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arXiv · 0712.2582

Minima in branching random walks

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.

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Louigi Addario-Berry, Bruce Reed. 2009-07-24. Minima in branching random walks. https://doi.org/10.1214/08-aop428

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