Related papers: Mixed thresholds in the Lonely Runner Conjecture

We study one-dimensional excited random walks with non-nearest neighbor jumps. When the process is at a vertex that has not been visited before, its next transition has a positive drift to the right, possibly with long jumps. Whenever the…

Motivated by the search for a counterexample to the Poincar\'e conjecture in three and four dimensions, the Andrews-Curtis conjecture was proposed in 1965. It is now generally suspected that the Andrews-Curtis conjecture is false, but small…

In this paper we study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball. We show that among all such paths,…

Under suitable moment assumptions, we show that a genuinely d-dimensional step-reinforced random walk undergoes a phase transition between recurrence and transience in dimensions $d=1,2$, and that it is transient for all reinforcement…

This work proposes a distance that combines Minkowski and Chebyshev distances and can be seen as an intermediary distance. This combination not only achieves efficient run times in neighbourhood iteration tasks in Z^2, but also obtains good…

Let $S$ be a set of points in $\mathbb{R}^2$ contained in a circle and $P$ an unrestricted point set in $\mathbb{R}^2$. We prove the number of distinct distances between points in $S$ and points in $P$ is at least…

Consider a sequence of independent random isometries of Euclidean space with a previously fixed probability law. Apply these isometries successively to the origin and consider the sequence of random points that we obtain this way. We prove…

Let $d_1\leq d_2\leq\ldots\leq d_{n\choose 2}$ denote the distances determined by $n$ points in the plane. It is shown that $\min\sum_i (d_{i+1}-d_i)^2=O(n^{-6/7})$, where the minimum is taken over all point sets with minimal distance $d_1…

We show that if $A=\{a_1,a_2,..., a_k\}$ is a monotone increasing set of numbers, and the differences of the consecutive elements are all distinct, then $|A+B|\geq c|A|^{1/2}|B|$ for any finite set of numbers $B$. The bound is tight up to…

We consider a Branching Random Walk on $\R$ whose step size decreases by a fixed factor, $0<b<1$, with each turn. This process generates a random probability measure on $\R$, that is, the limit of uniform distribution among the $2^n$…

We define a dynamical simple symmetric random walk in one dimension, and show that there almost surely exist exceptional times at which the walk tends to infinity. This is in contrast to the usual dynamical simple symmetric random walk in…

To assess whether a given time series can be modeled by a stochastic process possessing long range correlation one usually applies one of two types of analysis methods: the spectral method and the random walk analysis. The first objective…

Two Beatty sequences are recorded by athletes running in opposite directions in a round stadium. This approach suggests a nice interpretation for well known partitioning criteria: such sequences (eventually) partition the integers…

We consider a Random Walk in Random Environment (RWRE) moving in an i.i.d.\ random field of obstacles. When the particle hits an obstacle, it disappears with a positive probability. We obtain quenched and annealed bounds on the tails of the…

A $k$-modal sequence is a sequence of real numbers that can be partitioned into $k+1$ (possibly empty) monotone sections such that adjacent sections have opposite monotonicities. For every positive integer $k$, we prove that any sequence of…

Consider two entities with constant but not necessarily equal velocities, moving on two given piece-wise linear trajectories inside a simple polygon $P$. The Trajectory Range Visibility problem deals with determining the sub-trajectories on…

The lonely singles sequence represents the number of noncrossing partitions of the finite set {1,. .. , n} in which no pair of singletons {i} and {j} can be merged into the pair {i, j} so that the partition stays noncrossing. The…

Random walk on changing graphs is considered. For sequences of finite graphs increasing monotonically towards a limiting infinite graph, we establish transition probability upper bounds. It yields sufficient transience criteria for simple…

The statistics of persistent events, recently introduced in the context of phase ordering dynamics, is investigated in the case of the 1D lattice random walk in discrete time. We determine the survival probability of the random walker in…

The simple pairwise comparison is a method to provide different criteria with weights. We show that the values of those weights (in particular the maximum) depend just on the number of criteria. Additionally, it is shown that the distance…