1,189 research outputs found
Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile
The rotor-router model is a deterministic analogue of random walk. It can be
used to define a deterministic growth model analogous to internal DLA. We prove
that the asymptotic shape of this model is a Euclidean ball, in a sense which
is stronger than our earlier work. For the shape consisting of
sites, where is the volume of the unit ball in , we show that
the inradius of the set of occupied sites is at least , while the
outradius is at most for any . For a related
model, the divisible sandpile, we show that the domain of occupied sites is a
Euclidean ball with error in the radius a constant independent of the total
mass. For the classical abelian sandpile model in two dimensions, with particles, we show that the inradius is at least , and the
outradius is at most . This improves on bounds of Le Borgne
and Rossin. Similar bounds apply in higher dimensions.Comment: [v3] Added Theorem 4.1, which generalizes Theorem 1.4 for the abelian
sandpile. [v4] Added references and improved exposition in sections 2 and 4.
[v5] Final version, to appear in Potential Analysi
Colloid-Facilitated Radionuclide Transport in Fractured Carbonate Rock from Yucca Flat, Nevada National Security Site
On the freezing of variables in random constraint satisfaction problems
The set of solutions of random constraint satisfaction problems (zero energy
groundstates of mean-field diluted spin glasses) undergoes several structural
phase transitions as the amount of constraints is increased. This set first
breaks down into a large number of well separated clusters. At the freezing
transition, which is in general distinct from the clustering one, some
variables (spins) take the same value in all solutions of a given cluster. In
this paper we study the critical behavior around the freezing transition, which
appears in the unfrozen phase as the divergence of the sizes of the
rearrangements induced in response to the modification of a variable. The
formalism is developed on generic constraint satisfaction problems and applied
in particular to the random satisfiability of boolean formulas and to the
coloring of random graphs. The computation is first performed in random tree
ensembles, for which we underline a connection with percolation models and with
the reconstruction problem of information theory. The validity of these results
for the original random ensembles is then discussed in the framework of the
cavity method.Comment: 32 pages, 7 figure
Mutation, selection, and ancestry in branching models: a variational approach
We consider the evolution of populations under the joint action of mutation
and differential reproduction, or selection. The population is modelled as a
finite-type Markov branching process in continuous time, and the associated
genealogical tree is viewed both in the forward and the backward direction of
time. The stationary type distribution of the reversed process, the so-called
ancestral distribution, turns out as a key for the study of mutation-selection
balance. This balance can be expressed in the form of a variational principle
that quantifies the respective roles of reproduction and mutation for any
possible type distribution. It shows that the mean growth rate of the
population results from a competition for a maximal long-term growth rate, as
given by the difference between the current mean reproduction rate, and an
asymptotic decay rate related to the mutation process; this tradeoff is won by
the ancestral distribution.
Our main application is the quasispecies model of sequence evolution with
mutation coupled to reproduction but independent across sites, and a fitness
function that is invariant under permutation of sites. Here, the variational
principle is worked out in detail and yields a simple, explicit result.Comment: 45 pages,8 figure
Liquid ionization chamber calibrated gel dosimetry in conformal stereotactic radiotherapy of brain lesions
Determination of the Bending Rigidity of Graphene via Electrostatic Actuation of Buckled Membranes
The small mass and atomic-scale thickness of graphene membranes make them
highly suitable for nanoelectromechanical devices such as e.g. mass sensors,
high frequency resonators or memory elements. Although only atomically thick,
many of the mechanical properties of graphene membranes can be described by
classical continuum mechanics. An important parameter for predicting the
performance and linearity of graphene nanoelectromechanical devices as well as
for describing ripple formation and other properties such as electron
scattering mechanisms, is the bending rigidity, {\kappa}. In spite of the
importance of this parameter it has so far only been estimated indirectly for
monolayer graphene from the phonon spectrum of graphite, estimated from AFM
measurements or predicted from ab initio calculations or bond-order potential
models. Here, we employ a new approach to the experimental determination of
{\kappa} by exploiting the snap-through instability in pre-buckled graphene
membranes. We demonstrate the reproducible fabrication of convex buckled
graphene membranes by controlling the thermal stress during the fabrication
procedure and show the abrupt switching from convex to concave geometry that
occurs when electrostatic pressure is applied via an underlying gate electrode.
The bending rigidity of bilayer graphene membranes under ambient conditions was
determined to be eV. Monolayers have significantly lower
{\kappa} than bilayers
Scaling Limits for Internal Aggregation Models with Multiple Sources
We study the scaling limits of three different aggregation models on Z^d:
internal DLA, in which particles perform random walks until reaching an
unoccupied site; the rotor-router model, in which particles perform
deterministic analogues of random walks; and the divisible sandpile, in which
each site distributes its excess mass equally among its neighbors. As the
lattice spacing tends to zero, all three models are found to have the same
scaling limit, which we describe as the solution to a certain PDE free boundary
problem in R^d. In particular, internal DLA has a deterministic scaling limit.
We find that the scaling limits are quadrature domains, which have arisen
independently in many fields such as potential theory and fluid dynamics. Our
results apply both to the case of multiple point sources and to the
Diaconis-Fulton smash sum of domains.Comment: 74 pages, 4 figures, to appear in J. d'Analyse Math. Main changes in
v2: added "least action principle" (Lemma 3.2); small corrections in section
4, and corrected the proof of Lemma 5.3 (Lemma 5.4 in the new version);
expanded section 6.
On unified-entropy characterization of quantum channels
We consider properties of quantum channels with use of unified entropies.
Extremal unravelings of quantum channel with respect to these entropies are
examined. The concept of map entropy is extended in terms of the unified
entropies. The map -entropy is naturally defined as the unified
-entropy of rescaled dynamical matrix of given quantum channel.
Inequalities of Fannes type are obtained for introduced entropies in terms of
both the trace and Frobenius norms of difference between corresponding
dynamical matrices. Additivity properties of introduced map entropies are
discussed. The known inequality of Lindblad with the entropy exchange is
generalized to many of the unified entropies. For tensor product of a pair of
quantum channels, we derive two-sided estimating of the output entropy of a
maximally entangled input state.Comment: 12 pages, no figures. Typos are fixed. One lemma is extended and
removed to Appendi
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