Fully-connected bond percolation on $\mathbb{Z}^d$

TL;DR

This paper studies a fully-connected bond percolation model on the lattice 2^d, establishing a phase transition threshold below which the system is empty and above which an infinite cluster exists, with bounds significantly below the standard percolation threshold.

Contribution

It introduces and analyzes a conditioned bond percolation model with a unique connected component, providing bounds for the critical probability and demonstrating a drastically lower threshold than standard percolation.

Findings

Existence of a critical threshold p^*(d) for the model.

Below p^*(d), the system is almost surely empty.

Above p^*(d), an infinite cluster exists.

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 $0<p^*(d)<1$ 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 $1/2$.