First passage percolation for random colorings of \(\mathbb{Z}^ d\). (English) Zbl 0780.60101

Fontes, Luiz; Newman, Charles M.

Ann. Appl. Probab. 3, No. 3, 746-762 (1993).

Summary: Random colorings (independent or dependent) of \(\mathbb{Z}^ d\) give rise to dependent first-passage percolation in which the passage time along a path is the number of color changes. Under certain conditions, we prove strict positivity of the time constant (and a corresponding asymptotic shape result) by means of a theorem of J. T. Cox, A. Gandolfi, P. Griffin and H. Kesten [Greedy lattice animals. I: Upper bounds (ibid., to appear)] about “greedy” lattice animals. Of particular interest are i.i.d. colorings and the \(d=2\) Ising model. We also apply the greedy lattice animal theorem to prove a result on the omnipresence of the infinite cluster in high density independent bond percolation.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B43	Percolation
60G60	Random fields
82B20	Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics

Keywords:

Ising model; random colorings; first-passage percolation; lattice animals; bond percolation