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

2017 ◽  
Vol 22 (0) ◽  
Author(s):  
Jian Ding ◽  
Subhajit Goswami
1976 ◽  
Vol 13 (02) ◽  
pp. 290-300 ◽  
Author(s):  
R. T. Smythe

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.


2019 ◽  
Vol 23 ◽  
pp. 823-840
Author(s):  
Dariusz Buraczewski ◽  
Mariusz Maślanka

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.


1976 ◽  
Vol 13 (2) ◽  
pp. 290-300 ◽  
Author(s):  
R. T. Smythe

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.


1986 ◽  
Vol 23 (3) ◽  
pp. 670-678
Author(s):  
S. Lalley

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.


1986 ◽  
Vol 23 (03) ◽  
pp. 670-678
Author(s):  
S. Lalley

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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