First-passage percolation under weak moment conditions

1979 ◽  
Vol 16 (4) ◽  
pp. 750-763 ◽  
Author(s):  
Wolfgang Reh
Keyword(s):  
Time Constant ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Moment Conditions ◽  
Time Coordinate

Most research in first-passage percolation has been done under the assumption of a finite mean for the underlying time coordinate distribution. We demonstrate that the basic ergodic results can be derived under a weaker moment assumption, which still permits us to evaluate the time constant in the case where the atom at zero of the time coordinate distribution exceeds one-half. Further almost sure convergence is investigated more closely.

First-passage percolation under weak moment conditions

1979 ◽  
Vol 16 (04) ◽  
pp. 750-763 ◽  
Author(s):  
Wolfgang Reh
Keyword(s):  
Time Constant ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Moment Conditions ◽  
Time Coordinate

Most research in first-passage percolation has been done under the assumption of a finite mean for the underlying time coordinate distribution. We demonstrate that the basic ergodic results can be derived under a weaker moment assumption, which still permits us to evaluate the time constant in the case where the atom at zero of the time coordinate distribution exceeds one-half. Further almost sure convergence is investigated more closely.

Weak moment conditions for time coordinates in first-passage percolation models

1980 ◽  
Vol 17 (4) ◽  
pp. 968-978 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Time Constant ◽  
First Passage Times ◽  
First Passage Percolation ◽  
First Passage ◽  
Moment Conditions ◽  
Positive Order ◽  
Finite Moment ◽  
Time Coordinate ◽  
Point To Point ◽  
Passage Times

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. The existence of a time constant is proved by considering first-passage times between intervals of sites, rather than the usual point-to-point and point-to-line first-passage times. The basic limit theorems for the related stochastic processes follow easily by previous techniques. The time constant is evaluated as 0 when the atom at 0 of the time-coordinate distribution exceeds½.

Weak moment conditions for time coordinates in first-passage percolation models

1980 ◽  
Vol 17 (04) ◽  
pp. 968-978 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Time Constant ◽  
First Passage Times ◽  
First Passage Percolation ◽  
First Passage ◽  
Moment Conditions ◽  
Positive Order ◽  
Finite Moment ◽  
Time Coordinate ◽  
Point To Point ◽  
Passage Times

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. The existence of a time constant is proved by considering first-passage times between intervals of sites, rather than the usual point-to-point and point-to-line first-passage times. The basic limit theorems for the related stochastic processes follow easily by previous techniques. The time constant is evaluated as 0 when the atom at 0 of the time-coordinate distribution exceeds½.

scholarly journals On the Markov Transition Kernels for First Passage Percolation on the Ladder

2011 ◽  
Vol 48 (02) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Percolation Problem ◽  
Vertex Set ◽  
Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

On the continuity of the time constant of first-passage percolation

1981 ◽  
Vol 18 (4) ◽  
pp. 809-819 ◽  
Author(s):  
J. Theodore Cox ◽  
Harry Kesten
Keyword(s):  
Distribution Function ◽  
Time Constant ◽  
Passage Time ◽  
Square Lattice ◽  
First Passage Percolation ◽  
First Passage ◽  
Finite Constant

Let U be the distribution function of the non-negative passage time of an individual edge of the square lattice, and let a0n be the minimal passage time from (0, 0) to (n, 0). The process a0n/n converges in probability as n → ∞to a finite constant μ (U) called the time constant. It is proven that μ (Uk)→ μ(U) whenever Uk converges weakly to U.

On the time constant and path length of first-passage percolation

1980 ◽  
Vol 12 (04) ◽  
pp. 848-863 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Distribution Function ◽  
Time Constant ◽  
Path Length ◽  
Passage Time ◽  
Square Lattice ◽  
Bond Percolation ◽  
First Passage Percolation ◽  
Critical Probability ◽  
First Passage ◽  
Individual Bond

Let U be the distribution function of the passage time of an individual bond of the square lattice, and let pT be the critical probability above which the expected size of the open component of the origin (in the usual bond percolation) is infinite. It is shown that if (∗)U(0–) = 0, U(0) < pT , then there exist constants 0 < a, C 1 < ∞ such that a self-avoiding path of at least n steps starting at the origin and with passage time ≦ an} ≦ 2 exp (–C 1 n). From this it follows that under (∗) the time constant μ (U) of first-passage percolation is strictly positive and that for each c > 0 lim sup (1/n)Nn (c) <∞, where Nn (c) is the maximal number of steps in the paths starting at the origin with passage time at most cn.

First-passage percolation on the square lattice. I

1977 ◽  
Vol 9 (01) ◽  
pp. 38-54 ◽  
Author(s):  
R. T. Smythe ◽  
John C. Wierman
Keyword(s):  
Time Constant ◽  
First Passage Time ◽  
Passage Time ◽  
Square Lattice ◽  
Ergodic Theorems ◽  
First Passage Percolation ◽  
Critical Percolation ◽  
First Passage ◽  
Underlying Distribution ◽  
Percolation Probability

We consider several problems in the theory of first-passage percolation on the two-dimensional integer lattice. Our results include: (i) a mean ergodic theorem for the first-passage time from (0,0) to the line x = n; (ii) a proof that the time constant is zero when the atom at zero of the underlying distribution exceeds C, the critical percolation probability for the square lattice; (iii) a proof of the a.s. existence of routes for the unrestricted first-passage processes; (iv) a.s. and mean ergodic theorems for a class of reach processes; (v) continuity results for the time constant as a functional of the underlying distribution.

On the continuity of the time constant of first-passage percolation

1981 ◽  
Vol 18 (04) ◽  
pp. 809-819
Author(s):  
J. Theodore Cox ◽  
Harry Kesten
Keyword(s):  
Distribution Function ◽  
Time Constant ◽  
Passage Time ◽  
Square Lattice ◽  
First Passage Percolation ◽  
First Passage ◽  
Finite Constant

Let U be the distribution function of the non-negative passage time of an individual edge of the square lattice, and let a0n be the minimal passage time from (0, 0) to (n, 0). The process a0n/n converges in probability as n → ∞to a finite constant μ (U) called the time constant. It is proven that μ (Uk )→ μ(U) whenever Uk converges weakly to U.

scholarly journals First-Passage Percolation with Exponential Times on a Ladder

2010 ◽  
Vol 19 (4) ◽  
pp. 593-601 ◽  
Author(s):  
HENRIK RENLUND
Keyword(s):  
Markov Chain ◽  
Explicit Expression ◽  
Time Constant ◽  
First Passage Percolation ◽  
First Passage ◽  
Residual Time ◽  
Average Residual ◽  
The Times

We consider first-passage percolation on a ladder, i.e., the graph ℕ × {0, 1}, where nodes at distance 1 are joined by an edge, and the times are exponentially i.i.d. with mean 1. We find an appropriate Markov chain to calculate an explicit expression for the time constant whose numerical value is ≈0.6827. This time constant is the long-term average inverse speed of the process. We also calculate the average residual time.

Sign in / Sign up

Export Citation Format

Share Document