Multidimensional Contours \`a la Fr\"{o}hlich-Spencer and Boundary Conditions for Quantum Spin Systems

In this thesis, we present results from the investigation of two problems, one related to the phase transition of long-range Ising models and the other one associated with the characterization of equilibrium states in quantum spin systems. Due to the long-range nature of the interactions, $J|x-y|^{-\alpha}$, estimates using contours usually found in the literature have restrictions on the range of interactions ($\alpha>d+1$ in Ginibre, Grossmann, and Ruelle in 1966 and Park in 1988 for discrete spin systems and possibly non-symmetric situations but with the restrictions $\alpha>3d+1 $). We were able to extend the phase transition argument for long-range Ising-type models to the entire region $\alpha>d$ using the multi-scale arguments presented in the articles by Fr\"ohlich and Spencer. In quantum statistical mechanics, the KMS condition is used as a characterization for the equilibrium states of the system. Widely studied today, it is known to be equivalent to other equilibrium notions such as the variational principle. We present another possible characterization of equilibrium states in quantum spin systems by generalizing the DLR equations to the quantum context using Poisson point process representations. We also discuss the relationship of these quantum DLR equations with the KMS states of a subclass of interactions that contains the Ising model with a transverse field.