Open Access
May 2018 The minimum of a branching random walk outside the boundary case
Julien Barral, Yueyun Hu, Thomas Madaule
Bernoulli 24(2): 801-841 (May 2018). DOI: 10.3150/15-BEJ784


This paper is a complement to the studies on the minimum of a real-valued branching random walk. In the boundary case [Electron. J. Probab. 10 (2005) 609–631], Aïdékon in a seminal paper [Ann. Probab. 41 (2013) 1362–1426] obtained the convergence in law of the minimum after a suitable renormalization. We study here the situation when the log-generating function of the branching random walk explodes at some positive point and it cannot be reduced to the boundary case. In the associated thermodynamics framework, this corresponds to a first-order phase transition, while the boundary case corresponds to a second-order phase transition.


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Julien Barral. Yueyun Hu. Thomas Madaule. "The minimum of a branching random walk outside the boundary case." Bernoulli 24 (2) 801 - 841, May 2018.


Received: 1 October 2014; Revised: 1 October 2015; Published: May 2018
First available in Project Euclid: 21 September 2017

zbMATH: 06778348
MathSciNet: MR3706777
Digital Object Identifier: 10.3150/15-BEJ784

Keywords: Branching random walk , Minimal position , phase transition

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 2 • May 2018
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