Internal Erosion and the Exponent 3/4

Let m and n be positive integers. The internal erosion of the interval I = [−m,n] ⊂ Z is the random interval formed by starting a simple random walk at 0 and stopping when the walk hits one of the two endpoints {−m,n}, then removing that endpoint from I to obtain one of the intervals [1−m,n] or [−m,n − 1]. Suppose we iterate this erosion procedure until the origin itself is eroded; how large is the interval that remains? We can view the erosion process as a Markov chain whose state space is the set of intervals [−m,n] with m,n ≥ 0. The states with m = 0 or n = 0 are absorbing, and the transition probabilities for the remaining states are given by the classical gambler’s ruin problem: