Related papers: Late points for random walks in two dimensions

We find tight estimates for the minimum number of proper subspaces needed to cover all lattice points in an n-dimensional convex body symmetric about the origin. We also find the order of magnitude of the number of (n-1)-dimensional…

We consider a $d$-dimensional correlated percolation problem of sites {\em not} visited by a random walk on a hypercubic lattice $L^d$ for $d=3$, 4 and 5. The length of the random walk is ${\cal N}=uL^d$. Close to the critical value…

We consider the tree-reduced path of symmetric random walk on $\ZZ^{d}$. It is interesting to ask about the number of turns $T_n$ in the reduced path after $n$ steps. This question arises from inverting signature for lattice paths. We show…

We address the question of the time needed by $N$ particles, initially located on the first sites of a finite 1D lattice of size $L$, to exit that lattice when they move according to a TASEP transport model. Using analytical calculations…

We study a strongly Non-Markovian variant of random walk in which the probability of visiting a given site $i$ is a function $f$ of number of previous visits $v(i)$ to the site. If the probability is proportional to number of visits to the…

Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(\xi_y,y\in{\mathbb Z}^d)$ are two independent sequences of i.i.d. random variables with values in ${\mathbb Z}^d$ and…

We consider simple random walk on a discrete cylinder with base a large d-dimensional torus of side-length N, when d is two or more. We develop a stochastic domination control on the local picture left by the random walk in boxes of…

We consider large deviations of the cover time of the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, $d \geq 3$ by simple random walk. We prove a lower bound on the probability that the cover time is smaller than $\gamma\in (0,1)$ times its…

We study the search kinetics of an immobile target by a concentration of randomly moving searchers. The object of the study is to optimize the probability of detection within the constraints of our model. The target is hidden on a…

We consider the possible visits to visible points of a random walker moving up and right in the integer lattice (with probability $\alpha$ and $1-\alpha$, respectively) and starting from the origin. We show that, almost surely, the…

We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus (Z/NZ)^d, d >= 3, until u N^d time steps, u > 0, and the model of random interlacements recently introduced by Sznitman. In…

Let $M_n$ be the number of steps of the loop-erasure of a simple random walk on $\mathbb{Z}^2$ from the origin to the circle of radius $n$. We relate the moments of $M_n$ to $Es(n)$, the probability that a random walk and an independent…

We consider a simple random walk on a discrete torus (Z/NZ)^d with dimension d at least 3 and large side length N. For a fixed constant u > 0, we study the percolative properties of the vacant set, consisting of the set of vertices not…

We study the area distribution of closed walks of length $n$, beginning and ending at the origin. The concept of area of a walk in the square lattice is generalized and the usefulness of the new concept is demonstrated through a simple…

We consider a dynamic random graph on $n$ vertices that is obtained by starting from a random graph generated according to the configuration model with a prescribed degree sequence and at each unit of time randomly rewiring a fraction…

We study the full distribution $P_M(S)$ of the number of distinct sites $S$ visited by a random walker on a $d$-dimensional lattice after $M$ steps. We focus on the case $d \ge 2$, and we are interested in the long-time limit $M \gg 1$. Our…

We study the cover time $\tau_{\mathrm{cov}}$ by (continuous-time) random walk on the 2D box of side length $n$ with wired boundary or on the 2D torus, and show that in both cases with probability approaching 1 as $n$ increases,…

Using an inverse of the standard linear congruential random number generator, large randomly occupied lattices can be visited by a random walker without having to determine the occupation status of every lattice site in advance. In seven…

Let each of n particles starting at the origin in Z^2 perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set A(n) of n occupied sites is (with high…

We investigate the number $V_p(n)$ of distinct sites visited by an $n$-step resetting random walker on a $d$-dimensional hypercubic lattice with resetting probability $p$. In the case $p=0$, we recover the well-known result that the average…