Electronic Resource
Onedimensional random Ising systems with interaction decayr −(1+ɛ): A convergent cluster expansion
(1987)
Springer
ISSN:
14320916
Source:
Springer Online Journal Archives 18602000
Topics:
Mathematics
,
Physics
Notes:
Abstract WE consider a onedimensional 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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