Central limit theorems for associated random variables and the percolation model (Q792020)

The basic result (Th. 1.2) deals with triangular arrays of associated random variables \(X_ n(N)\), \(n\in k(N)\), \(N=1,2,...\), where k(N), \(N=1,2,..\). is a sequence of rectangles whose sides grow to infinity. The only assumptions are moment conditions, no stationarity is required. It is shown that \(\sum_{n}X_ n(N)\) satisfies a central limit theorem (CLT). This result is applied in two contexts: first a new proof of a CLT is given for the occupation time \(\int^{t}_{0}\eta_{s(0)}ds\) in the voter model in dimension \(d\geq 5\); then a CLT is established for expressions of the form \(\sum_{x\in k(N)}f_ N(W_ x)\) in a two dimensional bond percolation model away from the critical probability 1/2, where \(W_ x\) is the cluster containing the point x and \(f_ N\) a suitable decreasing function on subsets of \(Z^ 2\) (Th. 1.3). This last theorem is continued to show that the number of distinct clusters in a rectangle under growth conditions on its sides also obeys a CLT (Th.1.8).