Minima in branching random walks

Louigi Addario-Berry; Bruce Reed

doi:10.1214/08-AOP428

Minima in branching random walks

Probability 2009-07-24 v3 Combinatorics

Authors: Louigi Addario-Berry , Bruce Reed

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Abstract

Given a branching random walk, let $M_n$ be the minimum position of any member of the $n$ th generation. We calculate $\mathbf{E}M_n$ to within O(1) and prove exponential tail bounds for $\mathbf{P}\{|M_n-\mathbf{E}M_n|>x\}$ , under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89--108], our results fully characterize the possible behavior of $\mathbf {E}M_n$ when the branching random walk has bounded branching and step size.

Keywords

random walks random walk in random environment branching processes

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@article{arxiv.0712.2582,
  title  = {Minima in branching random walks},
  author = {Louigi Addario-Berry and Bruce Reed},
  journal= {arXiv preprint arXiv:0712.2582},
  year   = {2009}
}

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Published in at http://dx.doi.org/10.1214/08-AOP428 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Minima in branching random walks

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