One-dimensional random Ising systems with interaction decay \(r^{- (1+\epsilon)}:\) A convergent cluster expansion
We consider a one-dimensional random Ising model with Hamiltonian \[ H\quad =\sum_{i\neq j}\frac{J_{ij}}{| i-j|^{1+\epsilon}}s_ is_ j\quad +\quad h\sum_{i}s_ i, \] where \(\epsilon >0\) and \(J_{ij}\) are independent, identically distributed random variables with distribution dF(x) such that i) \(\int x dF(x)=0\), ii) \(\int e^{tx} dF(x)<\infty\forall t\in {\mathbb{R}}\). 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 are \(C^{\infty}\) functions of the temperature and of the magnetic field h. Moreover we can estimate the decay of truncated correlation functions. In particular, for every \(\epsilon '>0\) there exists a random variable c(\(\omega)\), finite almost everywhere, such that \[ | <s_ 0s_ j>_ H-<s_ 0>_ H<s_ j>_ H| \quad \leq \frac{c(\omega)}{| j|^{1+\epsilon -\epsilon '}}, \] where \(< >_ H\) denotes the Gibbs average with respect to the Hamiltonian H.
decay of truncated correlation functions
scientific article; zbMATH DE number 4046867