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Fully-connected bond percolation on \({\mathbb {Z}}^d\)

Abstract

We consider the bond percolation model on the lattice \({\mathbb {Z}}^d\) (\(d\ge 2\)) with the constraint to be fully connected. Each edge is open with probability \(p\in (0,1)\), closed with probability \(1-p\) and then the process is conditioned to have a unique open connected component (bounded or unbounded). The model is defined on \({\mathbb {Z}}^d\) by passing to the limit for a sequence of finite volume models with general boundary conditions. Several questions and problems are investigated: existence, uniqueness, phase transition, DLR equations. Our main result involves the existence of a threshold \(0<p^*(d)<1\) such that any infinite volume model is necessary the vacuum state in subcritical regime (no open edges) and is non trivial in the supercritical regime (existence of a stationary unbounded connected cluster). Bounds for \(p^*(d)\) are given and show that it is drastically smaller than the standard bond percolation threshold in \({\mathbb {Z}}^d\). For instance \(0.128<p^*(2)<0.202\) (rigorous bounds) whereas the 2D bond percolation threshold is equal to 1/2.

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Acknowledgements

The author would like to thank Vincent Beffara for the fruitful discussions on the topic. This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01), the ANR project PPPP (ANR-16-CE40-0016) and by the CNRS GdR 3477 GeoSto.

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Correspondence to David Dereudre.

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Dereudre, D. Fully-connected bond percolation on \({\mathbb {Z}}^d\). Probab. Theory Relat. Fields 183, 547–579 (2022). https://doi.org/10.1007/s00440-021-01088-8

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  • DOI: https://doi.org/10.1007/s00440-021-01088-8

Keywords

  • FK-percolation
  • Random cluster model
  • Phase transition
  • FKG inequalities
  • DLR equations

Mathematics Subject Classification

  • 05C80
  • 60D05
  • 60K35
  • 82B05
  • 82B26