scholarly journals The asymptotic shape theorem for generalized first passage percolation

2010 ◽  
Vol 38 (2) ◽  
pp. 632-660 ◽  
Author(s):  
Michael Björklund
Keyword(s):  
First Passage Percolation ◽  
First Passage ◽  
Asymptotic Shape ◽  
Shape Theorem

First Passage Percolation on Random Geometric Graphs and an Application to Shortest-Path Trees

2015 ◽  
Vol 47 (02) ◽  
pp. 328-354
Author(s):  
C. Hirsch ◽  
D. Neuhäuser ◽  
C. Gloaguen ◽  
V. Schmidt
Keyword(s):  
Shortest Path ◽  
Linear Growth ◽  
Telecommunication Networks ◽  
Geometric Graphs ◽  
First Passage Percolation ◽  
First Passage ◽  
Growth Property ◽  
Random Geometric Graphs ◽  
Shape Theorem ◽  
Path Lengths

We consider Euclidean first passage percolation on a large family of connected random geometric graphs in the d-dimensional Euclidean space encompassing various well-known models from stochastic geometry. In particular, we establish a strong linear growth property for shortest-path lengths on random geometric graphs which are generated by point processes. We consider the event that the growth of shortest-path lengths between two (end) points of the path does not admit a linear upper bound. Our linear growth property implies that the probability of this event tends to zero sub-exponentially fast if the direct (Euclidean) distance between the endpoints tends to infinity. Besides, for a wide class of stationary and isotropic random geometric graphs, our linear growth property implies a shape theorem for the Euclidean first passage model defined by such random geometric graphs. Finally, this shape theorem can be used to investigate a problem which is considered in structural analysis of fixed-access telecommunication networks, where we determine the limiting distribution of the length of the longest branch in the shortest-path tree extracted from a typical segment system if the intensity of network stations converges to 0.

First Passage Percolation on Random Geometric Graphs and an Application to Shortest-Path Trees

2015 ◽  
Vol 47 (2) ◽  
pp. 328-354 ◽  
Author(s):  
C. Hirsch ◽  
D. Neuhäuser ◽  
C. Gloaguen ◽  
V. Schmidt
Keyword(s):  
Shortest Path ◽  
Linear Growth ◽  
Telecommunication Networks ◽  
Geometric Graphs ◽  
First Passage Percolation ◽  
First Passage ◽  
Growth Property ◽  
Random Geometric Graphs ◽  
Shape Theorem ◽  
Path Lengths

We consider Euclidean first passage percolation on a large family of connected random geometric graphs in the d-dimensional Euclidean space encompassing various well-known models from stochastic geometry. In particular, we establish a strong linear growth property for shortest-path lengths on random geometric graphs which are generated by point processes. We consider the event that the growth of shortest-path lengths between two (end) points of the path does not admit a linear upper bound. Our linear growth property implies that the probability of this event tends to zero sub-exponentially fast if the direct (Euclidean) distance between the endpoints tends to infinity. Besides, for a wide class of stationary and isotropic random geometric graphs, our linear growth property implies a shape theorem for the Euclidean first passage model defined by such random geometric graphs. Finally, this shape theorem can be used to investigate a problem which is considered in structural analysis of fixed-access telecommunication networks, where we determine the limiting distribution of the length of the longest branch in the shortest-path tree extracted from a typical segment system if the intensity of network stations converges to 0.

scholarly journals On the Markov Transition Kernels for First Passage Percolation on the Ladder

2011 ◽  
Vol 48 (02) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Percolation Problem ◽  
Vertex Set ◽  
Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

First-passage percolation processes with finite height

1985 ◽  
Vol 22 (4) ◽  
pp. 766-775
Author(s):  
Norbert Herrndorf
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
First Passage Percolation ◽  
Horizontal Strip ◽  
First Passage ◽  
Time Constants ◽  
Finite Height ◽  
Special Cases ◽  
Percolation Processes ◽  
Passage Times

We consider first-passage percolation in an infinite horizontal strip of finite height. Using methods from the theory of Markov chains, we prove a central limit theorem for first-passage times, and compute the time constants for some special cases.

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