Correlation Functions of Complex Matrix Models

For a restricted class of potentials (harmonic+Gaussian potentials), we express the resolvent integral for the correlation functions of simple traces of powers of complex matrices of size $N$, in term of a determinant; this determinant is function of four kernels constructed from the orthogonal polynomials corresponding to the potential and from their Cauchy transform. The correlation functions are a sum of expressions attached to a set of fully packed oriented loops configurations; for rotational invariant systems, explicit expressions can be written for each configuration and more specifically for the Gaussian potential, we obtain the large $N$ expansion ('t Hooft expansion) and the so-called BMN limit.Comment: latex BMN.tex, 7 files, 6 figures, 30 pages (v2 for spelling mistake and added reference) [http://www-spht.cea.fr/articles/T05/174

Ratios of characteristic polynomials in complex matrix models

We compute correlation functions of inverse powers and ratios of characteristic polynomials for random matrix models with complex eigenvalues. Compact expressions are given in terms of orthogonal polynomials in the complex plane as well as their Cauchy transforms, generalizing previous expressions for real eigenvalues. We restrict ourselves to ratios of characteristic polynomials over their complex conjugate

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

Fractional statistic

We improve Haldane's formula which gives the number of configurations for $N$ particles on $d$ states in a fractional statistic defined by the coupling $g=l/m$. Although nothing is changed in the thermodynamic limit, the new formula makes sense for finite $N=pm+r$ with $p$ integer and $0<r\leq m.$ A geometrical interpretation of fractional statistic is given in terms of ''composite particles''.Comment: flatex hald.tex, 3 files Submitted to: Phys. Rev.

Renormalizability of Nonrenormalizable Field Theories

We give a simple and elegant proof of the Equivalence Theorem, stating that two field theories related by nonlinear field transformations have the same S matrix. We are thus able to identify a subclass of nonrenormalizable field theories which are actually physically equivalent to renormalizable ones. Our strategy is to show by means of the BRS formalism that the "nonrenormalizable" part of such fake nonrenormalizable theories, is a kind of gauge fixing, being confined in the cohomologically trivial sector of the theory.Comment: 3 pages, revtex, no figure

Higher Order Analogues of Tracy-Widom Distributions via the Lax Method

We study the distribution of the largest eigenvalue in formal Hermitian one-matrix models at multicriticality, where the spectral density acquires an extra number of k-1 zeros at the edge. The distributions are directly expressed through the norms of orthogonal polynomials on a semi-infinite interval, as an alternative to using Fredholm determinants. They satisfy non-linear recurrence relations which we show form a Lax pair, making contact to the string literature in the early 1990's. The technique of pseudo-differential operators allows us to give compact expressions for the logarithm of the gap probability in terms of the Painleve XXXIV hierarchy. These are the higher order analogues of the Tracy-Widom distribution which has k=1. Using known Backlund transformations we show how to simplify earlier equivalent results that are derived from Fredholm determinant theory, valid for even k in terms of the Painleve II hierarchy.Comment: 24 pages. Improved discussion of Backlund transformations, in addition to other minor improvements in text. Typos corrected. Matches published versio

`Composite particles' and the eigenstates of Calogero-Sutherland and Ruijsenaars-Schneider

We establish a one-to-one correspondance between the ''composite particles'' with $N$ particles and the Young tableaux with at most $N$ rows. We apply this correspondance to the models of Calogero-Sutherland and Ruijsenaars-Schneider and we obtain a momentum space representation of the ''composite particles'' in terms of creation operators attached to the Young tableaux. Using the technique of bosonisation, we obtain a position space representation of the ''composite particles'' in terms of products of vertex operators. In the special case where the ''composite particles'' are bosons and if we add one extra quasiparticle or quasihole, we construct the ground state wave functions corresponding to the Jain series $\nu =p/(2np\pm 1)$ of the fractional quantum Hall effect.Comment: latex calcomp2.tex, 5 files, 30 pages [SPhT-T99/080], submitted to J. Math. Phy

Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices

We apply the method of skew-orthogonal polynomials (SOP) in the complex plane to asymmetric random matrices with real elements, belonging to two different classes. Explicit integral representations valid for arbitrary weight functions are derived for the SOP and for their Cauchy transforms, given as expectation values of traces and determinants or their inverses, respectively. Our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product. Examples for the SOP are given in terms of Laguerre polynomials for the chiral ensemble (also called the non-Hermitian real Wishart-Laguerre ensemble), both without and with the insertion of characteristic polynomials. Such characteristic polynomials play the role of mass terms in applications to complex Dirac spectra in field theory. In addition, for the elliptic real Ginibre ensemble we recover the SOP of Forrester and Nagao in terms of Hermite polynomials