Abstract: In statistical mechanics, Gibbs states are probability distributions on the space of configurations of a system which minimize the mean "free energy", i.e. minimizes a potential and maximizes the entropy. At each temperature one finds different Gibbs states, reflecting the fact that the entropy contribution scales linearly with the temperature. It can happen that several Gibbs states exist at a given temperature (a "phase transition"); this is interpreted physically as saying that the material can take different stable forms, e.g. at room temperature Carbon is stable both as graphite and as diamond. Phase transitions typically happens in multidimensional models when the temperature is low, and there is a large literature on this subject. In this talk I will discuss a less known phenomenon, namely that as the temperature is decreased to 0, the corresponding Gibbs states may not converge. Although related to phase transitions, this behavior is of a slightly different nature, and can occur in systems without phase transitions, even in (infinite volume) one-diensional lattice models over discrete state spaces and fast-decaying potentials. This talk will not assume familiarity with statistical mechanics and is based on joint work with Jean-Rene Chazottes.