First passage percolation on the exponential of two-dimensional branching random walk

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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Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation

Stochastic Processes and their Applications ◽

10.1016/j.spa.2010.02.005 ◽

2010 ◽

Vol 120 (6) ◽

pp. 873-900

Author(s):

Raphaël Rossignol ◽

Marie Théret

Keyword(s):

Law Of Large Numbers ◽

Two Dimensional ◽

Maximal Flow ◽

First Passage Percolation ◽

First Passage ◽

Large Numbers ◽

Flow Through

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Random Walk in a Random Environment and First-Passage Percolation on Trees

The Annals of Probability ◽

10.1214/aop/1176989920 ◽

1992 ◽

Vol 20 (1) ◽

pp. 125-136 ◽

Cited By ~ 46

Author(s):

Russell Lyons ◽

Robin Pemantle

Keyword(s):

Random Walk ◽

Random Environment ◽

First Passage Percolation ◽

First Passage ◽

Percolation On Trees

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Large deviation estimates for branching random walks

ESAIM Probability and Statistics ◽

10.1051/ps/2019006 ◽

2019 ◽

Vol 23 ◽

pp. 823-840

Author(s):

Dariusz Buraczewski ◽

Mariusz Maślanka

Keyword(s):

Random Walk ◽

Law Of Large Numbers ◽

Large Deviation ◽

First Passage Time ◽

Passage Time ◽

Branching Random Walk ◽

First Passage ◽

Branching Random Walks ◽

The Central Limit Theorem ◽

Large Numbers

For the branching random walk drifting to −∞ we study large deviations-type estimates for the first passage time. We prove the corresponding law of large numbers and the central limit theorem.

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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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A first-passage problem for a two-dimensional controlled random walk

Journal of Applied Probability ◽

10.2307/3214005 ◽

1986 ◽

Vol 23 (3) ◽

pp. 670-678

Author(s):

S. Lalley

Keyword(s):

Random Walk ◽

First Passage Time ◽

Passage Time ◽

Two Dimensions ◽

Main Diagonal ◽

Two Dimensional ◽

First Passage ◽

Explicit Representations ◽

Mean Vectors ◽

First Passage Problem

The process of interest is a controlled random walk in two dimensions: whenever the walker is above the main diagonal, the next increment to his position is chosen from a distribution FA; whenever the walker is below the diagonal, the next increment comes from another distribution FB. The two distributions have mean vectors which tend to push the walker back toward the diagonal. We analyze the problem of first passage to the first quadrant, obtaining explicit representations for the limiting first-entry distribution and expected first-passage time.

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Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation

ESAIM Probability and Statistics ◽

10.1051/ps/2011109 ◽

2012 ◽

Vol 17 ◽

pp. 70-104

Author(s):

Raphaël Rossignol ◽

Marie Théret

Keyword(s):

Large Deviations ◽

Two Dimensional ◽

Maximal Flow ◽

First Passage Percolation ◽

First Passage ◽

Flow Through

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A first-passage problem for a two-dimensional controlled random walk

Journal of Applied Probability ◽

10.1017/s0021900200111829 ◽

1986 ◽

Vol 23 (03) ◽

pp. 670-678

Author(s):

S. Lalley

Keyword(s):

Random Walk ◽

First Passage Time ◽

Passage Time ◽

Two Dimensions ◽

Main Diagonal ◽

Two Dimensional ◽

First Passage ◽

Explicit Representations ◽

Mean Vectors ◽

First Passage Problem

The process of interest is a controlled random walk in two dimensions: whenever the walker is above the main diagonal, the next increment to his position is chosen from a distribution FA ; whenever the walker is below the diagonal, the next increment comes from another distribution FB. The two distributions have mean vectors which tend to push the walker back toward the diagonal. We analyze the problem of first passage to the first quadrant, obtaining explicit representations for the limiting first-entry distribution and expected first-passage time.

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