20,019 research outputs found
Law of large numbers for critical first-passage percolation on the triangular lattice
We study the site version of (independent) first-passage percolation on the
triangular lattice . Denote the passage time of the site in
by , and assume that . Denote by
the passage time from to , and by the
passage time from to the halfplane . We prove
that there exists a constant such that as ,
in probability and almost surely. This result confirms a prediction of Kesten and Zhang
(Probab. Theory Relat. Fields \textbf{107}: 137--160, 1997). The proof relies
on the existence of the full scaling limit of critical site percolation on
, established by Camia and Newman.Comment: 14 pages, 2 figure
Limit theorems for critical first-passage percolation on the triangular lattice
Consider (independent) first-passage percolation on the sites of the
triangular lattice . Denote the passage time of the site in
by , and assume that . Denote by
the passage time from 0 to the halfplane
\{v\in\mathbb{T}:\mbox{Re}(v)\geq n\}, and by the passage time from
0 to the nearest site to , where . We prove that as
, a.s.,
and Var; in probability but not a.s., and Var.
This answers a question of Kesten and Zhang (1997) and improves our previous
work (2014). From this result, we derive an explicit form of the central limit
theorem for and . A key ingredient for the proof is the
moment generating function of the conformal radii for conformal loop ensemble
CLE, given by Schramm, Sheffield and Wilson (2009).Comment: 18 pages, 1 figur
A CLT for winding angles of the arms for critical planar percolation
Consider critical percolation in two dimensions. Under the condition that
there are k disjoint alternating black and white arms crossing the annulus
A(l,n), we prove a central limit theorem and variance estimates for the winding
angles of the arms (as n\rightarrow \infty, l fixed). This result confirms a
prediction of Beffara and Nolin (Ann. Probab. 39: 1286--1304, 2011). Using this
theorem, we also get a CLT for the multiple-armed incipient infinite cluster
(IIC) measures.Comment: 20 pages, 3 figure
Asymptotics for 2D critical and near-critical first-passage percolation
We study Bernoulli first-passage percolation (FPP) on the triangular lattice
in which sites have 0 and 1 passage times with probability and
, respectively. Denote by the infinite cluster
with 0-time sites when , where is the critical probability.
Denote by the passage time from the origin 0 to
. First we obtain explicit limit theorem for
as . The proof relies on the limit
theorem in the critical case, the critical exponent for correlation length and
Kesten's scaling relations. Next, for the usual point-to-point passage time
in the critical case, we construct subsequences of sites with
different growth rate along the axis. The main tool involves the large
deviation estimates on the nesting of CLE loops derived by Miller, Watson
and Wilson (2016). Finally, we apply the limit theorem for critical Bernoulli
FPP to a random graph called cluster graph, obtaining explicit strong law of
large numbers for graph distance.Comment: 35 pages, 3 figure
Multi-arm incipient infinite clusters in 2D: scaling limits and winding numbers
We study the alternating -arm incipient infinite cluster (IIC) of site
percolation on the triangular lattice . Using Camia and Newman's
result that the scaling limit of critical site percolation on is
CLE, we prove the existence of the scaling limit of the -arm IIC for
. Conditioned on the event that there are open and closed arms
connecting the origin to , we show that the winding
number variance of the arms is as ,
which confirms a prediction of Wieland and Wilson (2003). Our proof uses
two-sided radial SLE and coupling argument. Using this result we get an
explicit form for the CLT of the winding numbers, and get analogous result for
the 2-arm IIC, thus improving our earlier result.Comment: 38 pages, 3 figures. arXiv admin note: text overlap with
arXiv:math/0605035 by other author
Complex Linear Physical-Layer Network Coding
This paper presents the results of a comprehensive investigation of complex
linear physical-layer network (PNC) in two-way relay channels (TWRC). A
critical question at relay R is as follows: "Given channel gain ratio , where and are the complex channel gains from nodes A and
B to relay R, respectively, what is the optimal coefficients
that minimizes the symbol error rate (SER) of when
we attempt to detect in the presence of noise?" Our contributions with
respect to this question are as follows: (1) We put forth a general
Gaussian-integer formulation for complex linear PNC in which , and are elements of a finite field of Gaussian integers, that is,
