Minima in branching random walks

Louigi Addario-Berry; Bruce Reed

doi:10.1214/08-AOP428

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Home > Journals > Ann. Probab. > Volume 37 > Issue 3 > Article

May 2009 Minima in branching random walks

Louigi Addario-Berry, Bruce Reed

Ann. Probab. 37(3): 1044-1079 (May 2009). DOI: 10.1214/08-AOP428

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Abstract

Given a branching random walk, let M_n be the minimum position of any member of the nth generation. We calculate EM_n to within O(1) and prove exponential tail bounds for P{|M_n−EM_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 EM_n when the branching random walk has bounded branching and step size.