Fully-connected bond percolation on Zd

David Dereudre
Université Lille 1
http://math.univ-lille1.fr/~dereudre/

Date(s) : 03/05/2022 iCal
14h30 - 15h30

We consider the bond percolation model on the lattice Zd 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 Zd 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 Zd. For instance 0.128<p*(2)<0.202 (rigorous bounds) whereas the 2D bond percolation threshold is equal to 1/2