Crossing walls and windows: the curious escape of Lyman-$\alpha$ photons through ionised channels

The diverse Lyman-alpha (Ly$\alpha$) line profiles are essential probes of gas in and around galaxies. While isotropic models can successfully reproduce a range of Ly$\alpha$ observables, the correspondence between the model and actual physical parameters remains uncertain. We investigate the effect of anisotropies of Ly$\alpha$ escape using a simplified setup: a hole (fractional size $\tilde s$) within a semi-infinite slab with constant column density. Due to the slab's high line-centre optical depth ($\tau_0\gtrsim 10^{5-6}$), most photons should escape through the empty channel. However, our numerical findings indicate that only a fraction $\sim \tilde s$ of photons exit through this channel. To explain this puzzle, we developed an analytical model describing the scattering and transmission behaviour of Ly$\alpha$ photons in an externally illuminated slab. Our findings show that the number of scatterings per reflection follows a L\'evy distribution ($\propto N^{-3/2}$). This means that the mean number of scatterings is orders of magnitude greater than expectations, facilitating a shift in frequency and the subsequent photon escape. Our results imply that Ly$\alpha$ photons are more prone to traverse high-density gas and are surprisingly less biased to the `path of least resistance'. Hence, Ly$\alpha$ can trace an averaged hydrogen distribution rather than only low-column density `channels'.