Internal configurations of span-constrained random walks

The spans of a random walk on a simple cubic lattice are the sides of the smallest rectangular box with sides parallel to the coordinate axes that entirely contain the random walk. We consider the position, at dimension-less time , of a random walker constrained by a set of spansS. We show in one dimension that ifS ² 4, the random walker tends to be located at the extremities of the span, while in the contrary case the random walker is most likely to be found halfway between the extremities. This is true whether the single-step transition probabilities have a finite or an infinite variance, as is shown by example. In higher dimensions the position of the random walker in the direction of the largest span tends to lie at the span extremities, while the position in the direction of the smallest span tends to be in the middle.