Related papers: Asymptotics for $2D$ Critical First Passage Percol…
Many scientific questions can be framed as asking for a first passage time (FPT), which generically describes the time it takes a random "searcher" to find a "target." The important timescale in a variety of biophysical systems is the time…
In majority bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: if at least half of the neighbours of a vertex v are already infected, then v is also infected, and infected vertices remain…
We consider first passage percolation with i.i.d. weights on edges of the d-dimensional cubic lattice. Under the assumptions that a weight is equal to zero with probability smaller than the critical probability of bond percolation in the…
A random hopping on a fractal network with dimension slightly above one, $d = 1 + \epsilon$, is considered as a model of transport for conducting polymers with nonmetallic conductivity. Within the real space renormalization group method of…
Percolation with edge-passage probability p and first-passage percolation are studied for the n-cube B_n ={0,1}^n with nearest neighbor edges. For oriented and unoriented percolation, p=e/n and p=1/n are the respective critical…
It is not known (and even physicists disagree) whether first passage percolation (FPP) on $\mathbb{Z}^d$ has an upper critical dimension $d_c$, such that the fluctuation exponent $\chi=0$ in dimensions $d>d_c$. In part to facilitate study…
First-passage percolation is a random growth model defined on $\mathbb{Z}^d$ using i.i.d. nonnegative weights $(\tau_e)$ on the edges. Letting $T(x,y)$ be the distance between vertices $x$ and $y$ induced by the weights, we study the random…
We consider a stochastic aggregation model on Z^d. Start with particles located at the vertices of the lattice, initially distributed according to the product Bernoulli measure with parameter \mu. In addition, there is an aggregate, which…
The $n$-dimensional binary hypercube is the graph whose vertices are the binary $n$-tuples $\{0, 1\}^n$ and where two vertices are connected by an edge if they differ at exactly one coordinate. We prove that if the edges are assigned…
In dynamical percolation, the status of every bond is refreshed according to an independent Poisson clock. For graphs which do not percolate at criticality, the dynamical sensitivity of this property was analyzed extensively in the last…
The study of first passage times for diffusing particles reaching target states is foundational in various practical applications, including diffusion-controlled reactions. In this work, we present a bi-scaling theory for the probability…
We investigate the moderate and large deviations in first-passage percolation (FPP) with bounded weights on $\mathbb{Z}^d$ for $d \geq 2$. Write $T(\mathbf{x}, \mathbf{y})$ for the first-passage time and denote by $\mu(\mathbf{u})$ the time…
The first passage time for a single diffusing particle has been studied extensively, but the first passage time of a system of many diffusing particles, as is often the case in physical systems, has received little attention until recently.…
We study intersection properties of two or more independent tree-like random graphs. Our setting encompasses critical, possibly long range, Bernoulli percolation clusters, incipient infinite clusters, as well as critical branching random…
We study planar first-passage percolation with independent weights whose common distribution is supported in $(0,\infty)$ and is absolutely continuous with respect to Lebesgue measure. We prove that the passage time from $x$ to $y$ denoted…
In a new type of percolation phase transition, which was observed in a set of non-equilibrium models, each new connection between vertices is chosen from a number of possibilities by an Achlioptas-like algorithm. This causes preferential…
Invasion percolation is a stochastic growth model that follows a greedy algorithm. After assigning i.i.d. uniform random variables (weights) to all edges of $\mathbb{Z}^d$, the growth starts at the origin. At each step, we adjoin to the…
The review is a brief description of the state of problems in percolation theory and their numerous applications, which are analyzed on base of interesting papers published in the last 15-20 years. At the submitted papers are studied both…
We consider oriented percolation on Z^d times Z_+ whose bond-occupation probability is pD(...), where p is the percolation parameter and D is a probability distribution on Z^d. Suppose that D(x) decays as |x|^{-d-\alpha} for some \alpha>0.…
We study existence of percolation in the hierarchical group of order $N$, which is an ultrametric space, and transience and recurrence of random walks on the percolation clusters. The connection probability on the hierarchical group for two…