David Dereudre (Univ. Lille): Fully-connected bond percolation on Z^d
3.5.2022

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