Related papers: The Number of Open Paths in Oriented Percolation
We study the random directed graph $\vec G(n,p)$ in which each of the $n(n-1)$ possible directed edges are present with probability $p$. We show that in the critical window the longest self avoiding oriented paths in $\vec G(n,p)$ have…
We consider an inhomogeneous oriented percolation model introduced by de Lima, Rolla and Valesin. In this model, the underlying graph is an oriented rooted tree in which each vertex points to each of its $d$ children with `short' edges, and…
We study the behavior of the optimal path between two sites separated by a distance $r$ on a $d$-dimensional lattice of linear size $L$ with weight assigned to each site. We focus on the strong disorder limit, i.e., when the weight of a…
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…
We consider the first passage percolation model on the square lattice with an edge weight distribution F. In this paper, we consider the number of optimal paths for two points separated by a long distance. We show that there is a phase…
Following H. Tomita and C. Murakami we propose an analytical model to predict critical probability of percolation. It is based on the excursion set theory which allows us to consider N-dimensional bounded regions. Details are given for the…
We study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-\beta\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}{\rm d} u{\rm d} v\}$ for $|i-j|>1$ for some fixed $\beta>0$…
We present an "ultimate" proof of Cardy's formula for the critical percolation on the hexagonal lattice \cite{Smirnov01criticalpercolation}, showing the existence of the universal and conformally invariant scaling limit of crossing…
We prove results for first-passage percolation on the configuration model with i.i.d. degrees having finite mean, infinite variance and i.i.d. weights with strictly positive support of the form Y=a+X, where a is a positive constant. We…
Let $\{Z_n\}_{n\geq 0 }$ be a $d$-dimensional supercritical branching random walk started from the origin. Write $Z_n(S)$ for the number of particles located in a set $S\subset\mathbb{R}^d$ at time $n$. Denote by…
Let Z_N be the number of self-avoiding paths of length N starting from the origin on the infinite cluster obtained after performing Bernoulli percolation on Z^d with parameter p>p_c(Z^d). The object of this paper is to study the connective…
We consider an i.i.d. supercritical bond percolation on Z^d , every edge is open with a probability p > p\_c (d), where p\_c (d) denotes the critical parameter for this percolation. We know that there exists almost surely a unique infinite…
In this paper we are concerned with contact processes on open clusters of oriented percolation in $Z^d$, where the disease spreads along the direction of open edges. We show that the two critical infection rates in the quenched and annealed…
We prove that, the diffusivity and conductivity on $\mathbb{Z}^d$-Bernoulli percolation ($d \geq 2$) are infinitely differentiable in supercritical regime. This extends a result by Kozlov [Uspekhi Mat. Nauk 44 (1989), no. 2(266), pp 79 -…
Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. To each vertex of a graph a random number is assigned and a path through the graph is called accessible if all numbers along the path are in…
We study subcritical two-dimensional oriented percolation seen from its rightmost point on the set of infinite configurations which are bounded above. This a Feller process whose state space is not compact and has no invariant measures. We…
We investigate oriented bond-site percolation on the planar lattice in which entire columns are stretched. Generalising recent results by Hil\'ario et al., we establish non-trivial percolation under a $(1+\varepsilon)$-th moment condition…
In this paper we study anisotropic oriented percolation on $\mathbb{Z}^d$ for $d\geq 4$ and show that the local condition for phase transition is closely related to the mean-field condition. More precisely, we show that if the sum of the…
In this paper, we study the maximal edge-traversal time (simply we call maximal weight hereafter) on the optimal paths in the first passage percolation for several edge distributions, including the Pareto and Weibull distributions. It is…
In this paper, we consider Bernoulli percolation on a locally finite, transitive and infinite graph (e.g. the hypercubic lattice $\mathbb{Z}^d$). We prove the following estimate, where $\theta_n(p)$ is the probability that there is a path…