2-Matrix versus Complex Matrix Model, Integrals over the Unitary Group as Triangular Integrals

2-matrix versus complex matrix model, integrals over the unitary group as triangular integrals

The Replica Limit of Unitary Matrix Integrals

We investigate the replica trick for the microscopic spectral density, ρs(x), of the Euclidean QCD Dirac operator. Our starting point is the low-energy limit of the QCD partition function for n fermionic flavors (or replicas) in the sector of topological charge ν. In the domain of the smallest eigenvalues, this partition function is simply given by a U(n) unitary matrix integral. We show that t...

Unitary Integrals and Related Matrix Models

Concise review of the basic properties of unitary matrix integrals. They are studied with the help of the three matrix models: the ordinary unitary model, Brezin-Gross-Witten model and the Harish-Charndra-Itzykson-Zuber model. Especial attention is paid to the tricky sides of the story, from De Wit-t’Hooft anomaly in unitary integrals to the problem of correlators with Itzykson-Zuber measure. O...

Matrix Models vs. Matrix Integrals

In a brief review, we discuss interrelations between arbitrary solutions of the loop equations that describe Hermitean one-matrix model and particular (multi-cut) solutions that describe concrete matrix integrals. These latter ones enjoy a series of specific properties and, in particular, are described in terms of Seiberg-Witten-Whitham theory. The simplest example of ordinary integral is consi...

Unitary Matrix Integrals, Primitive Factorizations, and Jucys-Murphy Elements

A factorization of a permutation into transpositions is called “primitive” if its factors are weakly ordered. We discuss the problem of enumerating primitive factorizations of permutations, and its place in the hierarchy of previously studied factorization problems. Several formulas enumerating minimal primitive and possibly non-minimal primitive factorizations are presented, and interesting co...