1,687 research outputs found
Asymptotics of a Class of Solutions to the Cylindrical Toda Equations
The small t asymptotics of a class of solutions to the 2D cylindrical Toda
equations is computed. The solutions, q_k(t), have the representation q_k(t) =
log det(I-lambda K_k) - log det(I-lambda K_{k-1}) where K_k are integral
operators. This class includes the n-periodic cylindrical Toda equations. For
n=2 our results reduce to the previously computed asymptotics of the 2D radial
sinh-Gordon equation and for n=3 (and with an additional symmetry contraint)
they reduce to earlier results for the radial Bullough-Dodd equation.Comment: 29 pages, no figures, LaTeX fil
Lower order terms in Szego type limit theorems on Zoll manifolds
This is a detailed version of the paper math.FA/0212273. The main motivation
for this work was to find an explicit formula for a "Szego-regularized"
determinant of a zeroth order pseudodifferential operator (PsDO) on a Zoll
manifold. The idea of the Szego-regularization was suggested by V. Guillemin
and K. Okikiolu. They have computed the second term in a Szego type expansion
on a Zoll manifold of an arbitrary dimension. In the present work we compute
the third asymptotic term in any dimension. In the case of dimension 2, our
formula gives the above mentioned expression for the Szego-redularized
determinant of a zeroth order PsDO. The proof uses a new combinatorial
identity, which generalizes a formula due to G.A.Hunt and F.J.Dyson. This
identity is related to the distribution of the maximum of a random walk with
i.i.d. steps on the real line. The proof of this combinatorial identity
together with historical remarks and a discussion of probabilistic and
algebraic connections has been published separately.Comment: 39 pages, full version, submitte
Asymptotics for the Fredholm Determinant of the Sine Kernel on a Union of Intervals
In the bulk scaling limit of the Gaussian Unitary Ensemble of Hermitian
matrices the probability that an interval of length contains no eigenvalues
is the Fredholm determinant of the sine kernel over
this interval. A formal asymptotic expansion for the determinant as tends
to infinity was obtained by Dyson. In this paper we replace a single interval
of length by where is a union of intervals and present a proof
of the asymptotics up to second order. The logarithmic derivative with respect
to of the determinant equals a constant (expressible in terms of
hyperelliptic integrals) times , plus a bounded oscillatory function of
(zero of , periodic if , and in general expressible in terms of the
solution of a Jacobi inversion problem), plus . Also determined are the
asymptotics of the trace of the resolvent operator, which is the ratio in the
same model of the probability that the set contains exactly one eigenvalue to
the probability that it contains none. The proofs use ideas from orthogonal
polynomial theory.Comment: 24 page
Entanglement entropy of fermions in any dimension and the Widom conjecture
We show that entanglement entropy of free fermions scales faster then area
law, as opposed to the scaling for the harmonic lattice, for example.
We also suggest and provide evidence in support of an explicit formula for the
entanglement entropy of free fermions in any dimension , as the size of a subsystem
, where is the Fermi surface and
is the boundary of the region in real space. The expression for the constant
is based on a conjecture due to H. Widom. We
prove that a similar expression holds for the particle number fluctuations and
use it to prove a two sided estimates on the entropy .Comment: Final versio
Trace functions as Laplace transforms
We study trace functions on the form t\to\tr f(A+tB) where is a
real function defined on the positive half-line, and and are
matrices such that is positive definite and is positive
semi-definite. If is non-negative and operator monotone decreasing, then
such a trace function can be written as the Laplace transform of a positive
measure. The question is related to the Bessis-Moussa-Villani conjecture.
Key words: Trace functions, BMV-conjecture.Comment: Minor change of style, update of referenc
Entanglement scaling in critical two-dimensional fermionic and bosonic systems
We relate the reduced density matrices of quadratic bosonic and fermionic
models to their Green's function matrices in a unified way and calculate the
scaling of bipartite entanglement of finite systems in an infinite universe
exactly. For critical fermionic 2D systems at T=0, two regimes of scaling are
identified: generically, we find a logarithmic correction to the area law with
a prefactor dependence on the chemical potential that confirms earlier
predictions based on the Widom conjecture. If, however, the Fermi surface of
the critical system is zero-dimensional, we find an area law with a
sublogarithmic correction. For a critical bosonic 2D array of coupled
oscillators at T=0, our results show that entanglement follows the area law
without corrections.Comment: 4 pages, 4 figure
Digital Cavities and Their Potential Applications
The concept of a digital cavity is presented. The functionality of a tunable
radio-frequency/microwave cavity with unrestricted Q-factor is implemented. The
theoretical aspects of the cavity and its potential applications in high
resolution spectroscopy and synchronization of clocks together with examples in
signal processing and data acquisition are discussed
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