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Random currents and continuity of Ising model’s spontaneous magnetization. (English) Zbl 1315.82004
The authors consider the ferromagnetic Ising model on the transitive graph $$\mathbb Z^d$$ and prove that the spontaneous magnetization vanish at the critical temperature if the coupling constants $$(J_{x,y})_{x,y\in \mathbb Z^d}$$ satisfy the conditions $J_{x,y}=J_{0,y-x}, J_{x,y}\geq 0,\, \sum_{x\in \mathbb Z^d}J_{0,x}<\infty,$ and for any $$x\in \mathbb Z^d,$$ there exist $$0=x_0,x_1,\dots,x_{m-1}$$, $$x_m=x$$ such that $J_{x_0,x_1}J_{x_1,x_2}\cdots J_{x_{m-1},x_m}>0.$

##### MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 82D40 Statistical mechanical studies of magnetic materials
##### Keywords:
Ising model; spontaneous magnetization; Gibbs state
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##### References:
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