Optimal Navigation in Stochastic and Disordered Gridworlds

TL;DR

This paper investigates how disorder affects optimal navigation policies in complex environments by analyzing a Brownian particle in a disordered landscape, revealing non-monotonic effects and deriving analytical insights.

Contribution

It introduces a novel framework to quantify how disorder reshapes optimal navigation policies and uncovers unexpected behaviors at low trap concentrations.

Findings

Optimal policies minimize mean first-passage time in disordered landscapes.

Disorder causes a non-monotonic change in navigation policies with trap concentration.

Analytical expression derived for policy change in fluctuation-dominated regimes.

Abstract

Navigation in complex and noisy environments is a key issue in diverse fields from biology to engineering. Despite extensive progress in numerical optimization methods for computing navigation policies, insights into how disorder reshapes optimal navigation remain elusive. To address this question, we investigate the navigation of a Brownian particle in a disordered energy landscape, modeled as a lattice with randomly distributed traps. Using dynamic programming, we compute the optimal navigation policies that minimize the mean first-passage time to a target site. To quantify the impact of disorder, we introduce a density of change from a Kullback-Leibler divergence, which captures how the optimal policy is reshaped by either the presence of disorder or the knowledge of its configuration. Our results reveal a non-monotonic dependence of the change of the policy on trap concentration, with a pronounced maximum. In the fluctuation-dominated regime where the navigation bias is weak, we derive an analytical expression for the density of change, and demonstrate that the maximum occurs unexpectedly at low trap concentrations.