Unitary integrals and related matrix models

This article examines the basic properties of unitary matrix integrals using three matrix models: the ordinary unitary model, the Brézin-Gross-Witten (BGW) model and the Harish-Chandra-Itzykson-Zuber (HCIZ) model. The tricky sides of the story are given special attention, such as the de Wit-’t Hooft anomaly in unitary integrals and the problem of correlators with Itzykson-Zuber measure. The method of character expansions is also emphasized as a technical tool. The article first provides an overview of the theory of the BGW model, taking into account the de Wit-’t Hooft anomaly and the M-theory of matrix models, before discussing the theory of the HCIZ integral. In particular, it describes the basics of character calculus, character expansion of the HCIZ integral, character expansion for the BGW model and Leutwyler-Smilga integral, and pair correlator in HCIZ theory.

PAIR CORRELATOR IN THE ITZYKSON-ZUBER INTEGRAL

An explicit expression is suggested for the average [Formula: see text] over the unitary group SU (N) with the Itzykson-Zuber measure [dU] exp tr ΦUΨU†.

UPPER BOUND ON THE TWO-POINT CORRELATION FUNCTION OF A SYSTEM OF COUPLED ANHARMONIC OSCILLATORS

A quantum system of coupled anharmonic oscillators on v-dimensional hypercubic lattice is considered. The upper bound on the pair correlator is obtained. For the representation of creation-annihilation operators this system corresponds to a lattice boson gas model with δ-type repulsion. We prove the existence of Bose condensation at sufficiently low temperatures, and hence of a phase transition in three or more dimensions. An estimate of the critical temperature is obtained.