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Some properties of random Ising models. (English) Zbl 0624.60116
We consider an Ising model with random magnetic field \(h_ i\) and random nearest-neighbor couplings \(J_{ij}\). The random variables \(h_ i\) and \(J_{ij}\) are independent and identically distributed with a nice enough distribution, e.g., Gaussian. We will prove that (i) at high temperature, infinite volume correlation functions are independent on the boundary conditions and decay exponentially fast with probability 1 and (ii) for any temperature with sufficiently strong magnetic field the correlation functions are again independent on the boundary conditions and decay exponentially fast with probability 1. We also prove that the averaged magnetization of the ground state configuration of the one-dimensional Ising model with random magnetic field is zero, no matter how small is the variance of the \(h_ i\).

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
81P20 Stochastic mechanics (including stochastic electrodynamics)
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