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 ζ n : = (log n) 2 L n ∗ converges to π. The author found numerical evidence that the average value of ζ n ( ζ n ¯) increases until a certain value of n , denoted as n c , after which it decreases and approaches π. Furthermore, estimates are presented for n c and ζ n c ¯. [ABSTRACT FROM AUTHOR]