Trees with exponential height dependent weight
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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−μh. Defining the total weight for such trees of fixed size to be Z(μ)N , we determine its asymptotic behaviour for large N, for arbitrary real values of μ. Based on this we identify the local limit of the corresponding probability measures and find a transition at μ = 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 μ < 0 to the familiar quadratic growth at μ = 0 and to cubic growth for μ > 0.
Originalsprog | Engelsk |
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Tidsskrift | Probability Theory and Related Fields |
Vol/bind | 186 |
Udgave nummer | 3-4 |
Sider (fra-til) | 999-1043 |
ISSN | 0178-8051 |
DOI | |
Status | Udgivet - 2023 |
Bibliografisk note
Funding Information:
We thank Mireille Bousquet-Mélou and Nicolas Curien for bringing to our attention the papers [] and [], respectively. We also thank an anonymous referee for valuable remarks that led to simplifications of some proofs. The authors acknowledge support from Villum Fonden via the QMATH Centre of Excellence (Grant No. 10059).
Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
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