1,042 research outputs found
Optimal construction of k-nearest neighbor graphs for identifying noisy clusters
We study clustering algorithms based on neighborhood graphs on a random
sample of data points. The question we ask is how such a graph should be
constructed in order to obtain optimal clustering results. Which type of
neighborhood graph should one choose, mutual k-nearest neighbor or symmetric
k-nearest neighbor? What is the optimal parameter k? In our setting, clusters
are defined as connected components of the t-level set of the underlying
probability distribution. Clusters are said to be identified in the
neighborhood graph if connected components in the graph correspond to the true
underlying clusters. Using techniques from random geometric graph theory, we
prove bounds on the probability that clusters are identified successfully, both
in a noise-free and in a noisy setting. Those bounds lead to several
conclusions. First, k has to be chosen surprisingly high (rather of the order n
than of the order log n) to maximize the probability of cluster identification.
Secondly, the major difference between the mutual and the symmetric k-nearest
neighbor graph occurs when one attempts to detect the most significant cluster
only.Comment: 31 pages, 2 figure
Blocking and Persistence in the Zero-Temperature Dynamics of Homogeneous and Disordered Ising Models
A ``persistence'' exponent theta has been extensively used to describe the
nonequilibrium dynamics of spin systems following a deep quench: for
zero-temperature homogeneous Ising models on the d-dimensional cubic lattice,
the fraction p(t) of spins not flipped by time t decays to zero like
t^[-theta(d)] for low d; for high d, p(t) may decay to p(infinity)>0, because
of ``blocking'' (but perhaps still like a power). What are the effects of
disorder or changes of lattice? We show that these can quite generally lead to
blocking (and convergence to a metastable configuration) even for low d, and
then present two examples --- one disordered and one homogeneous --- where p(t)
decays exponentially to p(infinity).Comment: 8 pages (LaTeX); to appear in Physical Review Letter
Glauber dynamics on trees:Boundary conditions and mixing time
We give the first comprehensive analysis of the effect of boundary conditions
on the mixing time of the Glauber dynamics in the so-called Bethe
approximation. Specifically, we show that spectral gap and the log-Sobolev
constant of the Glauber dynamics for the Ising model on an n-vertex regular
tree with plus-boundary are bounded below by a constant independent of n at all
temperatures and all external fields. This implies that the mixing time is
O(log n) (in contrast to the free boundary case, where it is not bounded by any
fixed polynomial at low temperatures). In addition, our methods yield simpler
proofs and stronger results for the spectral gap and log-Sobolev constant in
the regime where there are multiple phases but the mixing time is insensitive
to the boundary condition. Our techniques also apply to a much wider class of
models, including those with hard-core constraints like the antiferromagnetic
Potts model at zero temperature (proper colorings) and the hard--core lattice
gas (independent sets)
Dense packing on uniform lattices
We study the Hard Core Model on the graphs
obtained from Archimedean tilings i.e. configurations in with the nearest neighbor 1's forbidden. Our
particular aim in choosing these graphs is to obtain insight to the geometry of
the densest packings in a uniform discrete set-up. We establish density bounds,
optimal configurations reaching them in all cases, and introduce a
probabilistic cellular automaton that generates the legal configurations. Its
rule involves a parameter which can be naturally characterized as packing
pressure. It can have a critical value but from packing point of view just as
interesting are the noncritical cases. These phenomena are related to the
exponential size of the set of densest packings and more specifically whether
these packings are maximally symmetric, simple laminated or essentially random
packings.Comment: 18 page
Zero-Temperature Dynamics of Ising Spin Systems Following a Deep Quench: Results and Open Problems
We consider zero-temperature, stochastic Ising models with nearest-neighbor
interactions and an initial spin configuration chosen from a symmetric
Bernoulli distribution (corresponding physically to a deep quench). Whether a
final state exists, i.e., whether each spin flips only finitely many times as
time goes to infinity (for almost every initial spin configuration and
realization of the dynamics), or if not, whether every spin - or only a
fraction strictly less than one - flips infinitely often, depends on the nature
of the couplings, the dimension, and the lattice type. We review results,
examine open questions, and discuss related topics.Comment: 10 pages (LaTeX); to appear in Physica
The random geometry of equilibrium phases
This is a (long) survey about applications of percolation theory in
equilibrium statistical mechanics. The chapters are as follows:
1. Introduction
2. Equilibrium phases
3. Some models
4. Coupling and stochastic domination
5. Percolation
6. Random-cluster representations
7. Uniqueness and exponential mixing from non-percolation
8. Phase transition and percolation
9. Random interactions
10. Continuum modelsComment: 118 pages. Addresses: [email protected]
http://www.mathematik.uni-muenchen.de/~georgii.html [email protected]
http://www.math.chalmers.se/~olleh [email protected]
A spatial stochastic model for rumor transmission
We consider an interacting particle system representing the spread of a rumor
by agents on the -dimensional integer lattice. Each agent may be in any of
the three states belonging to the set {0,1,2}. Here 0 stands for ignorants, 1
for spreaders and 2 for stiflers. A spreader tells the rumor to any of its
(nearest) ignorant neighbors at rate \lambda. At rate \alpha a spreader becomes
a stifler due to the action of other (nearest neighbor) spreaders. Finally,
spreaders and stiflers forget the rumor at rate one. We study sufficient
conditions under which the rumor either becomes extinct or survives with
positive probability
- …