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We study isomorphism invariant point processes of R 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… (More)

- Ádám Timár
- 2004

We study isomorphism invariant point processes of R 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… (More)

- Ádám Timár
- 2015

We prove that the infinite components of the Free Uniform Spanning Forest of a Cayley graph are indistinguishable by any invariant property, given that the forest is different from its wired counterpart. Similar result is obtained for the Free Minimal Spanning Forest. We also show that with the above assumptions there can only be 0, 1 or infinitely many… (More)

- Ádám Timár
- Combinatorics, Probability & Computing
- 2007

We answer three questions posed in a paper by Babson and Benjamini. They introduced a parameter CG for Cayley graphs G that has significant application to percolation. For a minimal cutset of G and a partition of this cutset into two classes, take the minimal distance between the two classes. The supremum of this number over all minimal cutsets and all… (More)

We generalize theorems of Kesten and Deuschel-Pisztora about the connectedness of the exterior boundary of a connected subset of Zd, where “connectedness” and “boundary” are understood with respect to various graphs on the vertices of Zd. These theorems are widely used in statistical physics and related areas of probability. We provide simple and elementary… (More)

Consider Bernoulli(1/2) percolation on Z, and define a perfect matching between open and closed vertices in a way that is a deterministic equivariant function of the configuration. We want to find such matching rules that make the probability that the pair of the origin is at distance greater than r decay as fast as possible. For two dimensions, we give a… (More)

- Ádám Timár
- 2009

We show that every locally finite random graph embedded in the plane with an isometry-invariant distribution can be 5-colored in an invariant and deterministic way, under some nontriviality assumption and a mild assumption on the tail of edge lengths. The assumptions hold for any Voronoi map on a point process that has no nontrivial symmetries almost… (More)

We show that for any Cayley graph, the probability (at any p) that the cluster of the origin has size n decays at a well-defined exponential rate (possibly 0). For general graphs, we relate this rate being positive in the supercritical regime with the amenability/nonamenability of the underlying graph. AMS 2000 subject classification: 60K35, 82B43

- Ádám Timár
- Combinatorics, Probability & Computing
- 2012

- Ádám Timár
- 2008

We generalize theorems of Kesten and Deuschel-Pisztora about the connectedness of the exterior boundary of a connected subset of Z, where “connectedness” and “boundary” are understood with respect to various graphs on the vertices of Z. We provide simple and elementary proofs of their results. It turns out that the proper way of viewing these questions is… (More)