Bridge Scaling in Conditioned Henyey-Greenstein Random Walks

TL;DR

This study investigates three-dimensional Henyey-Greenstein random walks conditioned on fixed-length bridges, revealing anomalous scaling behaviors and structural effects that differ from classical Brownian excursion theory, through extensive Monte Carlo simulations.

Contribution

It uncovers novel anomalous scaling laws and structural features in conditioned Henyey-Greenstein random walks, highlighting the impact of a two-dimensional state space on their behavior.

Findings

Mean amplitude scales super-diffusively with path length

Diffusion coefficient scales with the mean free path to a fractional power

Midpoint depth distribution follows a Rayleigh distribution

Abstract

We study fixed-length bridge paths -- half-space excursions that start and end at a planar boundary -- for three-dimensional random walks with Henyey-Greenstein scattering angles and exponentially distributed step lengths, using Monte Carlo simulation over asymmetry parameter g from 0 to 0.95 and path lengths from 4 to 200 steps. The key structural feature is that the walk evolves on a two-dimensional Markovian state space (depth, direction cosine) rather than the scalar depth coordinate alone. Four anomalies with respect to classical Brownian-excursion theory are reported. The mean amplitude scales super-diffusively, as path length to a power of 0.57--0.58 for isotropic scattering, nine standard deviations above the Brownian prediction of 0.5, with no sign of convergence out to 200 steps. The diffusion coefficient scales as the transport mean free path to the power 0.415 rather than the predicted 1.0. The midpoint depth distribution is Rayleigh rather than half-normal, consistent with a two-dimensional Bessel process. The bridge-conditioned mean direction cosine converges to minus two-thirds at the final step, independently of the asymmetry parameter and initial direction -- the classical Milne result anchored by the H-function moment identity. All anomalies are attributed to the two-dimensional state-space structure. The two anomalous exponents sum to approximately unity, suggesting a common geometric origin. Whether this constitutes a permanent universality-class shift or an anomalously slow crossover to Brownian-excursion behaviour remains the primary open question.