  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.
