Stochastic topology aims at studying "expected" topological properties of random or partially known spaces. Such spaces arise naturally in areas such as shape recognition or in studying configuration spaces of large systems. A different motivation to study the topology of random spaces is the successful one dimensional version, the theory of random graphs, initiated in 1959 by Erdos and Rényi [ER]. Not only has this theory been relevant in several applications but also played (and still plays) a key role in the development of classical graph theory.
Recently several models of random simplicial complexes have been suggested and studied. We will focus on the one introduced by Linial and Meshulam in [LM] (later generalized by Meshulam and Wallach in [MW]) and analyse its "expected" topological properties, such as the asymptotics of the betti numbers. Several properties of these spaces were established in [BHK] and [CFK].

[BHK] E. Babson, C. Hoffman, M. Kahle, The fundamental group of random 2-complexes, preprint 2008. arxiv0711.2704v3.
[CFK] A. Costa, M. Farber, T. Kappeler, Topology of random 2-complexes, preprint 2010. arxiv:1006.4229v2.
[ER] P. Erdos, A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960), 17–61.
[LM] N. Linial, R. Meshulam, Homological connectivity of random 2-complexes, Combinatorica 26 (2006), 475–487.
[MW] R. Meshulam, N. Wallach, Homological connectivity of random k-complexes, Random Structures & Algorithms 34 (2009), 408–417.