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Phase transition in ferromagnetic Ising models with non-uniform external magnetic fields. (English) Zbl 1197.82038
The authors study the phase transition phenomenon for the Ising model under the action of a non-uniform external magnetic field $$\mathbf{h}=(h_i)_{i\in \mathbb Z^d}$$ and prove following theorems.
Theorem 1. If the magnetic field $$\mathbf{h}$$ belongs to $$l_1(\mathbb Z^2)$$, then the model presents a phase transition when $$J>3\|\mathbf{h}\|_1$$.
Theorem 2. If $$\mathbf{h}\in l_{\infty}(\mathbb Z^d)$$ is such that $$\liminf_{i\in\mathbb Z^d} h_i>0$$, then the Ising model with external field $$\mathbf{h}$$ has no phase transition.

##### MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 82B26 Phase transitions (general) in equilibrium statistical mechanics
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