On the d-dimensional lattice Zd with d ≥ 2, the phase tran-sition of the nearest-neighbor ferromagnetic Ising model can be proved by using Peierls argument, that requires a notion of contours, geometric curves on the dual of the lattice to study the spontaneous symmetry breaking.

It is known that the one-dimensional nearest-neighbor ferromagnetic Ising model does not undergo a phase transition at any temperature. On the other hand, if we add a polynomially decaying long-range interaction given

by Jxy = |x − y|−α for x, y ∈ Z, the works by Dyson and Fr¨ohlich-Spencer show the phase transition at low temperatures for 1 < α ≤ 2. Moreover, Fr¨ohlich and Spencer defined a notion of contours for α = 2. There are many other works that study and extend the notion of contours for other α. In this talk, we define a notion of contours for d-dimensional long-range Ising model with d ≥ 2, where the interactions are given by Jxy = |x − y|−α

where α > d and x, y ∈ Zd. As an application, we add non-homogeneous external fields hx = |x|−γ in the Hamiltonian and give conditions for γ and α so that the model undergoes a phase transition at low temperatures.

Joint work with Lucas Affonso, Rodrigo Bissacot, and Satoshi Handa.

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