Characteristic polynomials of random matrices

Number theorists have studied extensively the connections between the distribution of zeros of the Riemann $\zeta$-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices, which are the expectation values of the characteristic polynomials of the matrix. It turns out that these expectation values are quite interesting. For instance, the moments of order 2K scale, for unitary invariant ensembles, as the density of eigenvalues raised to the power $K^2$ ; the prefactor turns out to be a universal number, i.e. it is independent of the specific probability distribution. An equivalent behaviour and prefactor had been found, as a conjecture, within number theory. The moments of the characteristic determinants of random matrices are computed here as limits, at coinciding points, of multi-point correlators of determinants. These correlators are in fact universal in Dyson's scaling limit in which the difference between the points goes to zero, the size of the matrix goes to infinity, and their product remains finite.Comment: 30 pages,late

An extension of the HarishChandra-Itzykson-Zuber integral

The HarishChandra-Itzykson-Zuber integral over the unitary group U(k) (beta=2) is present in numerous problems involving Hermitian random matrices. It is well known that the result is semi-classically exact. This simple result does not extend to other symmetry groups, such as the symplectic or orthogonal groups. In this article the analysis of this integral is extended first to the symplectic group Sp(k) (beta=4). There the semi-classical approximation has to be corrected by a WKB expansion. It turns out that this expansion stops after a finite number of terms ; in other words the WKB approximation is corrected by a polynomial in the appropriate variables. The analysis is based upon new solutions to the heat kernel differential equation. We have also investigated arbitrary values of the parameter beta, which characterizes the symmetry group. Closed formulae are derived for arbitrary beta and k=3, and also for large beta and arbitrary k.Comment: 18 page

Pair Production in a Time Dependent Magnetic Field

The production of electron-positron pairs in a time-dependent magnetic field is estimated in the hypotheses that the magnetic field is uniform over large distances with respect to the pair localization and it is so strong that the spacing of the Landau levels is larger than the rest mass of the particles. This calculation is presented since it has been suggested that extremely intense and varying magnetic fields may be found around some astrophysical objects.Comment: 11 pages, Plain TeX, no figures. Submitted to Modern Physics Letter

The intersection numbers of the p-spin curves from random matrix theory

The intersection numbers of p-spin curves are computed through correlation functions of Gaussian ensembles of random matrices in an external matrix source. The p-dependence of intersection numbers is determined as polynomial in p; the large p behavior is also considered. The analytic continuation of intersection numbers to negative values of p is discussed in relation to SL(2,R)/U(1) black hole sigma model.Comment: 19 page

Characteristic polynomials of real symmetric random matrices

It is shown that the correlation functions of the random variables $\det(\lambda - X)$, in which $X$ is a real symmetric $N\times N$ random matrix, exhibit universal local statistics in the large $N$ limit. The derivation relies on an exact dual representation of the problem: the $k$-point functions are expressed in terms of finite integrals over (quaternionic) $k\times k$ matrices. However the control of the Dyson limit, in which the distance of the various parameters \la's is of the order of the mean spacing, requires an integration over the symplectic group. It is shown that a generalization of the Itzykson-Zuber method holds for this problem, but contrary to the unitary case, the semi-classical result requires a {\it finite} number of corrections to be exact. We have also considered the problem of an external matrix source coupled to the random matrix, and obtain explicit integral formulae, which are useful for the analysis of the large $N$ limit.Comment: 24 pages, late

Random super matrices with an external source

In the past we have considered Gaussian random matrix ensembles in the presence of an external matrix source. The reason was that it allowed, through an appropriate tuning of the eigenvalues of the source, to obtain results on non-trivial dual models, such as Kontsevich's Airy matrix models and generalizations. The techniques relied on explicit computations of the k-point functions for arbitrary N (the size of the matrices) and on an N-k duality. Numerous results on the intersection numbers of the moduli space of curves were obtained by this technique. In order to generalize these results to include surfaces with boundaries, we have extended these techniques to supermatrices. Again we have obtained quite remarkable explicit expressions for the k-point functions, as well as a duality. Although supermatrix models a priori lead to the same matrix models of 2d-gravity, the external source extensions considered in this article lead to new geometric results.Comment: 12 page

A matrix model for a quantum hall droplet with manifest particle-hole symmetry

We find that a gauged matrix model of rectangular fermionic matrices (a matrix version of the fermion harmonic oscillator) realizes a quantum hall droplet with manifest particle-hole symmetry. The droplet consists of free fermions on the topology of a sphere. It is also possible to deform the Hamiltonian by double trace operators, and we argue that this device can produce two body potentials which might lead the system to realize a fractional quantum hall state on the sphere. We also argue that a single gauged fermionic quantum mechanics of hermitian matrices realizes a droplet with an edge that has $c=1/2$ CFT on it.Comment: 25 pages, uses JHEP format, young.sty (included). v2: Updated references, typos correcte