Geometry of contours and Peierls estimates in d=1 Ising models

Details Geometry of contours and Peierls estimates in d=1 Ising models Geometry of contours and Peierls estimates in d=1 Ising models

2002-11-25

arxiv.org/abs/math-ph/0211062v2

J. Math. Phys. 46 (2005), no. 5, 053305, 22 pp.

Physics

Mathematical Physics

28 pages, 3 figures

Scientific paper

10.1063/1.1897644

Following Fr\"ohlich and Spencer, we study one dimensional Ising spin systems with ferromagnetic, long range interactions which decay as $|x-y|^{-2+\alpha}$, $0\leq \alpha\leq 1/2$. 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.

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