On correlated random walks and 21-cm fluctuations during cosmic reionization

Analytical approaches to galaxy formation and reionization are based on the mathematical problem of random walks with barriers. The statistics of a single random walk can be used to calculate one-point distributions ranging from the mass function of virialized haloes to the distribution of ionized bubble sizes during reionization. However, an analytical calculation of two-point correlation functions or of spatially dependent feedback processes requires the joint statistics of random walks at two different points. An accurate analytical expression for the statistics of two correlated random walks has been previously found only for the case of a constant barrier height. However, calculating bubble sizes or accurate statistics for halo formation involves more general barriers that can often be approximated as linear barriers. We generalize the two-point solution with constant barriers to linear barriers, and apply it as an illustration to calculate the correlation function of cosmological 21-cm fluctuations during reionization.