18 research outputs found
Double scaling limits of random matrices and minimal (2m,1) models: the merging of two cuts in a degenerate case
In this article, we show that the double scaling limit correlation functions
of a random matrix model when two cuts merge with degeneracy (i.e. when
for arbitrary values of the integer ) are the same as the
determinantal formulae defined by conformal models. Our approach
follows the one developed by Berg\`{e}re and Eynard in \cite{BergereEynard} and
uses a Lax pair representation of the conformal models (giving
Painlev\'e II integrable hierarchy) as suggested by Bleher and Eynard in
\cite{BleherEynard}. In particular we define Baker-Akhiezer functions
associated to the Lax pair to construct a kernel which is then used to compute
determinantal formulae giving the correlation functions of the double scaling
limit of a matrix model near the merging of two cuts.Comment: 37 pages, 4 figures. Presentation improved, typos corrected.
Published in Journal Of Statistical Mechanic
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 , 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 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
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 , (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
Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies
We compute the generating functions of a O(n) model (loop gas model) on a
random lattice of any topology. On the disc and the cylinder, they were already
known, and here we compute all the other topologies. We find that the
generating functions (and the correlation functions of the lattice) obey the
topological recursion, as usual in matrix models, i.e they are given by the
symplectic invariants of their spectral curve.Comment: pdflatex, 89 pages, 12 labelled figures (15 figures at all), minor
correction
Some properties of angular integrals
We find new representations for Itzykson-Zuber like angular integrals for
arbitrary beta, in particular for the orthogonal group O(n), the unitary group
U(n) and the symplectic group Sp(2n). We rewrite the Haar measure integral, as
a flat Lebesge measure integral, and we deduce some recursion formula on n. The
same methods gives also the Shatashvili's type moments. Finally we prove that,
in agreement with Brezin and Hikami's observation, the angular integrals are
linear combinations of exponentials whose coefficients are polynomials in the
reduced variables (x_i-x_j)(y_i-y_j).Comment: 43 pages, Late
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
Large N expansions and Painlev\'e hierarchies in the Hermitian matrix model
We present a method to characterize and compute the large N formal
asymptotics of regular and critical Hermitian matrix models with general even
potentials in the one-cut and two-cut cases. Our analysis is based on a method
to solve continuum limits of the discrete string equation which uses the
resolvent of the Lax operator of the underlying Toda hierarchy. This method
also leads to an explicit formulation, in terms of coupling constants and
critical parameters, of the members of the Painlev\'e I and Painlev\'e II
hierarchies associated with one-cut and two-cut critical models respectively
A Matrix model for plane partitions
We construct a matrix model equivalent (exactly, not asymptotically), to the
random plane partition model, with almost arbitrary boundary conditions.
Equivalently, it is also a random matrix model for a TASEP-like process with
arbitrary boundary conditions. Using the known solution of matrix models, this
method allows to find the large size asymptotic expansion of plane partitions,
to ALL orders. It also allows to describe several universal regimes.Comment: Latex, 41 figures. Misprints and corrections. Changing the term TASEP
to self avoiding particle porces
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
On the large N limit of matrix integrals over the orthogonal group
We reexamine the large N limit of matrix integrals over the orthogonal group
O(N) and their relation with those pertaining to the unitary group U(N). We
prove that lim_{N to infty} N^{-2} \int DO exp N tr JO is half the
corresponding function in U(N), and a similar relation for lim_{N to infty}
\int DO exp N tr(A O B O^t), for A and B both symmetric or both skew symmetric.Comment: 12 page