2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/56748This review describes quantum systems of bosonic particles moving on a lattice. These models are relevant in statistical physics, and have natural ties with probability theory. The general setting is recalled and the main questions about phase transitions are addressed. A lattice model with Lennard-Jones potential is studied as an example of a system where first-order phase transitions occur. A major interest of bosonic systems is the possibility of displaying a Bose-Einstein condensation. This is discussed in the light of the main existing rigorous result, namely its occurrence in the hard-core boson model. Finally, we consider another approach that involves the lengths of the cycles formed by the particles in the space-time representation; Bose-Einstein condensation should be related to positive probability of infinite cycles.22 pages, 8 figuresMathematical Physics82B10, 82B20, 82B26, 82B41, 60K40text