Search

We study matrix integration over the classical Lie groups U(N), Sp(2N), SO(2N) and SO(2N + 1), using symmetric function theory and the equivalent formulation in terms of determinants and minors of Toeplitz ± Hankel matrices. We establish a number of factorizations and expansions for such integrals, also with insertions of irreducible characters. As a specific example, we compute both at finite and large N the partition functions, Wilson loops and Hopf links of Chern–Simons theory on S3 with the aforementioned symmetry groups. The identities found for the general models translate in this context to relations between observables of the theory. Finally, we use character expansions to evaluate averages in random matrix ensembles of Chern–Simons type, describing the spectra of solvable fermionic models with matrix degrees of freedom. [ABSTRACT FROM AUTHOR]

The relationship between matrix integrals and integrable systems was revealed more than 20 years ago. As is known, matrix integrals over a Gaussian ensemble used in random matrix theory could act as the τ-function of several hierarchies of integrable systems. In this article, we will show that the time-dependent partition function of the Bures ensemble, whose measure has many interesting geometric properties, could act as the τ-function of BKP and DKP hierarchies. In addition, if discrete time variables are introduced, then this partition function could act as the τ-function of discrete BKP and DKP hierarchies. In particular, there are some links between the partition function of the Bures ensemble and Toda-type equations. [ABSTRACT FROM AUTHOR]

The partition function of the Penner matrix model for both positive and negative values of the coupling constant can be explicitly written in terms of the Barnes G function. In this paper we show that for negative values of the coupling constant this partition function can also be represented as the product of an holomorphic matrix integral by a nontrivial oscillatory function of n. We show that the planar limit of the free energy with ’t Hooft sequences does not exist. Therefore we use a certain modification that uses Kuijlaars–McLaughlin sequences instead of ’t Hooft sequences and leads to a well-defined planar free energy and to an associated two-dimensional phase space. We describe the different configurations of complex saddle points of the holomorphic matrix integral both to the left and to the right of the critical point, and interpret the phase transitions in terms of processes of gap closing, eigenvalue tunneling, and Bose condensation. [ABSTRACT FROM AUTHOR]

We compute the partition function of the q-states Potts model on a random planar lattice with allowed, equally weighted colours on a connected boundary. To this end, we employ its matrix model representation in the planar limit, generalising a result by Voiculescu for the addition of random matrices to a situation beyond free probability theory. We show that the partition functions with p and q − p colours on the boundary are related algebraically. Finally, we investigate the phase diagram of the model when and comment on the conformal field theory description of the critical points. [ABSTRACT FROM AUTHOR]

We present an irregular matrix model which has and Virasoro symmetry. The irregular matrix model is obtained using the colliding limit of the Toda field theories and produces the inner product between irregular modules of symmetry. We evaluate the partition function using the flow equation which is the realization of Virasoro and symmetry. [ABSTRACT FROM AUTHOR]