  This paper is a complement to the studies on the minimum of a real-valued
branching random walk. In the boundary case (Biggins, Kyprianou 2005),
A\"{i}d\'ekon in a seminal paper (2013) 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.
