Following Frohlich and Spencer, we study one dimensional Ising spin systems with ferromagnetic, long range interactions which decay as vertical bar x-y vertical bar(-2+alpha), 0 <=alpha <= 1/2. We introduce a geometric description of the spin configurations in terms of triangles which play the role of contours and for which we establish Peierls bounds. This in particular yields a direct proof of the well-known result by Dyson about phase transitions at low temperatures. (C) 2005 American Institute of Physics
We study, in the context of the Markov hyerarchical fields (d=2, 3), the role of the Markov property, of formal renormalization and of formal positivity. We determine upper and lower bounds for the ground state energy and discuss their relation with the perturbation theory series
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