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.