590 research outputs found
Anomalous heat-kernel decay for random walk among bounded random conductances
We consider the nearest-neighbor simple random walk on , ,
driven by a field of bounded random conductances . The
conductance law is i.i.d. subject to the condition that the probability of
exceeds the threshold for bond percolation on . For
environments in which the origin is connected to infinity by bonds with
positive conductances, we study the decay of the -step return probability
. We prove that is bounded by a random
constant times in , while it is in and
in . By producing examples with anomalous heat-kernel
decay approaching we prove that the bound in is the
best possible. We also construct natural -dependent environments that
exhibit the extra factor in . See also math.PR/0701248.Comment: 22 pages. Includes a self-contained proof of isoperimetric inequality
for supercritical percolation clusters. Version to appear in AIHP +
additional correction
Orbital order in classical models of transition-metal compounds
We study the classical 120-degree and related orbital models. These are the
classical limits of quantum models which describe the interactions among
orbitals of transition-metal compounds. We demonstrate that at low temperatures
these models exhibit a long-range order which arises via an "order by disorder"
mechanism. This strongly indicates that there is orbital ordering in the
quantum version of these models, notwithstanding recent rigorous results on the
absence of spin order in these systems.Comment: 7 pages, 1 eps fi
The Two-Screen Measurement Setup to Indirectly Measure Proton Beam Self-Modulation in AWAKE
The goal of the first phase of the AWAKE \cite{AWAKE1,AWAKE2} experiment at
CERN is to measure the self-modulation \cite{SMI} of the long SPS proton bunch into microbunches after traversing
of plasma with a plasma density of
. The two screen measurement setup
\cite{Turner2016} is a proton beam diagnostic that can indirectly prove the
successful development of the self-modulation of the proton beam by imaging
protons that got defocused by the transverse plasma wakefields after passing
through the plasma, at two locations downstream the end of the plasma. This
article describes the design and realization of the two screen measurement
setup integrated in the AWAKE experiment. We discuss the performance and
background response of the system based on measurements performed with an
unmodulated Gaussian SPS proton bunch during the AWAKE beam commissioning in
September and October 2016. We show that the system is fully commissioned and
adapted to eventually image the full profile of a self-modulated SPS proton
bunch in a single shot measurement during the first phase of the AWAKE
experiment.Comment: 5 pages 8 figure
A quantitative central limit theorem for the effective conductance on the discrete torus
We study a random conductance problem on a -dimensional discrete torus of
size . The conductances are independent, identically distributed random
variables uniformly bounded from above and below by positive constants. The
effective conductance of the network is a random variable, depending on
, and the main result is a quantitative central limit theorem for this
quantity as . In terms of scalings we prove that this nonlinear
nonlocal function essentially behaves as if it were a simple spatial
average of the conductances (up to logarithmic corrections). The main
achievement of this contribution is the precise asymptotic description of the
variance of .Comment: 37 page
Indirect Self-Modulation Instability Measurement Concept for the AWAKE Proton Beam
AWAKE, the Advanced Proton-Driven Plasma Wakefield Acceleration Experiment,
is a proof-of-principle R&D experiment at CERN using a 400 GeV/c proton beam
from the CERN SPS (longitudinal beam size sigma_z = 12 cm) which will be sent
into a 10 m long plasma section with a nominal density of approx. 7x10^14
atoms/cm3 (plasma wavelength lambda_p = 1.2mm). In this paper we show that by
measuring the time integrated transverse profile of the proton bunch at two
locations downstream of the AWAKE plasma, information about the occurrence of
the self-modulation instability (SMI) can be inferred. In particular we show
that measuring defocused protons with an angle of 1 mrad corresponds to having
electric fields in the order of GV/m and fully developed self-modulation of the
proton bunch. Additionally, by measuring the defocused beam edge of the
self-modulated bunch, information about the growth rate of the instability can
be extracted. If hosing instability occurs, it could be detected by measuring a
non-uniform defocused beam shape with changing radius. Using a 1 mm thick
Chromox scintillation screen for imaging of the self-modulated proton bunch, an
edge resolution of 0.6 mm and hence a SMI saturation point resolution of 1.2 m
can be achieved.Comment: 4 pages, 4 figures, EAAC conference proceeding
Colligative properties of solutions: I. Fixed concentrations
Using the formalism of rigorous statistical mechanics, we study the phenomena
of phase separation and freezing-point depression upon freezing of solutions.
