Tree and Grid factors of General Point processes
Adam Timar, Indiana University
Abstract
We study isomorphism invariant point processes of 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
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.
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