Related papers: Leading All The Way

Consider a simple random walk on the integers with the following transition mechanism. At each site $x$, the probability of jumping to the right is $\omega(x)\in[\frac12,1)$, until the first time the process jumps to the left from site $x$,…

The game of Hex has two players who take turns placing stones of their respective colors on the hexagons of a rhombus-shaped hexagonal grid. Black wins by completing a crossing between two opposite edges, while White wins by completing a…

We derive a universal, exact asymptotic form of the splitting probability for symmetric continuous jump processes, which quantifies the probability $ \pi_{0,\underline{x}}(x_0)$ that the process crosses $x$ before 0 starting from a given…

We study competition on scale-free random graphs, where the degree distribution satisfies an asymptotic power-law with infinite variance. Our competition process is such that the two types attempt at occupying vertices incident to the…

In this work, we consider "decision" variants of a monogamy-of-entanglement game by Tomamichel, Fehr, Kaniewski, and Wehner [New Journal of Physics '13]. In its original "search" variant, Alice prepares a (possibly entangled) state on…

To what extent is the underlying distribution of a finitely supported unbiased random walk on $\mathbb{Z}$ determined by the sequence of times at which the walk returns to the origin? The main result of this paper is that, in various…

We present some new results about the distribution of a random walk whose independent steps follow a $q-$Gaussian distribution with exponent $\frac{1}{1-q}; q \in \mathbb{R}$. In the case $q>1$ we show that a stochastic representation of…

It is well known that the ratio of two independent standard Gaussian random variables follows a Cauchy distribution. Any convex combination of independent standard Cauchy random variables also follows a Cauchy distribution. In a recent…

We consider a random trial-based telegraph process, which describes a motion on the real line with two constant velocities along opposite directions. At each epoch of the underlying counting process the new velocity is determined by the…

In the classical optimal stopping problem, a player is given a sequence of random variables $X_1\ldots X_n$ with known distributions. After observing the realization of $X_i$, the player can either accept the observed reward from $X_i$ and…

We consider a random walk on a homogeneous space $G/\Lambda$ where $G$ is a non-compact simple Lie group and $\Lambda$ is a lattice. The walk is driven by a probability measure $\mu$ on $G$ whose support generates a Zariski-dense subgroup.…

We study nearest neighbor random walks on fixed environments of $\mathbb{Z}$ composed of two point types : $(1/2,1/2)$ and $(p,1-p)$ for $p>1/2$. We show that for every environment with density of $p$ drifts bounded by $\lambda$ we have…

We propose a definition of generalized Cops and Robbers games where there are two players, the Pursuer and the Evader, who each move via prescribed rules. If the Pursuer can ensure that the game enters into a fixed set of final positions,…

We study the probability distribution $P(X_N=X,N)$ of the total displacement $X_N$ of an $N$-step run and tumble particle on a line, in presence of a constant nonzero drive $E$. While the central limit theorem predicts a standard Gaussian…

A neat question involving coin flips surfaced on $\Bbb X$, and generated an intensive `storm' of `social mathematics'. In a sequence of flips of a fair coin, Alice wins a point at each appearance of two consecutive heads, and Bob wins a…

Hit-and-Run is known to be one of the best random sampling algorithms, its mixing time is polynomial in dimension. Nevertheless, in practice the number of steps required to achieve uniformly distributed samples is rather high. We propose…

Suppose that $(X,Y,Z)$ is a random walk in $\mathbb{Z}^3$ that moves in the following way: on the first visit to a vertex only $Z$ changes by $\pm 1$ equally likely, while on later visits to the same vertex $(X,Y)$ performs a…

We study competing first passage percolation on graphs generated by the configuration model with infinite-mean degrees. Initially, two uniformly chosen vertices are infected with type 1 and type 2 infection, respectively, and the infection…

Building upon the knowledge of the distribution of the first positive position reached by a random walker starting from the origin, one can derive new results on the statistics of the gap between the largest and second-largest positions of…

The distribution of the first positive position reached by a random walker starting at the origin is central to the analysis of extremes and records in one-dimensional random walks. In this work, we present a detailed and self-contained…