713 research outputs found
Conformal Bootstrap Analysis for Yang-Lee Edge Singularity
The Yang-Lee edge singularity is investigated by the determinant method of
the conformal field theory. The critical dimension Dc, for which the scale
dimension of scalar Delta_phi is vanishing, is discussed by this determinant
method. The result is incorporated in the Pade analysis of epsilon expansion,
which leads to an estimation of the value Delta_phi between three and six
dimensions. The structure of the minors is viewed from the fixed points.Comment: 15 page, 8 figure
Conformal Bootstrap Analysis for Single and Branched Polymers
The determinant method in the conformal bootstrap is applied for the critical
phenomena of a single polymer in arbitrary dimensions. The scale dimensions
(critical exponents) of the polymer () and the branched polymer () are obtained from the small determinants. It is known that the
dimensional reduction of the branched polymer in dimensions to Yang-Lee
edge singularity in - dimensions holds exactly. We examine this
equivalence by the small determinant method.Comment: 13 pages, 5 figure
Diagrammatic analysis of the two-state quantum Hall system with chiral invariance
The quantum Hall system in the lowest Landau level with Zeeman term is
studied by a two-state model, which has a chiral invariance. Using a
diagrammatic analysis, we examine this two-state model with random impurity
scattering, and find the exact value of the conductivity at the Zeeman energy
. We further study the conductivity at the another extended state
(). We find that the values of the conductivities at
and do not depend upon the value of the Zeeman energy
. We discuss also the case where the Zeeman energy becomes a
random field.Comment: 14P, 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
Characteristic polynomials of real symmetric random matrices
It is shown that the correlation functions of the random variables
, in which is a real symmetric random
matrix, exhibit universal local statistics in the large limit. The
derivation relies on an exact dual representation of the problem: the -point
functions are expressed in terms of finite integrals over (quaternionic)
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 limit.Comment: 24 pages, late
Characteristic polynomials of random matrices
Number theorists have studied extensively the connections between the
distribution of zeros of the Riemann -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 ; 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
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
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