Daniel Ueltschi
Geometric and probabilistic aspects of boson lattice models
(143K, LaTeX2e with 8 postscript figures)
ABSTRACT. This 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.