Specifically, we devise an Ising-based model of a solvent-solute system and
show that, in the ensemble with a fixed amount of solute, a macroscopic phase
separation occurs in an interval of values of the chemical potential of the
solvent. The boundaries of the phase separation domain in the phase diagram are
characterized and shown to asymptotically agree with the formulas used in
heuristic analyses of freezing point depression. The limit of infinitesimal
concentrations is described in a subsequent paper.Comment: 28 pages, 1 fig; see also math-ph/0407035 (both to appear in JSP
Superconductivity and charge carrier localization in ultrathin bilayers
/ (LSCO15/LCO) bilayers
with a precisely controlled thickness of N unit cells (UCs) of the former and M
UCs of the latter ([LSCO15\_N/LCO\_M]) were grown on (001)-oriented {\slao}
(SLAO) substrates with pulsed laser deposition (PLD). X-ray diffraction and
reciprocal space map (RSM) studies confirmed the epitaxial growth of the
bilayers and showed that a [LSCO15\_2/LCO\_2] bilayer is fully strained,
whereas a [LSCO15\_2/LCO\_7] bilayer is already partially relaxed. The
\textit{in situ} monitoring of the growth with reflection high energy electron
diffraction (RHEED) revealed that the gas environment during deposition has a
surprisingly strong effect on the growth mode and thus on the amount of
disorder in the first UC of LSCO15 (or the first two monolayers of LSCO15
containing one plane each). For samples grown in pure
gas (growth type-B), the first LSCO15 UC next to the SLAO
substrate is strongly disordered. This disorder is strongly reduced if the
growth is performed in a mixture of and gas
(growth type-A). Electric transport measurements confirmed that the first UC of
LSCO15 next to the SLAO substrate is highly resistive and shows no sign of
superconductivity for growth type-B, whereas it is superconducting for growth
type-A. Furthermore, we found, rather surprisingly, that the conductivity of
the LSCO15 UC next to the LCO capping layer strongly depends on the thickness
of the latter. A LCO capping layer with 7~UCs leads to a strong localization of
the charge carriers in the adjacent LSCO15 UC and suppresses superconductivity.
The magneto-transport data suggest a similarity with the case of weakly hole
doped LSCO single crystals that are in a so-called {"{cluster-spin-glass
state}"
Optimal designs for rational function regression
We consider optimal non-sequential designs for a large class of (linear and
nonlinear) regression models involving polynomials and rational functions with
heteroscedastic noise also given by a polynomial or rational weight function.
The proposed method treats D-, E-, A-, and -optimal designs in a
unified manner, and generates a polynomial whose zeros are the support points
of the optimal approximate design, generalizing a number of previously known
results of the same flavor. The method is based on a mathematical optimization
model that can incorporate various criteria of optimality and can be solved
efficiently by well established numerical optimization methods. In contrast to
previous optimization-based methods proposed for similar design problems, it
also has theoretical guarantee of its algorithmic efficiency; in fact, the
running times of all numerical examples considered in the paper are negligible.
The stability of the method is demonstrated in an example involving high degree
polynomials. After discussing linear models, applications for finding locally
optimal designs for nonlinear regression models involving rational functions
are presented, then extensions to robust regression designs, and trigonometric
regression are shown. As a corollary, an upper bound on the size of the support
set of the minimally-supported optimal designs is also found. The method is of
considerable practical importance, with the potential for instance to impact
design software development. Further study of the optimality conditions of the
main optimization model might also yield new theoretical insights.Comment: 25 pages. Previous version updated with more details in the theory
and additional example
On the formation/dissolution of equilibrium droplets
We consider liquid-vapor systems in finite volume at parameter
values corresponding to phase coexistence and study droplet formation due to a
fixed excess of particles above the ambient gas density. We identify
a dimensionless parameter and a
\textrm{universal} value \Deltac=\Deltac(d), and show that a droplet of the
dense phase occurs whenever \Delta>\Deltac, while, for \Delta<\Deltac, the
excess is entirely absorbed into the gaseous background. When the droplet first
forms, it comprises a non-trivial, \textrm{universal} fraction of excess
particles. Similar reasoning applies to generic two-phase systems at phase
coexistence including solid/gas--where the ``droplet'' is crystalline--and
polymorphic systems. A sketch of a rigorous proof for the 2D Ising lattice gas
is presented; generalizations are discussed heuristically.Comment: An announcement of a forthcoming rigorous work on the 2D Ising model;
to appear in Europhys. Let
Monte Carlo study of the evaporation/condensation transition on different Ising lattices
In 2002 Biskup et al. [Europhys. Lett. 60, 21 (2002)] sketched a rigorous
proof for the behavior of the 2D Ising lattice gas, at a finite volume and a
fixed excess \delta M of particles (spins) above the ambient gas density
(spontaneous magnetisation). By identifying a dimensionless parameter \Delta
(\delta M) and a universal constant \Delta_c, they showed in the limit of large
system sizes that for \Delta < \Delta_c the excess is absorbed in the
background (``evaporated'' system), while for \Delta > \Delta_c a droplet of
the dense phase occurs (``condensed'' system).
To check the applicability of the analytical results to much smaller,
practically accessible system sizes, we performed several Monte Carlo
simulations for the 2D Ising model with nearest-neighbour couplings on a square
lattice at fixed magnetisation M. Thereby, we measured the largest minority
droplet, corresponding to the condensed phase, at various system sizes (L=40,
>..., 640). With analytic values for for the spontaneous magnetisation m_0, the
susceptibility \chi and the Wulff interfacial free energy density \tau_W for
the infinite system, we were able to determine \lambda numerically in very good
agreement with the theoretical prediction.
Furthermore, we did simulations for the spin-1/2 Ising model on a triangular
lattice and with next-nearest-neighbour couplings on a square lattice. Again,
finding a very good agreement with the analytic formula, we demonstrate the
universal aspects of the theory with respect to the underlying lattice. For the
case of the next-nearest-neighbour model, where \tau_W is unknown analytically,
we present different methods to obtain it numerically by fitting to the
distribution of the magnetisation density P(m).Comment: 14 pages, 17 figures, 1 tabl
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