Conditioning Galton–Watson Trees on Large Maximal Outdegree

Abstract

Abstract. We propose a new way to condition random trees, that is, condition random trees to have large maximal out-degree. Under this new conditioning, we show that conditioned critical Galton-Watson trees converge locally to size-biased trees with a unique infinite spine. For the sub-critical case, we obtain local convergence to size-biased trees with a unique infinite node. We also study tail of the maximal out-degree of sub-critical Galton-Watson trees, which is essential for the proof of the local convergence.

Critical Multi-type Galton–Watson Trees Conditioned to be Large

“…Since lim γ→0 h α(γ) = m α, we deduce that condition (8) implies h α(0+) < 1. A necessary and sufficient condition for the existence of a root to (9) is that lim γ↑R h α(γ) ≥ 1, with R = sup Q the radius of convergence of the series f q . A sufficient condition to get this latter condition is for example lim γ↑R f ′ q (γ) = +∞ or even the stronger condition R = +∞.…”

“…We now shall condition on a general asymptotic proportion α ∈ R d satisfying condition (8) and condition (9). We assume that there exists a root to the equation ( 9), say γ.…”

“…In Abraham and Delmas [2] a sufficient and necessary condition is given for a wide class of conditionings for a critical GW tree to converge locally to Kesten's tree under minimal hypotheses on the offspring distribution. Notice that condensation may arise when considering sub-critical GW trees, see Janson [12], Jonnson and Stefansson [13], He [9] or Abraham and Delmas [1] for results in this direction. When scaling limits of multi-type GW tree are considered, one obtains as a limit a continuous GW tree, see Miermont [17] or Gorostiza and Lopez-Mimbela [16] (when the probability to give birth to different types goes down to 0).…”

On the maximal offspring in a subcritical branching process

“…It was shown in [20,Prop. 3.2] that the maximal outdegree of the unconditioned Galton-Watson tree T satisfies…”