First-passage percolation on the square lattice. I

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.

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

Journal of Applied Probability ◽

10.2307/3213056 ◽

1981 ◽

Vol 18 (4) ◽

pp. 809-819 ◽

Cited By ~ 16

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.

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

Advances in Applied Probability ◽

10.1017/s0001867800020127 ◽

1980 ◽

Vol 12 (04) ◽

pp. 848-863 ◽

Cited By ~ 6

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.

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

Journal of Applied Probability ◽

10.1017/s0021900200034161 ◽

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.

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Remarks on renewal theory for percolation processes

Journal of Applied Probability ◽

10.1017/s0021900200094353 ◽

1976 ◽

Vol 13 (02) ◽

pp. 290-300 ◽

Cited By ~ 7

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.

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Remarks on renewal theory for percolation processes

Journal of Applied Probability ◽

10.2307/3212832 ◽

1976 ◽

Vol 13 (2) ◽

pp. 290-300 ◽

Cited By ~ 11

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.

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

Advances in Applied Probability ◽

10.2307/1426745 ◽

1980 ◽

Vol 12 (4) ◽

pp. 864-879 ◽

Cited By ~ 9

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.

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

Advances in Applied Probability ◽

10.1017/s0001867800020139 ◽

1980 ◽

Vol 12 (04) ◽

pp. 864-879 ◽

Cited By ~ 2

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.

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

Advances in Applied Probability ◽

10.2307/1426744 ◽

1980 ◽

Vol 12 (4) ◽

pp. 848-863 ◽

Cited By ~ 19

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, C1 < ∞ such that a self-avoiding path of at least n steps starting at the origin and with passage time ≦ an} ≦ 2 exp (–C1n).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.

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The density of interfaces: a new first-passage problem

Journal of Applied Probability ◽

10.1017/s0021900200044612 ◽

1993 ◽

Vol 30 (04) ◽

pp. 851-862 ◽

Cited By ~ 1

Author(s):

L. Chayes ◽

C. Winfield

Keyword(s):

Correlation Length ◽

First Passage Time ◽

Passage Time ◽

Scaling Behavior ◽

First Passage Percolation ◽

First Passage ◽

Site Percolation ◽

Percolation Problem ◽

First Passage Problem

We introduce and study a novel type of first-passage percolation problem onwhere the associated first-passage time measures the density of interface between two types of sites. If the types, designated + and –, are independently assigned their values with probabilitypand (1 —p) respectively, we show that the density of interface is non-zero provided that both species are subcritical with regard to percolation, i.e.pc>p> 1 –pc.Furthermore, we show that asp↑pcorp↓ (1 –pc), the interface density vanishes with scaling behavior identical to the correlation length of the site percolation problem.

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