## Bernoulli

• Bernoulli
• Volume 24, Number 2 (2018), 801-841.

### The minimum of a branching random walk outside the boundary case

#### Abstract

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.

#### Article information

Source
Bernoulli, Volume 24, Number 2 (2018), 801-841.

Dates
Revised: October 2015
First available in Project Euclid: 21 September 2017

https://projecteuclid.org/euclid.bj/1505980879

Digital Object Identifier
doi:10.3150/15-BEJ784

Mathematical Reviews number (MathSciNet)
MR3706777

Zentralblatt MATH identifier
06778348

#### Citation

Barral, Julien; Hu, Yueyun; Madaule, Thomas. The minimum of a branching random walk outside the boundary case. Bernoulli 24 (2018), no. 2, 801--841. doi:10.3150/15-BEJ784. https://projecteuclid.org/euclid.bj/1505980879

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