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

Yen-Chi Lee

TL;DR

This work characterizes how the First Arrival Position FAP channel transitions from a heavy-tailed Cauchy noise model in zero drift to an exponential tail under nonzero drift. A universal critical length scale $n_c = \sigma^2/v$ demarcates diffusion-dominated and drift-dominated regions, yielding a Truncated Cauchy channel and finite moments beyond $n_c$. Capacity analyses reveal that Gaussian approximations misestimate performance at low drift, while the zero-drift Cauchy baseline provides a robust lower bound; as drift grows, the channel behavior shifts toward Gaussian with optimal inputs transitioning from heavy-tailed to discrete. These insights guide the design of dense Molecular MIMO systems by quantifying tail interference and informing input shaping under drift-diffusion dynamics.

Abstract

While the zero-drift First Arrival Position (FAP) channel is rigorously known to be Cauchy-distributed, practical molecular communication systems typically operate with non-zero drift. This letter characterizes the transition from heavy-tailed Cauchy behavior to light-tailed exponential decay. Through asymptotic analysis, we identify a critical spatial scale $n_c=σ^2/v$ separating diffusion- and drift-dominated regimes, revealing that the channel effectively behaves as a ``Truncated Cauchy'' model. Numerical results show that Gaussian approximations severely underestimate capacity at low drift, while the zero-drift case provides the appropriate performance lower bound for systems where drift assists particle transport.

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

TL;DR

This work characterizes how the First Arrival Position FAP channel transitions from a heavy-tailed Cauchy noise model in zero drift to an exponential tail under nonzero drift. A universal critical length scale demarcates diffusion-dominated and drift-dominated regions, yielding a Truncated Cauchy channel and finite moments beyond . Capacity analyses reveal that Gaussian approximations misestimate performance at low drift, while the zero-drift Cauchy baseline provides a robust lower bound; as drift grows, the channel behavior shifts toward Gaussian with optimal inputs transitioning from heavy-tailed to discrete. These insights guide the design of dense Molecular MIMO systems by quantifying tail interference and informing input shaping under drift-diffusion dynamics.

Abstract

While the zero-drift First Arrival Position (FAP) channel is rigorously known to be Cauchy-distributed, practical molecular communication systems typically operate with non-zero drift. This letter characterizes the transition from heavy-tailed Cauchy behavior to light-tailed exponential decay. Through asymptotic analysis, we identify a critical spatial scale separating diffusion- and drift-dominated regimes, revealing that the channel effectively behaves as a ``Truncated Cauchy'' model. Numerical results show that Gaussian approximations severely underestimate capacity at low drift, while the zero-drift case provides the appropriate performance lower bound for systems where drift assists particle transport.

Paper Structure

This paper contains 17 sections, 10 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System Model and Exact Distribution
  3. The Physical Cauchy Baseline (Zero-Drift)
  4. The General Case (Non-Zero Drift)
  5. The Anatomy of the Tail: Asymptotic Analysis
  6. Regime 1: The Cauchy Core (Diffusion Dominated)
  7. Regime 2: The Exponential Tail (Drift Dominated)
  8. The Critical Length Scale
  9. Extension to 3D Systems (2D Receiving Plane)
  10. Capacity Analysis and Numerical Results
  11. Numerical Setup
  12. Results and Interpretation
  13. Impact on Spatial Interference
  14. Discussion: Optimal Inputs and Shaping Loss
  15. The Shaping Loss in Zero-Drift Regime
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Abstract FAP channel model featuring hyperplane-shaped transmitters (Tx) and receivers (Rx). The particle undergoes Brownian motion with diffusion coefficient $D$ and drift velocity $\mathbf{v}$ over a transmission distance $\lambda$. The Rx registers the arrival point on the transverse plane.
  • Figure 2: Capacity transition of the FAP channel from the Cauchy regime to the Gaussian regime. The significant gap at low drift velocities highlights the failure of standard variance-based Gaussian approximations.
  • Figure 3: Spatial interference probability versus separation distance $r$. The zero-drift baseline is computed analytically via the Cauchy CDF, while non-zero drift curves are obtained by numerically integrating the exact Bessel-based PDF. For the high-drift case (blue dashed), the exponential truncation begins visibly after the critical scale $n_c \approx 40\,\mu\text{m}$.