High radiative efficiency in moderately doped n-InP results in the transport of holes dominated by photon-assisted hopping, when radiative hole recombination at one spot produces a photon, whose interband absorption generates another hole, possibly far away. Due to "heavy tails" in the hop probability, this is a random walk with divergent diffusivity (process known as the Lévy flight). Our key evidence is derived from the ratio of transmitted and reflected luminescence spectra, measured in samples of different thicknesses. These experiments prove the non-exponential decay of the hole concentration from the initial photo-excitation spot. The power-law decay, characteristic of Lévy flights, is steep enough at short distances (steeper than an exponent) to fit the data for thin samples and slow enough at large distances to account for thick samples. The high radiative efficiency makes possible a semiconductor scintillator with efficient photon collection. It is rather unusual that the material is "opaque" at wavelengths of its own scintillation. Nevertheless, after repeated recycling most photons find their way to one of two photodiodes integrated on both sides of the semiconductor slab. We present an analytical model of photon collection in two-sided slab, which shows that the heavy tails of Lévy-flight transport lead to a high charge collection efficiency and hence high energy resolution. Finally, we discuss a possibility to increase the slab thickness while still quantifying the deposited energy and the interaction position within the slab. The idea is to use a layered semiconductor with photon-assisted collection of holes in narrow-bandgap layers spaced by distances far exceeding diffusion length. Holes collected in these radiative layers emit longwave radiation, to which the entire structure is transparent. Nearly-ideal calculated characteristics of a mm-thick layered scintillator can be scaled up to several centimeters.