Linear bounds on the North-East model and higher-dimensional analogs

Abstract

The North-East model is a combinatorial model arising from statistical physics in which counters are placed at or removed from lattice points in a quadrant, according to certain rules, while bounding the total number of occupied sites. We show that any site may be reached with a number of counters linear in the distance of the site from the origin. We also show that in contrast with the one-dimensional East model, a linear number of counters is necessary. We extend the North-East model to n dimensions with corresponding linear upper and lower bounds. In two dimensions, a polynomial number of steps are sufficient to achieve the linear upper bound.