Tree and grid factors for general point processes. (English) Zbl 1060.60050

Timár, Ádám

Electron. Commun. Probab. 9, 53-59 (2004).

Summary: We study isomorphism invariant point processes of \(\mathbb R^d\) whose groups of symmetries are almost surely trivial. We define a 1-ended, locally finite tree factor on the points of the process, that is, a mapping of the point configuration to a graph on it that is measurable and equivariant with the point process. This answers a question of Holroyd and Peres. The tree will be used to construct a factor isomorphic to \(\mathbb Z^n\). This perhaps surprising result (that any \(d\) and \(n\) works) solves a problem by Steve Evans. The construction, based on a connected clumping with \(2^i\) vertices in each clump of the \(i\)th partition, can be used to define various other factors.

MSC:

60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60K35	Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

point processes; factors; random tree; random rigid