Galton–Watson Trees with Vanishing Martingale Limit
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We show that an infinite Galton–Watson tree, conditioned on its martingale limit being smaller than \(\varepsilon \), agrees up to generation \(K\) with a regular \(\mu \)-ary tree, where \(\mu \) is the essential minimum of the offspring distribution and the random variable \(K\) is strongly concentrated near an explicit deterministic function growing like a multiple of \(\log (1/\varepsilon )\). More precisely, we show that if \(\mu \ge 2\) then with high probability, as \(\varepsilon \downarrow 0\), \(K\) takes exactly one or two values. This shows in particular that the conditioned trees converge to the regular \(\mu \)-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 .
KeywordsConditioning principle Large deviations Micro-canonical distribution Sharp thresholds Branching Entropic repulsion
Mathematics Subject Classification (2000)60J80 (Primary) 60F10 60K37
The first author is supported by EPSRC grants EP/G055068/1 and EP/IO3372X/1. The third author is supported by EPSRC grant EP/K016075/1. We would like to thank two anonymous referees for their careful reading of our manuscript.
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