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

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
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