Symmetry in stochasticity: Random walk models

of large-scale structure

RAVI K SHETH^1,2

¹Center for Particle Cosmology, University of Pennsylvania, Philadelphia, PA 19104, USA

²The Abdus Salam Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy

E-mail: shethrk@physics.upenn.edu

Abstract. 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.

Keywords. Galaxies – formation; large-scale structure; cosmology.

PACS Nos 98.65.Cw; 98.65.Dx