Probabilistic methods have become an important tool in many areas. The idea is to show the existence of structures satisfying certain properties by defining an appropriate random model of the structure, and show that the property holds with high probability. Examples include:
- Random regular graphs are expanders
- Gromov's notion of random groups, which have been used to construct hyperbolic groups with various properties, including property T, the Haagerup property, surface subgroups, and groups which do not embed uniformly in Hilbert space
- Ranks of elliptic curves
- Constructions of closed surfaces in hyperbolic 3-manifolds by gluing "random" pairs of pants together, where the pants are shown to satisfy certain geometric constraints with high probability and uniformly distributed by ergodic theoretic methods.
