Related papers: Random walk weakly attracted to a wall

We give sharp, uniform estimates for the probability that a random walk of n steps on the reals avoids a half-line [y,infinity) given that it ends at the point x. The estimates hold for general continuous or lattice distributions provided…

In this paper, a branching random walk $(V(x))$ in the boundary case is studied, where the associated one dimensional random walk is in the domain of attraction of an $\alpha-$stable law with $1<\alpha<2$. Let $M_n$ be the minimal position…

Random walks on the circle group $\mathbb{R}/\mathbb{Z}$ whose elementary steps are lattice variables with span $\alpha \not\in \mathbb{Q}$ or $p/q \in \mathbb{Q}$ taken mod $\mathbb{Z}$ exhibit delicate behavior. In the rational case we…

We prove that if $(X_n)_{n\geq 0}$ is a random walk on a transient graph such that the Green's function decays at least polynomially along the random walk, then $(X_n)_{n\geq 0}$ has infinitely many cut times almost surely. This condition…

Let p_n denote the persistence probability that the first n iterated partial sums of integrable, zero-mean, i.i.d. random variables X_k, are negative. We show that p_n is bounded above up to universal constant by the square root of the…

We study asymptotic behavior, for large time $n$, of the transition probability of a two-dimensional random walk killed when entering into a non-empty finite subset $A$. We show that it behaves like $4 \tilde u_A(x) \tilde u_{-A}(-y) (\lg…

We prove that an integrated simple random walk, where random walk and integrated random walk are conditioned to return to zero, has asymptotic probability $n^{-1/2}$ to stay positive. This question is motivated by so-called random polymer…

We consider random walk on a finite group $G$ as follows. We can consider $G$ as a group of substitutions. Randomly (i.e. with probability $U(g)=|G|^{-1}$ ) we choose a substitution $g \in G$ and execute it twice in a row, i.e. execute a…

We consider a process of noncolliding $q$-exchangeable random walks on $\mathbb{Z}$ making steps $0$ (straight) and $-1$ (down). A single random walk is called $q$-exchangeable if under an elementary transposition of the neighboring steps…

Suppose that $X$ is a simple random walk on $\Z_n^d$ for $d \geq 3$ and, for each $t$, we let $\U(t)$ consist of those $x \in \Z_n^d$ which have not been visited by $X$ by time $t$. Let $\tcov$ be the expected amount of time that it takes…

In this paper, we present a complete proof of the construction of graphs with bounded valency such that the simple random walk has a return probability at time $n$ at the origin of order $exp(-n^{\alpha}),$ for fixed $\alpha \in [0,1[$ and…

We study the asymptotic tail probability of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior…

We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on $Z^d \times Z_+$. In dimensions $d>6$, we obtain bounds on exit times, transition probabilities, and the range of the…

We study analytically a simple random walk model on a one-dimensional lattice, where at each time step the walker resets to the maximum of the already visited positions (to the rightmost visited site) with a probability $r$, and with…

We prove limit theorems for random walks with $n$ steps in the $d$-dimensional Euclidean space as both $n$ and $d$ tend to infinity. One of our results states that the path of such a random walk, viewed as a compact subset of the…

We study the behaviour of a sequence of biased random walks X(i), i>=0 on a sequence of random graphs, where the initial graph is Zd and otherwise the graph for the i-th walk is the trace of the (i - 1)-st walk. The sequence of bias vectors…

Spatially homogeneous random walks in $(\mathbb{Z}_{+})^{2}$ with non-zero jump probabilities at distance at most 1, with non-zero drift in the interior of the quadrant and absorbed when reaching the axes are studied. Absorption…

We study the mean first passage time of a one-dimensional random walker with step sizes decaying exponentially in discrete time. That is step sizes go like $\lambda^{n}$ with $\lambda\leq1$ . We also present, for pedagogical purposes, a…

Let $X_1, X_2, \ldots$ be i.i.d. random variables with values in $\mathbb{Z}^d$ satisfying $\mathbb{P} \left(X_1=x\right) = \mathbb{P} \left(X_1=-x\right) = \Theta \left(\|x\|^{-s}\right)$ for some $s>d$. We show that the random walk…

Several phase transitions for excited random walks on the integers are known to be characterized by a certain drift parameter delta. For recurrence/transience the critical threshold is |delta|=1, for ballisticity it is |delta|=2 and for…