1,438 research outputs found
Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to graphical enumeration
We study the partition function from random matrix theory using a well known
connection to orthogonal polynomials, and a recently developed Riemann-Hilbert
approach to the computation of detailed asymptotics for these orthogonal
polynomials. We obtain the first proof of a complete large N expansion for the
partition function, for a general class of probability measures on matrices,
originally conjectured by Bessis, Itzykson, and Zuber. We prove that the
coefficients in the asymptotic expansion are analytic functions of parameters
in the original probability measure, and that they are generating functions for
the enumeration of labelled maps according to genus and valence. Central to the
analysis is a large N expansion for the mean density of eigenvalues, uniformly
valid on the entire real axis.Comment: 44 pages, 4 figures. To appear, International Mathematics Research
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The Poisson Geometry of Plancherel Formulas for Triangular Groups
In this paper we establish the existence of canonical coordinates for generic
co-adjoint orbits on triangular groups. These orbits correspond to a set of
full Plancherel measure on the associated dual groups. This generalizes a
well-known coordinatization of co-adjoint orbits of a minimal (non-generic)
type originally discovered by Flaschka. The latter had strong connections to
the classical Toda lattice and its associated Poisson geometry. Our results
develop connections with the Full Kostant-Toda lattice and its Poisson
geometry. This leads to novel insights relating the details of Plancherel
theorems for Borel Lie groups to the invariant theory for Borels and their
subgroups. We also discuss some implications for the quantum integrability of
the Full Kostant Toda lattice
Testing for seasonal unit roots by frequency domain regression
This paper develops univariate seasonal unit root tests based on spectral regression estimators. An advantage of the frequency domain approach is that it enables serial correlation to be treated non-parametrically. We demonstrate that our proposed statistics have pivotal limiting distributions under both the null and near seasonally integrated alternatives when we allow for weak dependence in the driving shocks. This is in contrast to the popular seasonal unit root tests of, among others, Hylleberg et al. (1990) which treat serial correlation parametrically via lag augmentation of the test regression. Moreover, our analysis allows for (possibly infinite order) moving average behaviour in the shocks, while extant large sample results pertaining to the Hylleberg et al. (1990) type tests are based on the assumption of a finite autoregression. The size and power properties of our proposed frequency domain regression-based tests are explored and compared for the case of quarterly data with those of the tests of Hylleberg et al. (1990) in simulation experiments.Seasonal unit root tests; moving average; frequency domain regression; spectral density estimator; Brownian motion
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