D. Dereudre: Fully-connected bond percolation on \mbox{\huge{\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(d)<1 such that any infinite volume process 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.

During the talk, connections with the FK-percolation, the incipient cluster, the conjecture “\theta(p_c)=0” are also discussed.


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