The minimum of a branching random walk outside the boundary case

TL;DR

This paper investigates the behavior of the minimum of a branching random walk in non-boundary cases, especially when the log-generating function explodes, revealing a first order phase transition in the thermodynamic analogy.

Contribution

It extends the understanding of the minimum of branching random walks beyond the boundary case, analyzing scenarios with exploding log-generating functions and phase transitions.

Findings

Identifies the behavior of the minimum in non-boundary cases.

Connects phase transition types to the properties of the branching random walk.

Provides new insights into the thermodynamics analogy for branching processes.

Abstract

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.