Mixed Correlation Functions of the Two-Matrix Model

We compute the correlation functions mixing the powers of two non-commuting random matrices within the same trace. The angular part of the integration was partially known in the literature: we pursue the calculation and carry out the eigenvalue integration reducing the problem to the construction of the associated biorthogonal polynomials. The generating function of these correlations becomes then a determinant involving the recursion coefficients of the biorthogonal polynomials.Comment: 16 page

Moment determinants as isomonodromic tau functions

We consider a wide class of determinants whose entries are moments of the so-called semiclassical functionals and we show that they are tau functions for an appropriate isomonodromic family which depends on the parameters of the symbols for the functionals. This shows that the vanishing of the tau-function for those systems is the obstruction to the solvability of a Riemann-Hilbert problem associated to certain classes of (multiple) orthogonal polynomials. The determinants include Haenkel, Toeplitz and shifted-Toeplitz determinants as well as determinants of bimoment functionals and the determinants arising in the study of multiple orthogonality. Some of these determinants appear also as partition functions of random matrix models, including an instance of a two-matrix model.Comment: 24 page

Polar bulges and polar nuclear discs: the case of NGC 4698

The early-type spiral NGC 4698 is known to host a nuclear disc of gas and stars which is rotating perpendicularly with respect to the galaxy main disc. In addition, the bulge and main disc are characterised by a remarkable geometrical decoupling. Indeed they appear elongated orthogonally to each other. In this work the complex structure of the galaxy is investigated by a detailed photometric decomposition of optical and near-infrared images. The intrinsic shape of the bulge was constrained from its apparent ellipticity, its twist angle with respect to the major axis of the main disc, and the inclination of the main disc. The bulge is actually elongated perpendicular to the main disc and it is equally likely to be triaxial or axisymmetric. The central surface brightness, scalelength, inclination, and position angle of the nuclear disc were derived by assuming it is infinitesimally thin and exponential. Its size, orientation, and location do not depend on the observed passband. These findings support a scenario in which the nuclear disc is the end result of the acquisition of external gas by the pre-existing triaxial bulge on the principal plane perpendicular to its shortest axis and perpendicular to the galaxy main disc. The subsequent star formation either occurred homogeneously all over the extension of the nuclear disc or through an inside-out process that ended more than 5 Gyr ago.Comment: 6 pages, 3 figures. Accepted for publication in MNRAS Letter

Mixed correlation function and spectral curve for the 2-matrix model

We compute the mixed correlation function in a way which involves only the orthogonal polynomials with degrees close to $n$, (in some sense like the Christoffel Darboux theorem for non-mixed correlation functions). We also derive new representations for the differential systems satisfied by the biorthogonal polynomials, and we find new formulae for the spectral curve. In particular we prove the conjecture of M. Bertola, claiming that the spectral curve is the same curve which appears in the loop equations.Comment: latex, 1 figure, 55 page

The V_c-sigma_c relation in high and low surface brightness galaxies

We investigate the relation between the asymptotic circular velocity, V_c, and the central stellar velocity dispersion, sigma_c, in galaxies. We consider a new sample of high surface brightness spiral galaxies (HSB), low surface brightness spiral galaxies (LSB), and elliptical galaxies with HI-based V_c measurements. We find that: 1) elliptical galaxies with HI measurements fit well within the relation; 2) a linear law can reproduce the data as well as a power law (used in previous works) even for galaxies with sigma_c < 70 km/s; 3) LSB galaxies, considered for the first time with this respect, seem to behave differently, showing either larger V_c values or smaller sigma_c values.Comment: 2 pages, 2 figures, to appear in Proc. IAU Symp. 222, "The Interplay among Black Holes, Stars and ISM in Galactic Nuclei" eds. Th. Storchi Bergmann, L.C. Ho & H.R. Schmitt (Cambridge University Press

Second and Third Order Observables of the Two-Matrix Model

In this paper we complement our recent result on the explicit formula for the planar limit of the free energy of the two-matrix model by computing the second and third order observables of the model in terms of canonical structures of the underlying genus g spectral curve. In particular we provide explicit formulas for any three-loop correlator of the model. Some explicit examples are worked out.Comment: 22 pages, v2 with added references and minor correction

Minor-axis velocity gradients in spirals and the case of inner polar disks

We measured the ionized-gas and stellar kinematics along the major and minor axis of a sample of 10 early-type spirals. Much to our surprise we found a remarkable gas velocity gradient along the minor axis of 8 of them. According to the kinematic features observed in their ionized-gas velocity fields, we divide our sample galaxies in three classes of objects. (i) NGC 4984, NGC 7213, and NGC 7377 show an overall velocity curve along the minor axis without zero-velocity points, out to the last measured radius, which is interpreted as due to the warped structure of the gaseous disk. (ii) NGC 3885, NGC 4224, and NGC 4586 are characterized by a velocity gradient along both major and minor axis, although non-zero velocities along the minor axis are confined to the central regions. Such gas kinematics have been explained as being due to non-circular motions induced by a triaxial potential. (iii) NGC 2855 and NGC 7049 show a change of slope of the velocity gradient measured along the major axis (which is shallower in the center and steeper away from the nucleus), as well as non-zero gas velocities in the central regions of the minor axis. This has been attributed to the presence of a kinematically-decoupled gaseous component in orthogonal rotation with respect to the galaxy disk, namely an inner polar disk. The case and origin of inner polar disks are discussed and the list of their host galaxies is presented.Comment: 13 pages. 3 PostScript figures (Figs. 1 and 3 at lower resolution). Accepted for publication in A&

Loop equations for the semiclassical 2-matrix model with hard edges

The 2-matrix models can be defined in a setting more general than polynomial potentials, namely, the semiclassical matrix model. In this case, the potentials are such that their derivatives are rational functions, and the integration paths for eigenvalues are arbitrary homology classes of paths for which the integral is convergent. This choice includes in particular the case where the integration path has fixed endpoints, called hard edges. The hard edges induce boundary contributions in the loop equations. The purpose of this article is to give the loop equations in that semicassical setting.Comment: Latex, 20 page