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A factor of i.i.d. with uniform marginals and infinite clusters spanned by equal labels

Part of: Graph theory Noncompact transformation groups Ergodic theory Equilibrium statistical mechanics Stochastic processes

Published online by Cambridge University Press:  22 December 2022

PÉTER MESTER
PÉTER MESTER*
Affiliation:
Alfréd Rényi Institute of Mathematics, Budapest and Tomori Pál College, Budapest, Hungary
*
e-mail: pmester@alumni.iu.edu
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Abstract

We give an example of an FIID vertex-labeling of ${\mathbb T}_3$ whose marginals are uniform on $[0,1]$, and if we delete the edges between those vertices whose labels are different, then some of the remaining clusters are infinite. We also show that no such process can be finitary.

Keywords

regular treeshyperfinitenessnon-amenable groupsfactor of i.i.d

MSC classification

Primary: 60G10: Stationary processes 82B43: Percolation
Secondary: 37A50: Relations with probability theory and stochastic processes 22F10: Measurable group actions 05C78: Graph labelling (graceful graphs, bandwidth, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 11 , November 2023 , pp. 3707 - 3725
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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