  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 halos 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.
