Minima in branching random walks

Louigi Addario-Berry , Bruce Reed (1)

1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications

Given a branching random walk, let $M_n$ be the minimum position of any member of the $n$th generation. We calculate $\\mathbfEM_n$ to within O(1) and prove exponential tail bounds for $\\mathbfP{|M_n-\\mathbfEM_n|>x}$, 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 $\\mathbf EM_n$ when the branching random walk has bounded branching and step size.

Domaines: Informatique mobile; Réseaux et télécommunications [cs.NI]

Louigi Addario-Berry, Bruce Reed. Minima in branching random walks. Annals of Probability, 2009, 37, pp.1044―1079. ⟨hal-00795281⟩