Geometric and probabilistic aspects of boson lattice models

TL;DR

This review explores the geometric and probabilistic properties of boson lattice models, focusing on phase transitions and Bose-Einstein condensation, with emphasis on rigorous results and cycle representations.

Contribution

It provides a comprehensive overview of bosonic lattice models, highlighting the connection between phase transitions, probabilistic cycle analysis, and Bose-Einstein condensation.

Findings

First-order phase transitions in Lennard-Jones potential models

Occurrence of Bose-Einstein condensation in the hard-core boson model

Relation between infinite cycles and Bose-Einstein condensation

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.