Symmetry in stochasticity: Random walk models of large-scale structure

This paper describes the insights gained from the excursion set approach, in which various questions about the phenomenology of large-scale structure formation can be mapped to problems associated with the first crossing distribution of appropriately defined barriers by random walks. Much of this is summarized in R K Sheth, AIP Conf. Proc.1132, 158 (2009). So only a summary is given here, and instead a few new excursion set related ideas and results which are not published elsewhere are presented. One is a generalization of the formation time distribution to the case in which formation corresponds to the time when half the mass was first assembled in pieces, each of which was at least 1/n times the final mass, and where n ≥ 2; another is an analysis of the first crossing distribution of the Ornstein–Uhlenbeck process. The first derives from the mirror-image symmetry argument for random walks which Chandrasekhar described so elegantly in 1943; the second corrects a misuse of this argument. Finally, some discussion of the correlated steps and correlated walks assumptions associated with the excursion set approach, and the relation between these and peaks theory are also included. These are problems in which Chandra’s mirror-image symmetry is broken.