Stochastic growth models.

*(English)*Zbl 0628.60102
Percolation theory and ergodic theory of infinite particle systems, Proc. Workshop IMA, Minneapolis/Minn. 1984/85, IMA Vol. Math. Appl. 8, 85-119 (1987).

[For the entire collection see Zbl 0615.00015.]

In a paper by the first author and D. Griffeath [Ann. Probab. 11, 1-15 (1983; Zbl 0508.60080)] one-dimensional nearest neighbour additive growth models have been studied. The main purpose of the present paper is to show that most results of the just mentioned paper are also true for the corresponding class of finite range models, i.e., those which can be constructed from a percolation structure.

These generalized percolation processes (including e.g. oriented bond or site percolation) are constructed from a “graphical representation”. A critical probability is introduced and it is shown how it is related to the “edge speeds”.

Another basic key to the developments in this paper is the renormalized bond construction explained in Section 4. The interesting results include e.g. a complete (weak) convergence theorem (starting from any initial configuration) and a strong law for the number of particles at time n (as \(n\to \infty)\) for the process starting from a single particle at 0.

In a paper by the first author and D. Griffeath [Ann. Probab. 11, 1-15 (1983; Zbl 0508.60080)] one-dimensional nearest neighbour additive growth models have been studied. The main purpose of the present paper is to show that most results of the just mentioned paper are also true for the corresponding class of finite range models, i.e., those which can be constructed from a percolation structure.

These generalized percolation processes (including e.g. oriented bond or site percolation) are constructed from a “graphical representation”. A critical probability is introduced and it is shown how it is related to the “edge speeds”.

Another basic key to the developments in this paper is the renormalized bond construction explained in Section 4. The interesting results include e.g. a complete (weak) convergence theorem (starting from any initial configuration) and a strong law for the number of particles at time n (as \(n\to \infty)\) for the process starting from a single particle at 0.

Reviewer: K.Schürger

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B43 | Percolation |