A note about the maximum local time of a random walk on the square lattice

Abstract

This paper conducts a numerical analysis of the behavior of the average value of the Maximum Local Time, [Formula: see text], in the Simple Random Walk on the square lattice. It has been established in the literature that the sequence [Formula: see text] converges to [Formula: see text]. The author found numerical evidence that the average value of [Formula: see text] ([Formula: see text]) increases until a certain value of [Formula: see text], denoted as [Formula: see text], after which it decreases and approaches [Formula: see text]. Furthermore, estimates are presented for [Formula: see text] and [Formula: see text].