# Jamming and percolation of $k^2$-mers on simple cubic lattices

## TL;DR

This study investigates how the size of square objects affects their jamming coverage and percolation thresholds on simple cubic lattices, revealing size-dependent behaviors and confirming universality class consistency.

## Contribution

It provides the first comprehensive analysis of jamming and percolation of $k^2$-mers on cubic lattices across a wide size range, including critical exponents and universality.

## Key findings

- Jamming coverage decreases with increasing $k$, approaching 0.4285 for large $k$.
- Percolation threshold decreases then increases with $k$, showing a minimum around $k=18$.
- Percolation transition belongs to the 3D random percolation universality class.

## Abstract

Jamming and percolation of square objects of size $k \times k$ ($k^2$-mers) isotropically deposited on simple cubic lattices have been studied by numerical simulations complemented with finite-size scaling theory. The $k^2$-mers were irreversibly deposited into the lattice. Jamming coverage $\theta_{j,k}$ was determined for a wide range of $k$ ($2 \leq k \leq 200$). $\theta_{j,k}$ exhibits a decreasing behavior with increasing $k$, being $\theta_{j,k\rightarrow\infty}=0.4285(6)$ the limit value for large $k^2$-mer sizes. On the other hand, the obtained results shows that percolation threshold, $\theta_{c,k}$, has a strong dependence on $k$. It is a decreasing function in the range $2 \leq k \leq 18$ with a minimum around $k=18$ and, for $k \geq 18$, it increases smoothly towards a saturation value. Finally, a complete analysis of critical exponents and universality has been done, showing that the percolation phase transition involved in the system has the same universality class as the 3D random percolation, regardless of the size $k$ considered.