# WILBERT

## Wildauer Bücher+E-Medien Recherche-Tool

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• 1985-1989  (2)
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• 1
Electronic Resource
Springer
Communications in mathematical physics 111 (1987), S. 555-577
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract WE consider a one-dimensional random Ising model with Hamiltonian $$H = \sum\limits_{i\ddag j} {\frac{{J_{ij} }}{{\left| {i - j} \right|^{1 + \varepsilon } }}S_i S_j } + h\sum\limits_i {S_i }$$ , where ε〉0 andJ ij are independent, identically distributed random variables with distributiondF(x) such that i) $$\int {xdF\left( x \right) = 0}$$ , ii) $$\int {e^{tx} dF\left( x \right)〈 \infty \forall t \in \mathbb{R}}$$ . We construct a cluster expansion for the free energy and the Gibbs expectations of local observables. This expansion is convergent almost surely at every temperature. In this way we obtain that the free energy and the Gibbs expectations of local observables areC ∞ functions of the temperature and of the magnetic fieldh. Moreover we can estimate the decay of truncated correlation functions. In particular for every ε′〉0 there exists a random variablec(ω)m, finite almost everywhere, such that $$\left| {\left\langle {s_0 s_j } \right\rangle _H - \left\langle {s_0 } \right\rangle _H \left\langle {s_j } \right\rangle _H } \right| \leqq \frac{{c\left( \omega \right)}}{{\left| j \right|^{1 + \varepsilon - \varepsilon '} }}$$ , where 〈 〉 H denotes the Gibbs average with respect to the HamiltonianH.
Type of Medium: Electronic Resource
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• 2
Electronic Resource
Springer
Communications in mathematical physics 108 (1987), S. 241-255
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract One-dimensional Ising spin systems interacting via a two-body random potential are considered; a decay with the distance like 1/r 1+ε is assumed. We consider only boundary conditions independent of the random realization of the interactions and prove uniqueness and cluster properties of Gibbs states with probability one.
Type of Medium: Electronic Resource
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