Gaussian limit for critical oriented percolation in high dimensions

Abstract

In this paper, we consider the spread-out oriented bond percolation models inZ ^d ×Z withd>4 and the nearest-neighbor oriented bond percolation model in sufficiently high dimensions. Let η _n ,n=1, 2, ..., be the random measures defined onR ^d by

$$\eta _n (A) = \sum\limits_{x \in Z^d } {1_A (x/\sqrt n )1_{\{ (0,0) \to (x,n)\} } } $$

The mean of η _n , denoted by$\bar \eta _n$, is the measure defined by

$$\bar \eta _n (A) = E_p [\eta _n (A)]$$

We use the lace expansion method to show that the sequence of probability measures$[\bar \eta _n (R^d )]^{ - 1} \bar \eta _n$ converges weakly to a Gaussian limit asn→∞ for everyp in the subcritical regime as well as the critical regime of these percolation models. Also we show that for these models the parallel correlation length$\xi (p)~|p_c - p|^{ - 1}$ asp⇆p_c