444 research outputs found
Directed percolation effects emerging from superadditivity of quantum networks
Entanglement indcued non--additivity of classical communication capacity in
networks consisting of quantum channels is considered. Communication lattices
consisiting of butterfly-type entanglement breaking channels augmented, with
some probability, by identity channels are analyzed. The capacity
superadditivity in the network is manifested in directed correlated bond
percolation which we consider in two flavours: simply directed and randomly
oriented. The obtained percolation properties show that high capacity
information transfer sets in much faster in the regime of superadditive
communication capacity than otherwise possible. As a byproduct, this sheds
light on a new type of entanglement based quantum capacity percolation
phenomenon.Comment: 6 pages, 4 figure
Non-Existence of Positive Stationary Solutions for a Class of Semi-Linear PDEs with Random Coefficients
We consider a so-called random obstacle model for the motion of a
hypersurface through a field of random obstacles, driven by a constant driving
field. The resulting semi-linear parabolic PDE with random coefficients does
not admit a global nonnegative stationary solution, which implies that an
interface that was flat originally cannot get stationary. The absence of global
stationary solutions is shown by proving lower bounds on the growth of
stationary solutions on large domains with Dirichlet boundary conditions.
Difficulties arise because the random lower order part of the equation cannot
be bounded uniformly
Self-avoiding walks and connective constants
The connective constant of a quasi-transitive graph is the
asymptotic growth rate of the number of self-avoiding walks (SAWs) on from
a given starting vertex. We survey several aspects of the relationship between
the connective constant and the underlying graph .
We present upper and lower bounds for in terms of the
vertex-degree and girth of a transitive graph.
We discuss the question of whether for transitive
cubic graphs (where denotes the golden mean), and we introduce the
Fisher transformation for SAWs (that is, the replacement of vertices by
triangles).
We present strict inequalities for the connective constants
of transitive graphs , as varies.
As a consequence of the last, the connective constant of a Cayley
graph of a finitely generated group decreases strictly when a new relator is
added, and increases strictly when a non-trivial group element is declared to
be a further generator.
We describe so-called graph height functions within an account of
"bridges" for quasi-transitive graphs, and indicate that the bridge constant
equals the connective constant when the graph has a unimodular graph height
function.
A partial answer is given to the question of the locality of
connective constants, based around the existence of unimodular graph height
functions.
Examples are presented of Cayley graphs of finitely presented
groups that possess graph height functions (that are, in addition, harmonic and
unimodular), and that do not.
The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with
arXiv:1304.721
Percolation with Multiple Giant Clusters
We study the evolution of percolation with freezing. Specifically, we
consider cluster formation via two competing processes: irreversible
aggregation and freezing. We find that when the freezing rate exceeds a certain
threshold, the percolation transition is suppressed. Below this threshold, the
system undergoes a series of percolation transitions with multiple giant
clusters ("gels") formed. Giant clusters are not self-averaging as their total
number and their sizes fluctuate from realization to realization. The size
distribution F_k, of frozen clusters of size k, has a universal tail, F_k ~
k^{-3}. We propose freezing as a practical mechanism for controlling the gel
size.Comment: 4 pages, 3 figure
Strict inequalities of critical values in continuum percolation
We consider the supercritical finite-range random connection model where the
points of a homogeneous planar Poisson process are connected with
probability for a given . Performing percolation on the resulting
graph, we show that the critical probabilities for site and bond percolation
satisfy the strict inequality . We also show
that reducing the connection function strictly increases the critical
Poisson intensity. Finally, we deduce that performing a spreading
transformation on (thereby allowing connections over greater distances but
with lower probabilities, leaving average degrees unchanged) {\em strictly}
reduces the critical Poisson intensity. This is of practical relevance,
indicating that in many real networks it is in principle possible to exploit
the presence of spread-out, long range connections, to achieve connectivity at
a strictly lower density value.Comment: 38 pages, 8 figure
Vertex Models and Random Labyrinths: Phase Diagrams for Ice-type Vertex Models
We propose a simple geometric recipe for constructing phase diagrams for a
general class of vertex models obeying the ice rule. The disordered phase maps
onto the intersecting loop model which is interesting in its own right and is
related to several other statistical mechanical models. This mapping is also
useful in understanding some ordered phases of these vertex models as they
correspond to the polymer loop models with cross-links in their vulcanised
phase.Comment: 8 pages, 6 figure
Bond percolation on isoradial graphs: criticality and universality
In an investigation of percolation on isoradial graphs, we prove the
criticality of canonical bond percolation on isoradial embeddings of planar
graphs, thus extending celebrated earlier results for homogeneous and
inhomogeneous square, triangular, and other lattices. This is achieved via the
star-triangle transformation, by transporting the box-crossing property across
the family of isoradial graphs. As a consequence, we obtain the universality of
these models at the critical point, in the sense that the one-arm and
2j-alternating-arm critical exponents (and therefore also the connectivity and
volume exponents) are constant across the family of such percolation processes.
The isoradial graphs in question are those that satisfy certain weak conditions
on their embedding and on their track system. This class of graphs includes,
for example, isoradial embeddings of periodic graphs, and graphs derived from
rhombic Penrose tilings.Comment: In v2: extended title, and small changes in the tex
Electron Transport through Disordered Domain Walls: Coherent and Incoherent Regimes
We study electron transport through a domain wall in a ferromagnetic nanowire
subject to spin-dependent scattering. A scattering matrix formalism is
developed to address both coherent and incoherent transport properties. The
coherent case corresponds to elastic scattering by static defects, which is
dominant at low temperatures, while the incoherent case provides a
phenomenological description of the inelastic scattering present in real
physical systems at room temperature. It is found that disorder scattering
increases the amount of spin-mixing of transmitted electrons, reducing the
adiabaticity. This leads, in the incoherent case, to a reduction of conductance
through the domain wall as compared to a uniformly magnetized region which is
similar to the giant magnetoresistance effect. In the coherent case, a
reduction of weak localization, together with a suppression of spin-reversing
scattering amplitudes, leads to an enhancement of conductance due to the domain
wall in the regime of strong disorder. The total effect of a domain wall on the
conductance of a nanowire is studied by incorporating the disordered regions on
either side of the wall. It is found that spin-dependent scattering in these
regions increases the domain wall magnetoconductance as compared to the effect
found by considering only the scattering inside the wall. This increase is most
dramatic in the narrow wall limit, but remains significant for wide walls.Comment: 23 pages, 12 figure
First Passage Properties of the Erdos-Renyi Random Graph
We study the mean time for a random walk to traverse between two arbitrary
sites of the Erdos-Renyi random graph. We develop an effective medium
approximation that predicts that the mean first-passage time between pairs of
nodes, as well as all moments of this first-passage time, are insensitive to
the fraction p of occupied links. This prediction qualitatively agrees with
numerical simulations away from the percolation threshold. Near the percolation
threshold, the statistically meaningful quantity is the mean transit rate,
namely, the inverse of the first-passage time. This rate varies
non-monotonically with p near the percolation transition. Much of this behavior
can be understood by simple heuristic arguments.Comment: 10 pages, 9 figures, 2-column revtex4 forma
Quantum site percolation on amenable graphs
We consider the quantum site percolation model on graphs with an amenable
group action. It consists of a random family of Hamiltonians. Basic spectral
properties of these operators are derived: non-randomness of the spectrum and
its components, existence of an self-averaging integrated density of states and
an associated trace-formula.Comment: 10 pages, LaTeX 2e, to appear in "Applied Mathematics and Scientific
Computing", Brijuni, June 23-27, 2003. by Kluwer publisher
- …