Related papers: Passage time of the frog model has a sublinear var…
The fluctuations of the passage time in first passage percolation are of great interest. We show that the non-random fluctuations in planar FPP are at least of order $\log(n)^\alpha$ for any $\alpha<1/2$ under some conditions that are known…
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…
The first passage time (FPT) for random walks is a key indicator of how fast information diffuses in a given system. Despite the role of FPT as a fundamental feature in transport phenomena, its behavior, particularly in heterogeneous…
The frog model is a growing system of random walks where a particle is added whenever a new site is visited. A longstanding open question is how often the root is visited on the infinite $d$-ary tree. We prove the model undergoes a phase…
A rapidly increasing number of systems is identified in which the stochastic motion of tracer particles follows the Brownian law $\langle\mathbf{r}^2(t) \rangle\simeq Dt$ yet the distribution of particle displacements is strongly…
Consider the first passage percolation model on ${\bf Z}^d$ for $d\geq 2$. In this model we assign independently to each edge the value zero with probability $p$ and the value one with probability $1-p$. We denote by $T({\bf 0}, v)$ the…
In first-passage percolation, one places nonnegative i.i.d. random variables (T(e)) on the edges of Z^d. A geodesic is an optimal path for the passage times T(e). Consider a local property of the time environment. We call it a pattern. We…
We provide sufficient conditions for the validity of a dichotomy, i.e. zero-one law, between recurrence and transience of general frog models. In particular, the results cover frog models with i.i.d. numbers of frogs per site where the frog…
We study the behavior of random walk on dynamical percolation. In this model, the edges of a graph G are either open or closed and refresh their status at rate \mu\ while at the same time a random walker moves on G at rate 1 but only along…
In 1999, Zhang proved that, for first passage percolation on the square lattice $\mathbb{Z}^2$ with i.i.d. non-negative edge weights, if the probability that the passage time distribution of an edge $P(t_e = 0) =1/2 $, the critical value…
A simple lemma bounds $\mathrm{s.d.}(T)/\mathbb{E} T$ for hitting times $T$ in Markov chains with a certain strong monotonicity property. We show how this lemma may be applied to several increasing set-valued processes. Our main result…
We derive the asymptotic first passage time (FPT) distribution for space-dependent variable-order time-fractional diffusion, where the fractional exponent $\alpha(x)$ varies with position. For any sufficiently smooth $\alpha(x)$ on a finite…
As a first step in the first passage problem for passive tracer in stratified porous media, we consider the case of a two-dimensional system consisting of two layers with different convection velocities. Using a lattice generating function…
Commonly, normal diffusive behavior is characterized by a linear dependence of the second central moment on time, $< x^2(t) >\propto t$, while anomalous behavior is expected to show a different time dependence, $ < x^2(t) > \propto…
In one and two dimensions, the first-passage time for a diffusing particle in the presence of a radial potential flow to hit a sphere, conditioned on actually hitting the sphere, is independent of the sign of the drift. Moreover, the…
Attributing a positive value \tau_x to each x in Z^d, we investigate a nearest-neighbour random walk which is reversible for the measure with weights (\tau_x), often known as "Bouchaud's trap model". We assume that these weights are…
We introduce a new first passage percolation model in a Poissonian environment on $\mathbb{R}^{2}$. In this model, the action of a path depends on the geometry of the path and the travel time. We prove that the transversal fluctuation…
We study the first passage times of discrete-time branching random walks in ${\mathbb R}^d$ where $d\geq 1$. Here, the genealogy of the particles follows a supercritical Galton-Watson process. We provide asymptotics of the first passage…
We consider the Bernoulli first-passage percolation on $\mathbb Z^d (d\ge 2)$. That is, the edge passage time is taken independently to be 1 with probability $1-p$ and 0 otherwise. Let ${\mu(p)}$ be the time constant. We prove in this paper…
First passage percolation on $\mathbb{Z}^2$ is a model for describing the spread of an infection on the sites of the square lattice. The infection is spread via nearest neighbor sites and the time dynamic is specified by random passage…