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Digitale Medien

Comparison and disjoint-occurrence inequalities for random-cluster models (1995)

Grimmett, Geoffrey

Springer

Journal of statistical physics 78 (1995), S. 1311-1324

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ISSN: 1572-9613

Schlagwort(e): Random-cluster model ; Ising model ; Potts model ; comparison inequality ; BK inequality ; FKG inequality

Quelle: Springer Online Journal Archives 1860-2000

Thema: Physik

Notizen: Abstract A principal technique for studying percolation, (ferromagnetic) Ising, Potts, and random-cluster models is the FKG inequality, which implies certain stochastic comparison inequalities for the associated probability measures. The first result of this paper is a new comparison inequality, proved using an argument developed elsewhere in order to obtain strict inequalities for critical values. As an application of this inequality, we prove that the critical pointp c (q) of the random-cluster model with cluster-weighting factorq (≥1) is strictly monotone inq. Our second result is a “BK inequality” for the disjoint occurrence of increasing events, in a weaker form than that available in percolation theory.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1007/BF02180133

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Digitale Medien

Central limit theorems for percolation models (1981)

Cox, J. Theodore ; Grimmett, Geoffrey

Springer

Journal of statistical physics 25 (1981), S. 237-251

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ISSN: 1572-9613

Schlagwort(e): Percolation ; asymptotic normality ; circuits ; semi-invariants

Quelle: Springer Online Journal Archives 1860-2000

Thema: Physik

Notizen: 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).

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1007/BF01022185

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