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Abstract
We show that an infinite GaltonWatson 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 ].
Original language  English 

Pages (fromto)  737762 
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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Dive into the research topics of 'GaltonWatson trees with vanishing martingale limit'. Together they form a unique fingerprint.Projects
 2 Finished

Emergence of Condensation in Stochastic Systems
Morters, P.
Engineering and Physical Sciences Research Council
1/08/13 → 31/08/16
Project: Research council

INTERSECTION LOCAL TIMES AND STOCHASTIC PROCESSES IN RANDOM MEDIA
Morters, P.
Engineering and Physical Sciences Research Council
1/09/05 → 31/08/10
Project: Research council