# Jamming and percolation of $k^3$-mers on simple cubic lattices

## TL;DR

This study investigates how cubic objects of various sizes jam and percolate on simple cubic lattices, revealing size-dependent jamming coverage, a critical correlation length exponent, and the disappearance of percolation at large sizes, with results consistent with 3D percolation universality.

## Contribution

The paper provides a comprehensive numerical analysis of jamming and percolation of $k^3$-mers in 3D, including critical exponents and the size limit for percolation transition.

## Key findings

- Jamming coverage decreases with increasing $k$, approaching 0.4204 for large $k$.
- Percolation threshold increases with $k$ up to 16, then percolation ceases for larger $k$.
- The critical exponents match those of 3D random percolation, indicating universality.

## Abstract

Jamming and percolation of three-dimensional (3D) $k \times k \times k $ cubic objects ($k^3$-mers) deposited on simple cubic lattices have been studied by numerical simulations complemented with finite-size scaling theory. The $k^3$-mers were irreversibly deposited into the lattice. Jamming coverage $\theta_{j,k}$ was determined for a wide range of $k$ ($2 \leq k \leq 40$). $\theta_{j,k}$ exhibits a decreasing behavior with increasing $k$, being $\theta_{j,k=\infty}=0.4204(9)$ the limit value for large $k^3$-mer sizes. In addition, a finite-size scaling analysis of the jamming transition was carried out, and the corresponding spatial correlation length critical exponent $\nu_j$ was measured, being $\nu_j \approx 3/2$. On the other hand, the obtained results for the percolation threshold $\theta_{p,k}$ showed that $\theta_{p,k}$ is an increasing function of $k$ in the range $2 \leq k \leq 16$. For $k \geq 17$, all jammed configurations are non-percolating states, and consequently, the percolation phase transition disappears. The interplay between the percolation and the jamming effects is responsible for the existence of a maximum value of $k$ (in this case, $k = 16$) from which the percolation phase transition no longer occurs. 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.