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

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

Asymptotics of First-Passage Percolation on One-Dimensional Graphs

2015 ◽  
Vol 47 (01) ◽  
pp. 182-209
Author(s):  
Daniel Ahlberg
Keyword(s):  
Law Of Large Numbers ◽  
Nearest Neighbour ◽  
First Passage Percolation ◽  
First Passage ◽  
Strong Law ◽  
One Dimensional ◽  
Minimal Weight ◽  
Large Numbers ◽  
Mean And Variance ◽  
The Mean

In this paper we consider first-passage percolation on certain one-dimensional periodic graphs, such as thenearest neighbour graph ford,K≥ 1. We expose a regenerative structure within the first-passage process, and use this structure to show that both length and weight of minimal-weight paths present a typical one-dimensional asymptotic behaviour. Apart from a strong law of large numbers, we derive a central limit theorem, a law of the iterated logarithm, and a Donsker theorem for these quantities. In addition, we prove that the mean and variance of the length and weight of minimizing paths are monotone in the distance between their end-points, and further show how the regenerative idea can be used to couple two first-passage processes to eventually coincide. Using this coupling we derive a 0–1 law.

Asymptotics of First-Passage Percolation on One-Dimensional Graphs

2015 ◽  
Vol 47 (1) ◽  
pp. 182-209 ◽  
Author(s):  
Daniel Ahlberg
Keyword(s):  
Law Of Large Numbers ◽  
Nearest Neighbour ◽  
First Passage Percolation ◽  
First Passage ◽  
Strong Law ◽  
One Dimensional ◽  
Minimal Weight ◽  
Large Numbers ◽  
Mean And Variance ◽  
The Mean

In this paper we consider first-passage percolation on certain one-dimensional periodic graphs, such as the nearest neighbour graph for d, K ≥ 1. We expose a regenerative structure within the first-passage process, and use this structure to show that both length and weight of minimal-weight paths present a typical one-dimensional asymptotic behaviour. Apart from a strong law of large numbers, we derive a central limit theorem, a law of the iterated logarithm, and a Donsker theorem for these quantities. In addition, we prove that the mean and variance of the length and weight of minimizing paths are monotone in the distance between their end-points, and further show how the regenerative idea can be used to couple two first-passage processes to eventually coincide. Using this coupling we derive a 0–1 law.

scholarly journals Law of large numbers for the drift of the two-dimensional wreath product

2021 ◽  
Author(s):  
Anna Erschler ◽  
Tianyi Zheng
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Wreath Product ◽  
Law Of Large Numbers ◽  
Abelian Groups ◽  
Limiting Distribution ◽  
Two Dimensional ◽  
Lamplighter Group ◽  
Large Numbers ◽  
Asymptotic Geometry

AbstractWe prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $$(2+\epsilon )$$ ( 2 + ϵ ) -moment. This result is in contrast with classical examples of abelian groups, where the displacement after n steps, normalised by its mean, does not concentrate, and the limiting distribution of the normalised n-step displacement admits a density whose support is $$[0,\infty )$$ [ 0 , ∞ ) . We study further examples of groups, some with random walks satisfying LLN for drift and other examples where such concentration phenomenon does not hold, and study relation of this property with asymptotic geometry of groups.

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