First-passage percolation on the square lattice. III

1978 ◽  
Vol 10 (01) ◽  
pp. 155-171 ◽  
Author(s):  
R. T. Smythe ◽  
John C. Wierman
Keyword(s):  
Asymptotic Behavior ◽  
Almost Sure Convergence ◽  
Square Lattice ◽  
First Passage Percolation ◽  
First Passage ◽  
Underlying Distribution ◽  
Connectivity Constant ◽  
Route Length ◽  
Percolation Processes ◽  
Negative Time

The principal results of this paper concern the asymptotic behavior of the number of arcs in the optimal routes of first-passage percolation processes on the square lattice. Assuming that the underlying distribution has an atom at zero less than λ–1, where λ is the connectivity constant, Lp and (in some cases) almost sure convergence theorems are proved for the normalized route length processes. The proofs involve the extension of much of the existing theory of first-passage percolation to the case where negative time coordinates are permitted.

First-passage percolation on the square lattice. III

1978 ◽  
Vol 10 (1) ◽  
pp. 155-171 ◽  
Author(s):  
R. T. Smythe ◽  
John C. Wierman
Keyword(s):  
Asymptotic Behavior ◽  
Almost Sure Convergence ◽  
Square Lattice ◽  
First Passage Percolation ◽  
First Passage ◽  
Underlying Distribution ◽  
Connectivity Constant ◽  
Route Length ◽  
Percolation Processes ◽  
Negative Time

The principal results of this paper concern the asymptotic behavior of the number of arcs in the optimal routes of first-passage percolation processes on the square lattice. Assuming that the underlying distribution has an atom at zero less than λ–1, where λ is the connectivity constant, Lp and (in some cases) almost sure convergence theorems are proved for the normalized route length processes. The proofs involve the extension of much of the existing theory of first-passage percolation to the case where negative time coordinates are permitted.

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.

Remarks on renewal theory for percolation processes

1976 ◽  
Vol 13 (02) ◽  
pp. 290-300 ◽  
Author(s):  
R. T. Smythe
Keyword(s):  
Ergodic Theorem ◽  
First Passage Time ◽  
Renewal Theory ◽  
Passage Time ◽  
Integer Lattice ◽  
Two Dimensional ◽  
First Passage Percolation ◽  
First Passage ◽  
Underlying Distribution ◽  
Percolation Processes

We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) an L 1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the line X = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.

First-passage percolation on the square lattice. I

1977 ◽  
Vol 9 (1) ◽  
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.

Remarks on renewal theory for percolation processes

1976 ◽  
Vol 13 (2) ◽  
pp. 290-300 ◽  
Author(s):  
R. T. Smythe
Keyword(s):  
Ergodic Theorem ◽  
First Passage Time ◽  
Renewal Theory ◽  
Passage Time ◽  
Integer Lattice ◽  
Two Dimensional ◽  
First Passage Percolation ◽  
First Passage ◽  
Underlying Distribution ◽  
Percolation Processes

We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) an L1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the line X = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.

The time constant of first-passage percolation on the square lattice

1980 ◽  
Vol 12 (4) ◽  
pp. 864-879 ◽  
Author(s):  
J. Theodore Cox
Keyword(s):  
Distribution Function ◽  
Weak Convergence ◽  
Time Constant ◽  
Square Lattice ◽  
First Passage Percolation ◽  
First Passage ◽  
Underlying Distribution

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 varies continuously in F with respect to weak convergence. These results are improvements of existing continuity theorems.

The time constant of first-passage percolation on the square lattice

1980 ◽  
Vol 12 (04) ◽  
pp. 864-879 ◽  
Author(s):  
J. Theodore Cox
Keyword(s):  
Distribution Function ◽  
Weak Convergence ◽  
Time Constant ◽  
Square Lattice ◽  
First Passage Percolation ◽  
First Passage ◽  
Underlying Distribution

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 varies continuously in F with respect to weak convergence. These results are improvements of existing continuity theorems.

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.

First-passage percolation processes with finite height

1985 ◽  
Vol 22 (4) ◽  
pp. 766-775
Author(s):  
Norbert Herrndorf
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
First Passage Percolation ◽  
Horizontal Strip ◽  
First Passage ◽  
Time Constants ◽  
Finite Height ◽  
Special Cases ◽  
Percolation Processes ◽  
Passage Times

We consider first-passage percolation in an infinite horizontal strip of finite height. Using methods from the theory of Markov chains, we prove a central limit theorem for first-passage times, and compute the time constants for some special cases.

Sign in / Sign up

Export Citation Format

Share Document