The Pendulum Arrangement: Maximizing the Escape Time of Heterogeneous Random Walks

TL;DR

This paper demonstrates that arranging transition probabilities in a pyramid-like structure, called the pendulum arrangement, maximizes escape time in heterogeneous one-dimensional random walks, with implications across various scientific fields.

Contribution

It proves that the pendulum arrangement uniquely maximizes escape time in heterogeneous random walks, linking combinatorial optimization to stochastic process design.

Findings

Pyramid-like transition probability arrangements maximize escape time.

The problem reduces to a sum over products, combinatorial optimization.

The structure's optimality applies broadly to models in epidemiology, biology, and computer science.

Abstract

We identify a fundamental phenomenon of heterogeneous one dimensional random walks: the escape (traversal) time is maximized when the heterogeneity in transition probabilities forms a pyramid-like potential barrier. This barrier corresponds to a distinct arrangement of transition probabilities, sometimes referred to as the pendulum arrangement. We reduce this problem to a sum over products, combinatorial optimization problem, proving that this unique structure always maximizes the escape time. This general property may influence studies in epidemiology, biology, and computer science to better understand escape time behavior and construct intruder-resilient networks.