To do so we need to control the occurrence of large degrees uniformly in large subtrees of the Galton–Watson tree. Secondly, we let be a critical Galton-Watson tree. We first find an asymptotic estimate for the probability of a Galton-Watson tree having leaves. For classical proofs of Galton-Watson limit theorems by means of generating functions and for background material on Galton-Watson processes we refer to 2, 1. A GaltonWatson branching process can be represented by a tree in which each node represents an individual, and is linked to its parent as well as its. Progress in probability, Birkhäuser, Basel, 2021) can be pushed through. We are interested in the asymptotic behavior of critical Galton-Watson trees whose offspring distribution may have infinite variance, which are conditioned on having a large fixed number of leaves. as it is conducted according to the latest Watson is. (The parabolic Anderson model on a Galton-Watson tree, to appear in in and out of equilibrium 3: celebrating Vladas Sidoravicius. programme He was one of the men of whom Galton And, when Bishop Watson. We iterated the procedure, and obtain a decomposition of the size-biased Galton Watson tree into an (infinite) spine and i.i.d. We identify the weakest condition on the tail of the degree distribution under which the arguments in den Hollander et al. The present paper extends the analysis to degree distributions with unbounded support. Start with a root vertex, and attach edges from to D 0 first-generation vertices. The second term contains a variational formula indicating that the solution concentrates on a subtree with minimal degree according to a computable profile. The Galton-Watson tree with initial degree distribution D 0 and general degree distribution D g is constructed as follows. In the standard case of a critical Galton-Watson tree, the limit tree has an infinite spine, where the offspring distribution is size- biased. Under the assumption that the degree distribution has bounded support, two terms in the asymptotic expansion were identified under the quenched law, i.e., conditional on the realisation of the random tree and the random potential. To get a nice tree, children interact with. random potential whose marginal distribution is double-exponential. Each individual has an independent, identically distributed number of children. Progress in probability, Birkhäuser, Basel, 2021) a detailed analysis was given of the large-time asymptotics of the total mass of the solution to the parabolic Anderson model on a supercritical Galton–Watson random tree with an i.i.d. (The parabolic Anderson model on a Galton-Watson tree, to appear in in and out of equilibrium 3: celebrating Vladas Sidoravicius.
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