387 research outputs found
Phase transition in a log-normal Markov functional model
We derive the exact solution of a one-dimensional Markov functional model
with log-normally distributed interest rates in discrete time. The model is
shown to have two distinct limiting states, corresponding to small and
asymptotically large volatilities, respectively. These volatility regimes are
separated by a phase transition at some critical value of the volatility. We
investigate the conditions under which this phase transition occurs, and show
that it is related to the position of the zeros of an appropriately defined
generating function in the complex plane, in analogy with the Lee-Yang theory
of the phase transitions in condensed matter physics.Comment: 9 pages, 5 figures. v2: Added asymptotic expressions for the
convexity-adjusted Libors in the small and large volatility limits. v3: Added
one reference. Final version to appear in Journal of Mathematical Physic
Multipartite minimum uncertainty products
In our previous work we have found a lower bound for the multipartite
uncertainty product of the position and momentum observables over all separable
states. In this work we are trying to minimize this uncertainty product over a
broader class of states to find the fundamental limits imposed by nature on the
observable quantites. We show that it is necessary to consider pure states only
and find the infimum of the uncertainty product over a special class of pure
states (states with spherically symmetric wave functions). It is shown that
this infimum is not attained. We also explicitly construct a parametrized
family of states that approaches the infimum by varying the parameter. Since
the constructed states beat the lower bound for separable states, they are
entangled. We thus show that there is a gap that separates the values of a
simple measurable quantity for separable states from entangled ones and we also
try to find the size of this gap.Comment: 18 pages, 5 figure
Chern-Simons matrix models and Stieltjes-Wigert polynomials
Employing the random matrix formulation of Chern-Simons theory on Seifert
manifolds, we show how the Stieltjes-Wigert orthogonal polynomials are useful
in exact computations in Chern-Simons matrix models. We construct a
biorthogonal extension of the Stieltjes-Wigert polynomials, not available in
the literature, necessary to study Chern-Simons matrix models when the geometry
is a lens space. We also discuss several other results based on the properties
of the polynomials: the equivalence between the Stieltjes-Wigert matrix model
and the discrete model that appears in q-2D Yang-Mills and the relationship
with Rogers-Szego polynomials and the corresponding equivalence with an unitary
matrix model. Finally, we also give a detailed proof of a result that relates
quantum dimensions with averages of Schur polynomials in the Stieltjes-Wigert
ensemble.Comment: 25 pages, AMS-LaTe
Asymptotic corrections to the eigenvalue density of the GUE and LUE
We obtain correction terms to the large N asymptotic expansions of the
eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of
random N by N matrices, both in the bulk of the spectrum and near the spectral
edge. This is achieved by using the well known orthogonal polynomial expression
for the kernel to construct a double contour integral representation for the
density, to which we apply the saddle point method. The main correction to the
bulk density is oscillatory in N and depends on the distribution function of
the limiting density, while the corrections to the Airy kernel at the soft edge
are again expressed in terms of the Airy function and its first derivative. We
demonstrate numerically that these expansions are very accurate. A matching is
exhibited between the asymptotic expansion of the bulk density, expanded about
the edge, and the asymptotic expansion of the edge density, expanded into the
bulk.Comment: 14 pages, 4 figure
Analytic solutions of the 1D finite coupling delta function Bose gas
An intensive study for both the weak coupling and strong coupling limits of
the ground state properties of this classic system is presented. Detailed
results for specific values of finite are given and from them results for
general are determined. We focus on the density matrix and concomitantly
its Fourier transform, the occupation numbers, along with the pair correlation
function and concomitantly its Fourier transform, the structure factor. These
are the signature quantities of the Bose gas. One specific result is that for
weak coupling a rational polynomial structure holds despite the transcendental
nature of the Bethe equations. All these new results are predicated on the
Bethe ansatz and are built upon the seminal works of the past.Comment: 23 pages, 0 figures, uses rotate.sty. A few lines added. Accepted by
Phys. Rev.
The resistance of randomly grown trees
Copyright @ 2011 IOP Publishing Ltd. This is a preprint version of the published article which can be accessed from the link below.An electrical network with the structure of a random tree is considered: starting from a root vertex, in one iteration each leaf (a vertex with zero or one adjacent edges) of the tree is extended by either a single edge with probability p or two edges with probability 1 â p. With each edge having a resistance equal to 1 omega, the total resistance Rn between the root vertex and a busbar connecting all the vertices at the nth level is considered. A dynamical system is presented which approximates Rn, it is shown that the mean value (Rn) for this system approaches (1 + p)/(1 â p) as n â â, the distribution of Rn at large n is also examined. Additionally, a random sequence construction akin to a random Fibonacci sequence is used to approximate Rn; this sequence is shown to be related to the Legendre polynomials and its mean is shown to converge with |(Rn) â (1 + p)/(1 â p)| ⌠nâ1/2.Engineering and Physical Sciences Research Council (EPSRC
Entanglement in Quantum Spin Chains, Symmetry Classes of Random Matrices, and Conformal Field Theory
We compute the entropy of entanglement between the first spins and the
rest of the system in the ground states of a general class of quantum
spin-chains. We show that under certain conditions the entropy can be expressed
in terms of averages over ensembles of random matrices. These averages can be
evaluated, allowing us to prove that at critical points the entropy grows like
as , where and are determined explicitly. In an important class of systems,
is equal to one-third of the central charge of an associated Virasoro algebra.
Our expression for therefore provides an explicit formula for the
central charge.Comment: 4 page
The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source
In classical random matrix theory the Gaussian and chiral Gaussian random
matrix models with a source are realized as shifted mean Gaussian, and chiral
Gaussian, random matrices with real , complex ( and
real quaternion ) elements. We use the Dyson Brownian motion model
to give a meaning for general . In the Gaussian case a further
construction valid for is given, as the eigenvalue PDF of a
recursively defined random matrix ensemble. In the case of real or complex
elements, a combinatorial argument is used to compute the averaged
characteristic polynomial. The resulting functional forms are shown to be a
special cases of duality formulas due to Desrosiers. New derivations of the
general case of Desrosiers' dualities are given. A soft edge scaling limit of
the averaged characteristic polynomial is identified, and an explicit
evaluation in terms of so-called incomplete Airy functions is obtained.Comment: 21 page
Explicit Integration of the Full Symmetric Toda Hierarchy and the Sorting Property
We give an explicit formula for the solution to the initial value problem of
the full symmetric Toda hierarchy. The formula is obtained by the
orthogonalization procedure of Szeg\"{o}, and is also interpreted as a
consequence of the QR factorization method of Symes \cite{symes}. The sorting
property of the dynamics is also proved for the case of a generic symmetric
matrix in the sense described in the text, and generalizations of tridiagonal
formulae are given for the case of matrices with nonzero diagonals.Comment: 13 pages, Latex
State estimation in quantum homodyne tomography with noisy data
In the framework of noisy quantum homodyne tomography with efficiency
parameter , we propose two estimators of a quantum state whose
density matrix elements decrease like , for
fixed known and . The first procedure estimates the matrix
coefficients by a projection method on the pattern functions (that we introduce
here for ), the second procedure is a kernel estimator of the
associated Wigner function. We compute the convergence rates of these
estimators, in risk
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