52,361 research outputs found
Trans-Planckian Issues for Inflationary Cosmology
The accelerated expansion of space during the period of cosmological
inflation leads to trans-Planckian issues which need to be addressed. Most
importantly, the physical wavelength of fluctuations which are studied at the
present time by means of cosmological observations may well originate with a
wavelength smaller than the Planck length at the beginning of the inflationary
phase. Thus, questions arise as to whether the usual predictions of
inflationary cosmology are robust considering our ignorance of physics on
trans-Planckian scales, and whether the imprints of Planck-scale physics are at
the present time observable. These and other related questions are reviewed in
this article.Comment: 32 pages, 11 figures; invited review for "Classical and Quantum
Gravity
The DUNE-ALUGrid Module
In this paper we present the new DUNE-ALUGrid module. This module contains a
major overhaul of the sources from the ALUgrid library and the binding to the
DUNE software framework. The main changes include user defined load balancing,
parallel grid construction, and an redesign of the 2d grid which can now also
be used for parallel computations. In addition many improvements have been
introduced into the code to increase the parallel efficiency and to decrease
the memory footprint.
The original ALUGrid library is widely used within the DUNE community due to
its good parallel performance for problems requiring local adaptivity and
dynamic load balancing. Therefore, this new model will benefit a number of DUNE
users. In addition we have added features to increase the range of problems for
which the grid manager can be used, for example, introducing a 3d tetrahedral
grid using a parallel newest vertex bisection algorithm for conforming grid
refinement. In this paper we will discuss the new features, extensions to the
DUNE interface, and explain for various examples how the code is used in
parallel environments.Comment: 25 pages, 11 figure
Exponential tail bounds for loop-erased random walk in two dimensions
Let be the number of steps of the loop-erasure of a simple random walk
on from the origin to the circle of radius . We relate the
moments of to , the probability that a random walk and an
independent loop-erased random walk both started at the origin do not intersect
up to leaving the ball of radius . This allows us to show that there exists
such that for all and all and hence to establish exponential moment bounds for
. This implies that there exists such that for all and all
,
Using similar techniques, we then establish a second moment result for a
specific conditioned random walk which enables us to prove that for any
such that for all and ,
Comment: Published in at http://dx.doi.org/10.1214/10-AOP539 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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