Trees with exponential height dependent weight
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Trees with exponential height dependent weight. / Durhuus, Bergfinnur; Ünel, Meltem.
I: Probability Theory and Related Fields, Bind 186, Nr. 3-4, 2023, s. 999-1043.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Trees with exponential height dependent weight
AU - Durhuus, Bergfinnur
AU - Ünel, Meltem
N1 - Publisher Copyright: © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2023
Y1 - 2023
N2 - 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 ZN(μ), 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.
AB - 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 ZN(μ), 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.
KW - Height coupled trees
KW - Local limits of BGW trees
KW - Random trees
U2 - 10.1007/s00440-023-01188-7
DO - 10.1007/s00440-023-01188-7
M3 - Journal article
AN - SCOPUS:85147372161
VL - 186
SP - 999
EP - 1043
JO - Probability Theory and Related Fields
JF - Probability Theory and Related Fields
SN - 0178-8051
IS - 3-4
ER -
ID: 372959166