Gaussian limit for critical oriented percolation in high dimensions |
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Authors: | Bao Gia Nguyen Wei-Shih Yang |
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Institution: | 1. Department of Mathematics, Illinois Institute of Technology, 60616, Chicago, Illinois 2. Department of Mathematics, Temple University Philadelphia, 19122, Pennsylvania
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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?pc |
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