ISSN:
1572-9613
Keywords:
Percolation
;
asymptotic normality
;
circuits
;
semi-invariants
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
Abstract Letp ≠ 1/2 be the open-bond probability in Broadbent and Hammersley's percolation model on the square lattice. LetW x be the cluster of sites connected tox by open paths, and letγ(n) be any sequence of circuits with interiors $$|\mathop \gamma \limits^ \circ (n)| \to \infty $$ . It is shown that for certain sequences of functions {f n }, $$S_n = \sum _{x \in \mathop \gamma \limits^ \circ (n)} f_n (W_x )$$ converges in distribution to the standard normal law when properly normalized. This result answers a problem posed by Kunz and Souillard, proving that the numberS n of sites insideγ(n) which are connected by open paths toγ(n) is approximately normal for large circuitsγ(n).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01022185
Permalink
