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  • Ising model  (2)
  • BK inequality  (1)
  • asymptotic normality  (1)
  • 1
    Electronic Resource
    Electronic Resource
    Comparison and disjoint-occurrence inequalities for random-cluster models (1995)
    Springer
    Journal of statistical physics 78 (1995), S. 1311-1324 
    Details
    ISSN: 1572-9613
    Keywords: Random-cluster model ; Ising model ; Potts model ; comparison inequality ; BK inequality ; FKG inequality
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02180133
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Potts models and random-cluster processes with many-body interactions (1994)
    Springer
    Journal of statistical physics 75 (1994), S. 67-121 
    Details
    ISSN: 1572-9613
    Keywords: Potts model ; Ising model ; random-cluster process ; critical temperature ; critical exponent ; differential inequality ; universality
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Known differential inequalities for certain ferromagnetic Potts models with pair interactions may be extended to Potts models with many-body interactions. As a major application of such differential inequalities, we obtain necessary and sufficient conditions on the set of interactions of such a Potts model in order that its critical point be astrictly monotonic function of the strengths of interactions. The method yields some ancillary information concerning the equality of certain critical exponents for Potts models; this amounts to a small amount of rigorous universality. These results are achieved in the context of a “Fortuin-Kasteleyn representation” of Potts models with many-body interactions. For such a Potts model, the corresponding random-cluster process is a (random) hypergraph.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02186281
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Central limit theorems for percolation models (1981)
    Springer
    Journal of statistical physics 25 (1981), S. 237-251 
    Details
    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
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    Paper (German National Licenses)
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