Geometry of contours and Peierls estimates in d=1 Ising models with long range interactions

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.