First-return statistics and the point-spread function of a Henyey-Greenstein random walk - a far-field stable law

Abstract

A particle launched normally into a half space, scattering by the Henyey-Greenstein (HG) phase function, exits after a first-return time whose count of flights n approaches the distribution-free Sparre Andersen exponent, f(n) ~ n^{-3/2}, with an amplitude fixed by the launch rather than by the theorem. Conditioned on n, and past a correlation-set transient, the lateral exit position is Gaussian with variance proportional to n, a scaling we test directly. Subordinating that Gaussian displacement to the one-sided 1/2-stable first-return time gives, in the diffusion limit, an isotropic 1-stable scaling limit: the far tail of the point-spread function (PSF) for a normally incident pencil beam is the two-dimensional Cauchy tail p(r) ~ r^{-3}, the large-radius limit of the half-space Poisson kernel. The index alpha = 1 is the product of the central limit theorem and the distribution-free first-return exponent, so no phase function admitting the same diffusion limit - finite step variance, short-range angular correlation - can move it. Absorption weights each path by exp(-mu_a L) in its total length L, truncating the subordinator and producing the screened radial structure (mu_eff + 1/r) exp(-mu_eff r)/r^2 carried by each source term of the Farrell-Patterson-Wilson (1992) diffusion dipole; the finite-depth two-source dipole is not derived here. Keeping the launch-depth factor that the tail calculation discards continues the same kernel into a finite core of radius 1.7 l*, with no additional fitted parameter. A scale-dependent index alpha_eff from the empirical characteristic function tracks the similarity parameter gamma, separating phase functions matched in g_1 by 0.15-0.22 near one transport mean free path. Results are Monte Carlo at g = 0 to 0.9 with up to 10^6 walkers.