598,249 research outputs found
Two-dimensional random interlacements and late points for random walks
We define the model of two-dimensional random interlacements using simple
random walk trajectories conditioned on never hitting the origin, and then
obtain some properties of this model. Also, for random walk on a large torus
conditioned on not hitting the origin up to some time proportional to the mean
cover time, we show that the law of the vacant set around the origin is close
to that of random interlacements at the corresponding level. Thus, this new
model provides a way to understand the structure of the set of late points of
the covering process from a microscopic point of view.Comment: Final version, to appear in Commun. Math. Phys. 49 pages, 5 figure
Hausdorff dimension of affine random covering sets in torus
We calculate the almost sure Hausdorff dimension of the random covering set
in -dimensional torus ,
where the sets are parallelepipeds, or more generally,
linear images of a set with nonempty interior, and are
independent and uniformly distributed random points. The dimension formula,
derived from the singular values of the linear mappings, holds provided that
the sequences of the singular values are decreasing.Comment: 16 pages, 1 figur
Choose Outsiders First: a mean 2-approximation random algorithm for covering problems
A high number of discrete optimization problems, including Vertex Cover, Set
Cover or Feedback Vertex Set, can be unified into the class of covering
problems. Several of them were shown to be inapproximable by deterministic
algorithms. This article proposes a new random approach, called Choose
Outsiders First, which consists in selecting randomly ele- ments which are
excluded from the cover. We show that this approach leads to random outputs
which mean size is at most twice the optimal solution.Comment: 8 pages The paper has been withdrawn due to an error in the proo
Understanding Preservation Theorems, II
This is an exposition of much of Sections VI.3 and XVIII.3 of "Proper and
Improper Forcing", including preservations for "no random reals over V", "reals
of V form a non-meager set", "every dense open set contains a dense open set in
V", weak bounding, and weak -bounding. The current version of
part I covering Sections VI.1 and VI.2 is available from the author.Comment: To appear in ML
A Bose-Einstein Approach to the Random Partitioning of an Integer
Consider N equally-spaced points on a circle of circumference N. Choose at
random n points out of on this circle and append clockwise an arc of
integral length k to each such point. The resulting random set is made of a
random number of connected components. Questions such as the evaluation of the
probability of random covering and parking configurations, number and length of
the gaps are addressed. They are the discrete versions of similar problems
raised in the continuum. For each value of k, asymptotic results are presented
when n,N both go to infinity according to two different regimes. This model may
equivalently be viewed as a random partitioning problem of N items into n
recipients. A grand-canonical balls in boxes approach is also supplied, giving
some insight into the multiplicities of the box filling amounts or spacings.
The latter model is a k-nearest neighbor random graph with N vertices and kn
edges. We shall also briefly consider the covering problem in the context of a
random graph model with N vertices and n (out-degree 1) edges whose endpoints
are no more bound to be neighbors
A note on the hitting probabilities of random covering sets
Let be the random covering set on
the torus , where is a sequence of ball-like sets and
is a sequence of independent random variables uniformly distributed on
\T^d. We prove that almost surely whenever
is an analytic set with Hausdorff dimension,
, where is the almost sure Hausdorff dimension of
. Moreover, examples are given to show that the condition on
cannot be replaced by the packing dimension of .Comment: 11 page
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