A counterexample to a conjecture of J. M. Hammersley and D. J. A. Welsh concerning first-passage percolation

Consider first-passage percolation on the square lattice. Hammersley and Welsh, who introduced the subject in 1965, conjectured that the expected minimum travel time from (0, 0) to (n, 0) along paths contained in the cylinder is always non-decreasing in n. However, when the bonds have time-coordinate 1 with probability p and 0 with probability 1 – p (0 < P < 1), then, for p sufficiently small, we get a counterexample.

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A Note on a Problem by Welsh in First-Passage Percolation

Combinatorics Probability Computing ◽

10.1017/s0963548397003301 ◽

1998 ◽

Vol 7 (1) ◽

pp. 11-15 ◽

Cited By ~ 6

Author(s):

SVEN ERICK ALM

Keyword(s):

First Passage Times ◽

Square Lattice ◽

Trivial Case ◽

First Passage Percolation ◽

First Passage ◽

General Proof ◽

Mean First Passage Times ◽

The Subject ◽

Passage Times

Consider first-passage percolation on the square lattice. Welsh, who together with Hammersley introduced the subject in 1963, has formulated a problem about mean first-passage times, which, although seemingly simple, has not been proved in any non-trivial case. In this paper we give a general proof of Welsh's problem.

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First-passage percolation on the square lattice

Advances in Mathematics ◽

10.1016/0001-8708(85)90059-3 ◽

1985 ◽

Vol 57 (2) ◽

pp. 208

Author(s):

Gian-Carlo Rota

Keyword(s):

Square Lattice ◽

First Passage Percolation ◽

First Passage

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

Journal of Applied Probability ◽

10.2307/3213206 ◽

1980 ◽

Vol 17 (4) ◽

pp. 968-978 ◽

Cited By ~ 5

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.

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First-passage percolation on the square lattice. I

Advances in Applied Probability ◽

10.1017/s000186780004310x ◽

1977 ◽

Vol 9 (01) ◽

pp. 38-54 ◽

Cited By ~ 5

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.

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

Journal of Applied Probability ◽

10.1017/s0021900200097254 ◽

1980 ◽

Vol 17 (04) ◽

pp. 968-978 ◽

Cited By ~ 2

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.

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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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First-passage percolation on the square lattice, II

Advances in Applied Probability ◽

10.2307/1426387 ◽

1977 ◽

Vol 9 (2) ◽

pp. 283-295 ◽

Cited By ~ 5

Author(s):

John C. Wierman

Keyword(s):

Sufficient Conditions ◽

Initial Result ◽

First Passage Times ◽

Square Lattice ◽

Maximum Height ◽

First Passage Percolation ◽

First Passage ◽

Necessary And Sufficient ◽

Passage Times

Several problems are considered in the theory of first-passage percolation on the two-dimensional integer lattice. The results include: (i) necessary and sufficient conditions for the existence of moments of first-passage times; (ii) determination of an upper bound for the time constant; (iii) an initial result concerning the maximum height of routes for first-passage times; (iv) ergodic theorems for a class of reach processes.

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