the field of where is a Gaussian prime. Previous vector
formulation, in which , , and were represented by
-dimensional vectors and and were represented by matrices, corresponds to a subcase of our Gaussian-integer formulation where
is real prime only. Extension to Gaussian prime , where can be
complex, gives us a larger set of signal constellations to achieve different
rates at different SNR. (2) We show how to divide the complex plane of
into different Voronoi regions such that the within each Voronoi region
share the same optimal PNC mapping . We uncover the
structure of the Voronoi regions that allows us to compute a minimum-distance
metric that characterizes the SER of under optimal PNC mapping
. Overall, the contributions in (1) and (2) yield a
toolset for a comprehensive understanding of complex linear PNC in
. We believe investigation of linear PNC beyond
can follow the same approach.Comment: submitted to IEEE Transactions on Information Theor
On bifurcation of eigenvalues along convex symplectic paths
We consider a continuously differentiable curve in the
space of real symplectic matrices, which is the solution of the
following ODE:
,
where and is a continuous in the space of real matrices which are
symmetric. Under certain convexity assumption (which includes the particular
case that is strictly positive definite for all ), we
investigate the dynamics of the eigenvalues of when varies,
which are closely related to the stability of such Hamiltonian dynamical
systems. We rigorously prove the qualitative behavior of the branching of
eigenvalues and explicitly give the first order asymptotics of the eigenvalues.
This generalizes classical Krein-Lyubarskii theorem on the analytic bifurcation
of the Floquet multipliers under a linear perturbation of the Hamiltonian. As a
corollary, we give a rigorous proof of the following statement of Ekeland:
is a discrete set.Comment: 8 figure
Dynamics of domain wall in charged AdS dilaton black hole spacetime
For the dimensional FRW domain wall universe induced by dimensional
charged dilaton black hole, its movement formula in the bulk can be rewrite as
the expansion or collapsing of domain wall. By analysing, we found that in this
static AdS space, the cosmologic behaviour of domain wall is particularly
single. Even more surprising, it exists an anomaly that the domain wall has a
motion area outside of horizon, in which it cannot be explained by our
classical theory.Comment: 6 pages, 10figure
Scalable Bayesian Variable Selection for Structured High-dimensional Data
Variable selection for structured covariates lying on an underlying known
graph is a problem motivated by practical applications, and has been a topic of
increasing interest. However, most of the existing methods may not be scalable
to high dimensional settings involving tens of thousands of variables lying on
known pathways such as the case in genomics studies. We propose an adaptive
Bayesian shrinkage approach which incorporates prior network information by
smoothing the shrinkage parameters for connected variables in the graph, so
that the corresponding coefficients have a similar degree of shrinkage. We fit
our model via a computationally efficient expectation maximization algorithm
which scalable to high dimensional settings (p~100,000). Theoretical properties
for fixed as well as increasing dimensions are established, even when the
number of variables increases faster than the sample size. We demonstrate the
advantages of our approach in terms of variable selection, prediction, and
computational scalability via a simulation study, and apply the method to a
cancer genomics study
A self-organized particle moving model on scale free network with behavior
In this paper we propose a self-organized particle moving model on scale free
network with the algorithm of the shortest path and preferential walk. The
over-capacity property of the vertices in this particle moving system on
complex network is studied from the holistic point of view. Simulation results
show that the number of over-capacity vertices forms punctuated equilibrium
processes as time elapsing, that the average number of over-capacity vertices
under each local punctuated equilibrium process has power law relationship with
the local punctuated equilibrium value. What's more, the number of
over-capacity vertices has the bell-shaped temporal correlation and
behavior. Finally, the average lifetime of particles accumulated before
time is analyzed to find the different roles of the shortest path algorithm
and the preferential walk algorithm in our model.Comment: 8 pages, 5 figure
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