Fully-connected bond percolation on Zd

We consider the bond percolation model on the lattice Z(d) (d >= 2) with the constraint to be fully connected. Each edge is open with probability p is an element of (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 Z(d) by passing to the limit for a sequence of finite volume modelswith 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 Z(d). For instance 0.128 < p* (2) < 0.202 (rigorous bounds) whereas the 2D bond percolation threshold is equal to 1/2.