D. Dereudre: [latexpage]Fully-connected bond percolation on Zd

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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D. Dereudre: Fully-connected bond percolation on Z^d

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