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 -