### Abstract

Original language | English |
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Pages (from-to) | 737-762 |

Journal | Journal of Statistical Physics |

Volume | 155 |

Issue number | 4 |

Early online date | 23 Mar 2014 |

DOIs | |

Publication status | Published - May 2014 |

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### Cite this

*Journal of Statistical Physics*,

*155*(4), 737-762. https://doi.org/10.1007/s10955-014-0975-1

**Galton-Watson trees with vanishing martingale limit.** / Berestycki, Nathanael; Gantert, Nina; Morters, Peter; Sidorova, Nadia.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 155, no. 4, pp. 737-762. https://doi.org/10.1007/s10955-014-0975-1

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TY - JOUR

T1 - Galton-Watson trees with vanishing martingale limit

AU - Berestycki, Nathanael

AU - Gantert, Nina

AU - Morters, Peter

AU - Sidorova, Nadia

PY - 2014/5

Y1 - 2014/5

N2 - We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than (Formula presented.), agrees up to generation (Formula presented.) with a regular (Formula presented.)-ary tree, where (Formula presented.) is the essential minimum of the offspring distribution and the random variable (Formula presented.) is strongly concentrated near an explicit deterministic function growing like a multiple of (Formula presented.). More precisely, we show that if (Formula presented.) then with high probability, as (Formula presented.), (Formula presented.) takes exactly one or two values. This shows in particular that the conditioned trees converge to the regular (Formula presented.)-ary tree, providing an example of entropic repulsion where the limit has vanishing entropy. Our proofs are based on recent results on the left tail behaviour of the martingale limit obtained by Fleischmann and Wachtel [ 11 ].

AB - We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than (Formula presented.), agrees up to generation (Formula presented.) with a regular (Formula presented.)-ary tree, where (Formula presented.) is the essential minimum of the offspring distribution and the random variable (Formula presented.) is strongly concentrated near an explicit deterministic function growing like a multiple of (Formula presented.). More precisely, we show that if (Formula presented.) then with high probability, as (Formula presented.), (Formula presented.) takes exactly one or two values. This shows in particular that the conditioned trees converge to the regular (Formula presented.)-ary tree, providing an example of entropic repulsion where the limit has vanishing entropy. Our proofs are based on recent results on the left tail behaviour of the martingale limit obtained by Fleischmann and Wachtel [ 11 ].

UR - http://www.scopus.com/inward/record.url?scp=84896406485&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1007/s10955-014-0975-1

U2 - 10.1007/s10955-014-0975-1

DO - 10.1007/s10955-014-0975-1

M3 - Article

VL - 155

SP - 737

EP - 762

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 4

ER -