.~Brzeski [Phys. Rev. E 96, 022154 (2017)] has been generalized to the case of periodic boundary conditions. We have proved that any cluster at jamming is percolating cluster and that percolation occurs before jamming.