Some limit theorems for voter model occupation times. (English) Zbl 0656.60105

Cox, J. T.

Ann. Probab. 16, No. 4, 1559-1569 (1988).

The author studies the occupation time functionals \(\int^{t}_{0}f(\eta_ s)ds\) of the voter model \(\{\eta_ t\), \(t\geq 0\}\) on \({\mathbb{Z}}^ d\). He first extends a pointwise ergodic theorem, which was proven for \(d\geq 3\) by E. D. Andjel and C. P. Kipnis [Probab. Theory Relat. Fields 75, 545-550 (1987; Zbl 0621.60114)], to the case \(d=2\). (It is known that the statement is false for \(d=1).\)
Secondly, he proves a central limit theorem for the case \(f(\eta)=\eta (0)\) and initial distributions being either a fixed \(\eta\) of a certain class of states for \(d\geq 2\) or an extremal invariant measure for \(d\geq 3\). As is usual in the study of the voter model, the duality with a coalescing random walk is basically used.

Reviewer: M.Mürmann

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60F05	Central limit and other weak theorems
60F15	Strong limit theorems

Keywords:

occupation time functionals; voter model; central limit theorem; coalescing random walk

Citations:

Zbl 0621.60114