Random connected subgraph on Zd: A phase transition result

26.04.2021, 14:00 - 15:00
Forschungsseminar Wahrscheinlichkeitstheorie

David Dereudre (Lille, Frankreich)

We consider the bond percolation model on the lattice Zd (d≥2) with the constraint to be fully connected. Each edge is open with probability p∈(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 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 Zd. For instance 0.128<p∗(2)<0.202 (rigorous bounds) whereas the 2D bond percolation threshold is equal to 1/2.

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