Aspects of first passage percolation

  title={Aspects of first passage percolation},
  author={Harry Kesten},
On a Lower Bound for the Time Constant of First-Passage Percolation
It is proved in this paper that the Bernoulli first-passage percolation on $\mathbb Z^d (d\ge 2)$ is considered and the edge passage time is taken independently to be 1 with probability $1-p$ and 0 otherwise.
Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation
. We consider first-passage percolation on Z 2 with independent and identically distributed weights whose common distribution is absolutely continuous with a finite exponential moment. Under the
Continuity of the time constant in a continuous model of first passage percolation
For a given dimension d ≥ 2 and a finite measure ν on (0,+∞), we consider ξ a Poisson point process on R× (0,+∞) with intensity measure dc⊗ ν where dc denotes the Lebesgue measure on R. We consider
A geometric property for optimal paths and its applications in first passage percolation.
We consider the first passage percolation model in ${\bf Z}^d$ with a weight distribution $F$ for $0 < F(0) < p_c$. In this paper, we derive a geometric property for optimal paths to show that all of
Course notes on geodesics in first-passage percolation
These notes accompany courses given in University of Bath and Northwestern University during their summer schools in 2016. The main topic is geodesics in first-passage percolation, specifically
Concentrations for the simple random walk in unbounded nonnegative potentials
We consider the simple random walk in i.i.d. nonnegative potentials on the multidimensional cubic lattice. Our goal is to investigate the cost paid by the simple random walk for traveling from the
Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation
We consider the standard first passage percolation model in $\mathbb{Z}^d$ for $d\geq 2$. We are interested in two quantities, the maximal flow $\tau$ between the lower half and the upper half of the
The Time Constant Vanishes Only on the Percolation Cone in Directed First Passage Percolation
  • Yu Zhang
  • Mathematics, Computer Science
  • 2008
The shape of the directed growth model on the distribution of $F$ is described and a phase transition is given for the shape at $\vec{p}_c$ is shown.
The divergence of fluctuations for shape in first passage percolation
AbstractWe consider the first passage percolation model on Zd for d ≥ 2. In this model, we assign independently to each edge the value zero with probability p and the value one with probability 1−p.


On the continuity of the time constant of first-passage percolation
Let U be the distribution function of the non-negative passage time of an individual edge of the square lattice, and let a 0n be the minimal passage time from (0, 0) to (n, 0). The process a 0n /n
The time constant of first-passage percolation on the square lattice
  • J. T. Cox
  • Mathematics
    Advances in Applied Probability
  • 1980
Let μ (F) be the time constant of first-passage percolation on the square lattice with underlying distribution function F. Two theorems are presented which show, under some restrictions, that μ
Weak moment conditions for time coordinates in first-passage percolation models
A generalization of first-passage percolation theory proves that the fundamental convergence theorems hold provided only that the time coordinate distribution has a finite moment of a positive order.
Percolation Models in two and Three Dimensions
Let Zd, d = 2,3, denote the d-dimensional integer lattice. Call x, y € Zd neighbors if ∥x — y∥ = 1 (here and in what follows ∥x — y∥ = ∣x1 — y1∣ + |x2 — y2∣ for d = 2, with the corresponding
Oriented Percolation in Dimensions D ≥ 4: Bounds and Asymptotic Formulas
Let p c (d) be the critical probability for oriented percolation in ℤ d and let μ(d) be the time constant for the first passage process based on the exponential distribution. In this paper we show
A lower bound for the critical probability in a certain percolation process
Consider a lattice L in the Cartesian plane consisting of all points (x, y) such that either x or y is an integer. Points with integer coordinates (positive, negative, or zero) are called vertices