# Jammed systems of oriented needles always percolate on square lattices

## TL;DR

This paper proves that in jammed configurations of fixed-length, nonoverlapping needles on a square lattice, all clusters percolate, contradicting previous conjectures that long needles do not percolate.

## Contribution

The paper provides a rigorous proof that all jammed configurations of oriented needles on a square lattice percolate, refuting prior conjectures about non-percolation for long needles.

## Key findings

- All clusters in jammed needle systems percolate.
- Percolation occurs regardless of needle length in jammed states.
- Contradicts previous conjectures about non-percolation of long needles.

## Abstract

Random sequential adsorption (RSA) is a standard method of modeling adsorption of large molecules at the liquid-solid interface. Several studies have recently conjectured that in the RSA of rectangular needles, or $k$-mers, on a square lattice the percolation is impossible if the needles are sufficiently long ($k$ of order of several thousand). We refute these claims and present a strict proof that in any jammed configuration of nonoverlapping, fixed-length, horizontal or vertical needles on a square lattice, all clusters are percolating clusters.