Abstract
Consider percolation on Z d with parameter p. Let K n be the number of occupied clusters in [−n, n]d. Here we use a martingale method to show that if p≠0, 1, K n satisfies the CLT for all d>1. Furthermore, we investigate the large deviations and concentration property for K n . Besides K n , we also consider the distribution of the number Λ n of such vertices connected by the infinite occupied cluster in a large box [−n, n]d. We show that Λ n satisfies the CLT and investigate the concentration property for Λ n , by using the martingale method in the supercritical phase.
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REFERENCES
Akcoglu, M., and Krengel, U. (1981). Ergodic theorems for superadditive processes. J. Reine Angew. Math. 323, 53–67.
Cox, T., and Grimmett G. (1984). Central limit theorems for associated random variables and the percolation model. Ann. Probab. 12, 514-528.
Durrett, R., and Schonmann, R. Large deviations for the contact process and two-dimension percolation. Ann. Probab. 77, 583û603.
Grimmett, G. (1989). Percolation, Berlin, Springer.
Kesten, H. (1986). The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields 73, 369–394.
Kesten, H., and Zhang, Y. (1990). The probability of a large finite cluster in super critical Bernoulli percolation. Ann. Probab. 18, 537–555; Related Fields 107, 137û160 (1997).
Kesten, H., and Zhang, Y. (1997). A central limit theorem for critical first passage percolation in two dimensions. Probab. Theory Relat. Fields 107, 137–160.
Mcleish, D. L. (1974). Dependent central limit theorems and invariance principles. Ann. Probab. 2, 620–628.
Smythe, R. (1976). Multiparameter subadditive processes. Ann. Probab. 4, 772–782.
Williams, D. (1991). Probability with Martingale, Cambridge Univ. Press.
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Zhang, Y. A Martingale Approach in the Study of Percolation Clusters on the Z d Lattice. Journal of Theoretical Probability 14, 165–187 (2001). https://doi.org/10.1023/A:1007877216583
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DOI: https://doi.org/10.1023/A:1007877216583