Trees with exponential height dependent weight

Abstract

We consider planar rooted random trees whose distribution is even for fixed height $h$ and size $N$ and whose height dependence is of exponential form $e^{-\mu h}$. Defining the total weight for such trees of fixed size to be $Z_N^{(\mu )}$, we determine its asymptotic behaviour for large $N$, for arbitrary real values of $\mu$. Based on this we evaluate the local limit of the corresponding probability measures and find a transition at $\mu =0$ from a single spine phase to a multi-spine phase. Correspondingly, there is a transition in the volume growth rate of balls around the root as a function of radius from linear growth for $\mu <0$ to the familiar quadratic growth at $\mu =0$ and to cubic growth for $\mu >0$.

arXiv comment40 pages, 3 figures

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Cited by in corpus (2)

• Progressive and Rushed Dyck Paths
• Conditioning Bienaym{é}-Galton-Watson trees to have large sub-populations