19,249 research outputs found
The correspondence between Tracy-Widom (TW) and Adler-Shiota-van Moerbeke (ASvM) approaches in random matrix theory: the Gaussian case
Two approaches (TW and ASvM) to derivation of integrable differential
equations for random matrix probabilities are compared. Both methods are
rewritten in such a form that simple and explicit relations between all TW
dependent variables and -functions of ASvM are found, for the example of
finite size Gaussian matrices. Orthogonal function systems and Toda lattice are
seen as the core structure of both approaches and their relationship.Comment: 20 pages, submitted to Journal of Mathematical Physic
Mesoscopic conductance fluctuations in a coupled quantum dot system
We study the transport properties of an Aharonov-Bohm ring containing two
quantum dots. One of the dots has well-separated resonant levels, while the
other is chaotic and is treated by random matrix theory. We find that the
conductance through the ring is significantly affected by mesoscopic
fluctuations. The Breit-Wigner resonant peak is changed to an antiresonance by
increasing the ratio of the level broadening to the mean level spacing of the
random dot. The asymmetric Fano form turns into a symmetric one and the
resonant peak can be controlled by magnetic flux. The conductance distribution
function clearly shows the influence of strong fluctuations.Comment: 4 pages, 4 figures; revised for publicatio
Calculation of some determinants using the s-shifted factorial
Several determinants with gamma functions as elements are evaluated. This
kind of determinants are encountered in the computation of the probability
density of the determinant of random matrices. The s-shifted factorial is
defined as a generalization for non-negative integers of the power function,
the rising factorial (or Pochammer's symbol) and the falling factorial. It is a
special case of polynomial sequence of the binomial type studied in
combinatorics theory. In terms of the gamma function, an extension is defined
for negative integers and even complex values. Properties, mainly composition
laws and binomial formulae, are given. They are used to evaluate families of
generalized Vandermonde determinants with s-shifted factorials as elements,
instead of power functions.Comment: 25 pages; added section 5 for some examples of application
Probability density of determinants of random matrices
In this brief paper the probability density of a random real, complex and
quaternion determinant is rederived using singular values. The behaviour of
suitably rescaled random determinants is studied in the limit of infinite order
of the matrices
Generalization of the Poisson kernel to the superconducting random-matrix ensembles
We calculate the distribution of the scattering matrix at the Fermi level for
chaotic normal-superconducting systems for the case of arbitrary coupling of
the scattering region to the scattering channels. The derivation is based on
the assumption of uniformly distributed scattering matrices at ideal coupling,
which holds in the absence of a gap in the quasiparticle excitation spectrum.
The resulting distribution generalizes the Poisson kernel to the nonstandard
symmetry classes introduced by Altland and Zirnbauer. We show that unlike the
Poisson kernel, our result cannot be obtained by combining the maximum entropy
principle with the analyticity-ergodicity constraint. As a simple application,
we calculate the distribution of the conductance for a single-channel chaotic
Andreev quantum dot in a magnetic field.Comment: 7 pages, 2 figure
Why the Universe Started from a Low Entropy State
We show that the inclusion of backreaction of massive long wavelengths
imposes dynamical constraints on the allowed phase space of initial conditions
for inflation, which results in a superselection rule for the initial
conditions. Only high energy inflation is stable against collapse due to the
gravitational instability of massive perturbations. We present arguments to the
effect that the initial conditions problem {\it cannot} be meaningfully
addressed by thermostatistics as far as the gravitational degrees of freedom
are concerned. Rather, the choice of the initial conditions for the universe in
the phase space and the emergence of an arrow of time have to be treated as a
dynamic selection.Comment: 12 pages, 2 figs. Final version; agrees with accepted version in
Phys. Rev.
Two photon annihilation operators and squeezed vacuum
Inverses of the harmonic oscillator creation and annihilation operators by their actions on the number states are introduced. Three of the two photon annihilation operators, viz., a(sup +/-1)a, aa(sup +/-1), and a(sup 2), have normalizable right eigenstates with nonvanishing eigenvalues. The eigenvalue equation of these operators are discussed and their normalized eigenstates are obtained. The Fock state representation in each case separates into two sets of states, one involving only the even number states while the other involving only the odd number states. It is shown that the even set of eigenstates of the operator a(sup +/-1)a is the customary squeezed vacuum S(sigma) O greater than
Subnormalized states and trace-nonincreasing maps
We investigate the set of completely positive, trace-nonincreasing linear
maps acting on the set M_N of mixed quantum states of size N. Extremal point of
this set of maps are characterized and its volume with respect to the
Hilbert-Schmidt (Euclidean) measure is computed explicitly for an arbitrary N.
The spectra of partially reduced rescaled dynamical matrices associated with
trace-nonincreasing completely positive maps belong to the N-cube inscribed in
the set of subnormalized states of size N. As a by-product we derive the
measure in M_N induced by partial trace of mixed quantum states distributed
uniformly with respect to HS-measure in .Comment: LaTeX, 21 pages, 4 Encapsuled PostScript figures, 1 tabl
Level number variance and spectral compressibility in a critical two-dimensional random matrix model
We study level number variance in a two-dimensional random matrix model
characterized by a power-law decay of the matrix elements. The amplitude of the
decay is controlled by the parameter b. We find analytically that at small
values of b the level number variance behaves linearly, with the
compressibility chi between 0 and 1, which is typical for critical systems. For
large values of b, we derive that chi=0, as one would normally expect in the
metallic phase. Using numerical simulations we determine the critical value of
b at which the transition between these two phases occurs.Comment: 6 page
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