Percolation of even sites for enhanced random sequential adsorption

TL;DR

This paper studies a percolation model on a chequerboard lattice with random sequential adsorption, revealing a critical surface of parameters where phase percolation transitions occur for black and white sites.

Contribution

It introduces a new percolation model with diagonal connections and identifies a critical surface of parameters dictating phase percolation behavior.

Findings

Existence of a critical surface of parameters $(\\lambda, p)$

Phase percolation depends on the position relative to the critical surface

At the critical surface, neither phase percolates

Abstract

Consider random sequential adsorption on a chequerboard lattice with arrivals at rate $1$ on light squares and at rate $\lambda$ on dark squares. Ultimately, each square is either occupied, or blocked by an occupied neighbour. Colour the occupied dark squares and blocked light sites {\em black}, and the remaining squares {\em white}. Independently at each meeting-point of four squares, allow diagonal connections between black squares with probability $p$; otherwise allow diagonal connections between white squares. We show that there is a critical surface of pairs $(\lambda, p)$, containing the pair $(1,0.5)$, such that for $(\lambda, p)$ lying above (respectively, below) the critical surface the black (resp. white) phase percolates, and on the critical surface neither phase percolates.