Branching Random Walks Conditioned on Particle Numbers.

In this paper, we consider a pruned Galton–Watson tree conditioned to have k particles in generation n, i.e. we take a Galton–Watson tree satisfying Z n = k , and delete all branches that die before generation n. We show that with k fixed and n → ∞ , the first n generations of this tree can be described by an explicit probability measure P k st . As an application, we study a branching random walk (V u) u ∈ T indexed by such a pruned Galton–Watson tree T, and give the asymptotic tail behavior of the span and gap statistics of its k particles in generation n, (V u) | u | = n . This is the discrete version of Ramola et al. (Chaos Solitons Fractals 74:79–88, 2015), generalized to arbitrary offspring and displacement distributions with moment constraints. [ABSTRACT FROM AUTHOR]