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Discontinuity of the magnetization in one-dimensional \(1/| x-y| ^ 2\) Ising and Potts models. (English) Zbl 1084.82514
Summary: Results from percolation theory are used to study phase transitions in one-dimensional Ising and \(q\)-state Potts models with couplings of the asymptotic form \(J_{x,y} \approx \text{const}/|x-y|^2\). For translation-invariant systems with well-defined \(\lim_{x\to \infty} x^2 J_x =J^+\) (possibly \(0\) or \(\infty\)) we establish: (1) There is no long-range order at inverse temperatures \(\beta\) with \(\beta J^+ \leq 1\). (2) If \(\beta J^+>q\), then by sufficiently increasing \(J_1\) the spontaneous magnetization \(M\) is made positive. (3) In models with \(0<J^+<\infty\) the magnetization is discontinuous at the transition point (as originally predicted by Thouless), and obeys \(M(\beta_c )\geq 1/(\beta_c J^+)^{1/2}\). (4) For Ising \((q=2)\) models with \(J^+<\infty\), it is noted that the correlation function decays as \(\langle \sigma_x \sigma_y\rangle (\beta) \approx c(\beta)/|x-y|^2\). Points 1–3 are deduced from previous percolation results by utilizing the Fortuin-Kasteleyn representation, which also yields other results of independent interest relating Potts models with different values of \(q\).

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
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