Quasi-additive estimates on the Hamiltonian for the one-dimensional long range Ising model

International audienceIn this work, we study the problem of getting quasi-additive bounds for the Hamiltonian of the long range Ising model, when the two-body interaction term decays proportionally to 1/d(2-alpha) , alpha is an element of (0, 1). We revisit the paper by Cassandro et al. [J. Math. Phys. 46, 053305 (2005)] where they extend to the case alpha is an element of[0, ln2/ln2 - 1) the result of the existence of a phase transition by using a Peierls argument given by Fr " ohlich and Spencer [Commun. Math. Phys. 84, 87-101 (1982)] for alpha= 0. The main arguments of Cassandro et al. [J. Math. Phys. 46, 053305 (2005)] are based in a quasi-additive decomposition of the Hamiltonian in terms of hierarchical structures called triangles and contours, which are related to the original definition of contours introduced by Fr " ohlich and Spencer [Commun. Math. Phys. 84, 87-101 (1982)]. In this work, we study the existence of a quasi-additive decomposition of the Hamiltonian in terms of the contours defined in thework of Cassandro et al. [J. Math. Phys. 46, 053305 (2005)]. The most relevant result obtained is Theorem 4.3 where we show that there is a quasiadditive decomposition for the Hamiltonian in terms of contours when alpha is an element of [0, 1) but not in terms of triangles. The fact that it cannot be a quasi-additive bound in terms of triangles lead to a very interesting maximization problem whose maximizer is related to a discrete Cantor set. As a consequence of the quasi-additive bounds, we prove that we can generalise the [Cassandro et al., J. Math. Phys. 46, 053305 (2005)] result, that is, a Peierls argument, to the whole interval alpha is an element of [0, 1). We also state here the result of Cassandro et al. [Commun. Math. Phys. 327, 951-991 (2014)] about cluster expansions which implies that Theorem 2.4 that concerns interfaces and Theorem 2.5 that concerns n point truncated correlation functions in Cassandro et al. [Commun. Math. Phys. 327, 951-991 (2014)] are valid for all ff 2 [0, 1) instead of only alpha is an element of[0, ln3/ln2 - 1)