On the Tail Transition of First Arrival Position Channels: From Cauchy to Exponential Decay

TL;DR

This paper analyzes how nonzero drift affects the distribution of first arrival positions, transitioning from heavy-tailed Cauchy to exponential decay, with implications for communication system performance.

Contribution

It characterizes the drift-induced transition of FAP distribution and introduces a characteristic propagation distance to distinguish regimes.

Findings

Drift causes a transition from Cauchy to exponential FAP distribution.

Variance-matched Gaussian models underestimate communication potential in low-drift environments.

Zero-drift Cauchy law offers a robust baseline for performance evaluation.

Abstract

While the zero-drift first arrival position (FAP) channel exhibits a Cauchy-distributed lateral displacement, nonzero drift in practical systems introduces advective transport that regularizes this singular limit. This letter characterizes the drift-induced transition of FAP distribution from heavy-tailed algebraic regime to exponential regularization. By asymptotically examining the exact FAP density, we identify a characteristic propagation distance (CPD) that serves as the fundamental boundary separating diffusion-dominated and drift-dominated regimes. Numerical experiments demonstrate that in low-drift environments, variance-matched Gaussian approximations severely underestimate the true communication potential, whereas the zero-drift Cauchy law provides a robust, physically grounded performance baseline.