Jammed systems of oriented needles always percolate on square lattices

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.