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

This paper conducts a numerical analysis of the behavior of the average value of the Maximum Local Time, L-n*, in the Simple Random Walk on the square lattice. It has been established in the literature that the sequence ?(n):=((logn)2)(Ln)* converges to p. The author found numerical evidence that the average value of ?(n) (?n over bar ) increases until a certain value of n, denoted as n(c), after which it decreases and approaches p. Furthermore, estimates are presented for nc and ?(n)c .

Automated summary

This paper presents a numerical analysis of the average value of the Maximum Local Time, L-n*, in the Simple Random Walk on the square lattice. It is known from previous studies that the sequence ?(n):=((logn)2)(Ln)* converges to p. The author found numerical evidence showing that the average value of ?(n) (?n over bar ) increases until a certain value of n, referred to as n(c), after which it decreases and approaches p. Furthermore, estimates for nc and ?(n)c are provided.