Phase transition of the long range Ising model in lower dimensions, for d d + 1, with a Peierls argument

We extend previous results due to Ding and Zhuang in order to prove that a phase transition occurs for the long range, random field Ising model (RFIM) in lower dimensions. By making use of a 2022 argument due to Affonso, Bissacot and Maia which establishes that a phase transition occurs for the long range, random-field Ising model in higher dimensions from a suggestion of the authors we demonstrate that a phase transition occurs in the lower dimensional state space from a set of appropriately defined contours through a Peierls argument. In comparison to the higher dimensional contour system, the lower dimensional counterpart: is dependent upon fewer degrees of freedom encoded through an Formula: see text parameter in the coupling constants of the Hamiltonian; exhibits qualitatively different behaviors under Formula: see text, or −, boundary conditions at the inverse temperature; can be characterized through a novel combinatorial enumeration of the number of suitable paths within the contour system; can be examined through a coarse-graining approach which shares in many similarities with the action of the Renormalization group for closely related systems appearing in Statistical Physics and Mathematical Physics. The forthcoming approach in the lower dimensional, long-range, setting is novel through the action of a suitably defined flipping procedure applied to the spins contained within the interior of a contour. For a RFIM spin configuration Formula: see text, through a lower dimensional spin reversal procedure from the operation Formula: see text, we distinguish whether Formula: see text depending upon whether Formula: see text has a special label from the interior of the contour, or whether Formula: see text depending upon whether Formula: see text belongs to the upper half of a contour. Moreover, the lower-dimensional contour, in comparison to the higher-dimensional one, not only takes into account how large collections of Formula: see text, or −, spins aggregate, but also how interactions typically occur in lower-dimensional lattices.