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 $\limsup_{n\to\infty}(g_n + \xi_n)$ in $d$-dimensional torus $\mathbb T^d$, where the sets $g_n\subset\mathbb T^d$ are parallelepipeds, or more generally, linear images of a set with nonempty interior, and $\xi_n\in\mathbb T^d$ 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leads to random outputs which mean size is at most twice the optimal solution.Comment: 8 pages The paper has been withdrawn due to an error in the proo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 $\omega^\omega$-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 $N$ 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 $E=\limsup\limits_{n\to\infty}(g_n+\xi_n)$ be the random covering set on the torus $\mathbb{T}^d$, where $\{g_n\}$ is a sequence of ball-like sets and $\xi_n$ is a sequence of independent random variables uniformly distributed on \T^d. We prove that $E\cap F\neq\emptyset$ almost surely whenever $F\subset\mathbb{T}^d$ is an analytic set with Hausdorff dimension, $\dim_H(F)>d-\alpha$, where $\alpha$ is the almost sure Hausdorff dimension of $E$. Moreover, examples are given to show that the condition on $\dim_H(F)$ cannot be replaced by the packing dimension of $F$.Comment: 11 page