scholarly journals First-passage percolation on the square lattice

1985 ◽  
Vol 57 (2) ◽  
pp. 208
Author(s):  
Gian-Carlo Rota
Keyword(s):  
Square Lattice ◽  
First Passage Percolation ◽  
First Passage

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.

First-passage percolation on the square lattice, II

1977 ◽  
Vol 9 (2) ◽  
pp. 283-295 ◽  
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.

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.

An upper bound for the velocity of first-passage percolation

1981 ◽  
Vol 18 (01) ◽  
pp. 256-262 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Lower Bound ◽  
Time Constant ◽  
Upper Bound ◽  
Square Lattice ◽  
First Passage Percolation ◽  
First Passage ◽  
Numerical Examples ◽  
Asymptotic Velocity

An upper bound for the asymptotic velocity in various directions of first-passage percolation on the square lattice is derived. In particular this gives a lower bound for the so-called time constant. The result is generalized to other lattices. Numerical examples are included.

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

1983 ◽  
Vol 15 (02) ◽  
pp. 465-467 ◽  
Author(s):  
J. Van Den Berg
Keyword(s):  
Travel Time ◽  
Square Lattice ◽  
First Passage Percolation ◽  
First Passage ◽  
Time Coordinate ◽  
The Subject

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 cylinderis always non-decreasing inn.However, when the bonds have time-coordinate 1 with probabilitypand 0 with probability 1 –p(0 <P< 1), then, forpsufficiently small, we get a counterexample.

Lower and Upper Bounds for the Time Constant of First-Passage Percolation

2002 ◽  
Vol 11 (5) ◽  
pp. 433-445 ◽  
Author(s):  
SVEN ERICK ALM ◽  
ROBERT PARVIAINEN
Keyword(s):  
Time Constant ◽  
Transition Matrix ◽  
Upper Bounds ◽  
Square Lattice ◽  
Confidence Limits ◽  
First Passage Percolation ◽  
Lower And Upper Bounds ◽  
First Passage ◽  
Time Constants ◽  
Uniform Distributions

We present improved lower and upper bounds for the time constant of first-passage percolation on the square lattice. For the case of lower bounds, a new method, using the idea of a transition matrix, has been used. Numerical results for the exponential and uniform distributions are presented. A simulation study is included, which results in new estimates and improved upper confidence limits for the time constants.

An upper bound for the velocity of first-passage percolation

1981 ◽  
Vol 18 (1) ◽  
pp. 256-262 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Lower Bound ◽  
Time Constant ◽  
Upper Bound ◽  
Square Lattice ◽  
First Passage Percolation ◽  
First Passage ◽  
Numerical Examples ◽  
Asymptotic Velocity

An upper bound for the asymptotic velocity in various directions of first-passage percolation on the square lattice is derived. In particular this gives a lower bound for the so-called time constant. The result is generalized to other lattices. Numerical examples are included.

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