Multiscale phenomena in quantum and classical models

In this thesis we present several works dealing with multiscale phenomena. First, we treat a model of pattern formation: the Ising model with competing short-range and long-range interactions. We compute the energy asymptotic of excitations above the periodic ground states. In a second work we study a model of Quantum Hall Effect with a single edge-mode and subject to quasi-periodic disorder: through a combination of rigorous Renormalization Group techniques and \textsc{kam}-like estimates we compute the long distance asymptotic of all the correlations and use this asymptotic to in turn compute Kubo transport coefficients, as the condutivity, which turns to be quantized as expected. In the last work we consider a one-dimensional fermionic system again subject to quasi-periodic disorder, as a first step to treat two-dimensional systems with multiple edge-modes. We develop a new Renormalization Group scheme and report on progress in its applications.