133 research outputs found
A note on the Gauss decomposition of the elliptic Cauchy matrix
Explicit formulas for the Gauss decomposition of elliptic Cauchy type
matrices are derived in a very simple way. The elliptic Cauchy identity is an
immediate corollary.Comment: 5 page
Particles in a magnetic field and plasma analogies: doubly periodic boundary conditions
The -particle free fermion state for quantum particles in the plane
subject to a perpendicular magnetic field, and with doubly periodic boundary
conditions, is written in a product form. The absolute value of this is used to
formulate an exactly solvable one-component plasma model, and further motivates
the formulation of an exactly solvable two-species Coulomb gas. The large
expansion of the free energy of both these models exhibits the same O(1) term.
On the basis of a relationship to the Gaussian free field, this term is
predicted to be universal for conductive Coulomb systems in doubly periodic
boundary conditions.Comment: 12 page
Entanglement-Saving Channels
The set of Entanglement Saving (ES) quantum channels is introduced and
characterized. These are completely positive, trace preserving transformations
which when acting locally on a bipartite quantum system initially prepared into
a maximally entangled configuration, preserve its entanglement even when
applied an arbitrary number of times. In other words, a quantum channel
is said to be ES if its powers are not entanglement-breaking for all
integers . We also characterize the properties of the Asymptotic
Entanglement Saving (AES) maps. These form a proper subset of the ES channels
that is constituted by those maps which, not only preserve entanglement for all
finite , but which also sustain an explicitly not null level of entanglement
in the asymptotic limit~. Structure theorems are provided
for ES and for AES maps which yield an almost complete characterization of the
former and a full characterization of the latter.Comment: 26 page
Random transition-rate matrices for the master equation
Random-matrix theory is applied to transition-rate matrices in the Pauli
master equation. We study the distribution and correlations of eigenvalues,
which govern the dynamics of complex stochastic systems. Both the cases of
identical and of independent rates of forward and backward transitions are
considered. The first case leads to symmetric transition-rate matrices, whereas
the second corresponds to general, asymmetric matrices. The resulting matrix
ensembles are different from the standard ensembles and show different
eigenvalue distributions. For example, the fraction of real eigenvalues scales
anomalously with matrix dimension in the asymmetric case.Comment: 15 pages, 12 figure
Two-Center Integrals for r_{ij}^{n} Polynomial Correlated Wave Functions
All integrals needed to evaluate the correlated wave functions with
polynomial terms of inter-electronic distance are included. For this form of
the wave function, the integrals needed can be expressed as a product of
integrals involving at most four electrons
Integrable Time-Discretisation of the Ruijsenaars-Schneider Model
An exactly integrable symplectic correspondence is derived which in a
continuum limit leads to the equations of motion of the relativistic
generalization of the Calogero-Moser system, that was introduced for the first
time by Ruijsenaars and Schneider. For the discrete-time model the equations of
motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2
Heisenberg magnet. We present a Lax pair, the symplectic structure and prove
the involutivity of the invariants. Exact solutions are investigated in the
rational and hyperbolic (trigonometric) limits of the system that is given in
terms of elliptic functions. These solutions are connected with discrete
soliton equations. The results obtained allow us to consider the Bethe Ansatz
equations as ones giving an integrable symplectic correspondence mixing the
parameters of the quantum integrable system and the parameters of the
corresponding Bethe wavefunction.Comment: 27 pages, latex, equations.st
Quantum chaos, random matrix theory, and statistical mechanics in two dimensions - a unified approach
We present a theory where the statistical mechanics for dilute ideal gases
can be derived from random matrix approach. We show the connection of this
approach with Srednicki approach which connects Berry conjecture with
statistical mechanics. We further establish a link between Berry conjecture and
random matrix theory, thus providing a unified edifice for quantum chaos,
random matrix theory, and statistical mechanics. In the course of arguing for
these connections, we observe sum rules associated with the outstanding
counting problem in the theory of braid groups. We are able to show that the
presented approach leads to the second law of thermodynamics.Comment: 23 pages, TeX typ
BGWM as Second Constituent of Complex Matrix Model
Earlier we explained that partition functions of various matrix models can be
constructed from that of the cubic Kontsevich model, which, therefore, becomes
a basic elementary building block in "M-theory" of matrix models. However, the
less topical complex matrix model appeared to be an exception: its
decomposition involved not only the Kontsevich tau-function but also another
constituent, which we now identify as the Brezin-Gross-Witten (BGW) partition
function. The BGW tau-function can be represented either as a generating
function of all unitary-matrix integrals or as a Kontsevich-Penner model with
potential 1/X (instead of X^3 in the cubic Kontsevich model).Comment: 42 page
Mixed Hodge polynomials of character varieties
We calculate the E-polynomials of certain twisted GL(n,C)-character varieties
M_n of Riemann surfaces by counting points over finite fields using the
character table of the finite group of Lie-type GL(n,F_q) and a theorem proved
in the appendix by N. Katz. We deduce from this calculation several geometric
results, for example, the value of the topological Euler characteristic of the
associated PGL(n,C)-character variety. The calculation also leads to several
conjectures about the cohomology of M_n: an explicit conjecture for its mixed
Hodge polynomial; a conjectured curious Hard Lefschetz theorem and a conjecture
relating the pure part to absolutely indecomposable representations of a
certain quiver. We prove these conjectures for n = 2.Comment: with an appendix by Nicholas M. Katz; 57 pages. revised version: New
definition for homogeneous weight in Definition 4.1.6, subsequent arguments
modified. Some other minor changes. To appear in Invent. Mat
Controllability on infinite-dimensional manifolds
Following the unified approach of A. Kriegl and P.W. Michor (1997) for a
treatment of global analysis on a class of locally convex spaces known as
convenient, we give a generalization of Rashevsky-Chow's theorem for control
systems in regular connected manifolds modelled on convenient
(infinite-dimensional) locally convex spaces which are not necessarily
normable.Comment: 19 pages, 1 figur
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