Winning quick and dirty: the greedy random walk

TL;DR

This paper analyzes a one-dimensional random walk with increasing step lengths upon each return to the origin, revealing linear displacement growth, exponential displacement distribution, and algebraic escape probabilities, with implications for quick game completion strategies.

Contribution

It introduces a novel random walk model with deterministic step length increases and derives asymptotic behaviors for displacement and escape times, extending to power-law scaling.

Findings

Displacement grows linearly with time.

Displacement distribution decays exponentially.

Escape probability decreases algebraically over time.

Abstract

As a strategy to complete games quickly, we investigate one-dimensional random walks where the step length increases deterministically upon each return to the origin. When the step length after the kth return equals k, the displacement of the walk x grows linearly in time. Asymptotically, the probability distribution of displacements is a purely exponentially decaying function of |x|/t. The probability E(t,L) for the walk to escape a bounded domain of size L at time t decays algebraically in the long time limit, E(t,L) ~ L/t^2. Consequently, the mean escape time <t> ~ L ln L, while <t^n> ~ L^{2n-1} for n>1. Corresponding results are derived when the step length after the kth return scales as k^alpha$ for alpha>0.