Dyson Models Under Renormalization and in Weak Fields

We consider one-dimensional long-range spin models (usually called Dyson models), consisting of Ising ferromagnets with slowly decaying long-range pair potentials of the form $$ {1}/{| i-j |^\alpha }$$, mainly focusing on the range of slow decays $$1 <\alpha \le 2$$. We describe two recent results, one about renormalization and one about the effect of external fields at low temperature