1205.3145 (Igor Kortchemski)
Igor Kortchemski
We study a particular type of subcritical Galton-Watson trees, which are called non-generic trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in large conditioned non-generic trees, meaning that with high probability there exists a unique vertex with macroscopic degree comparable to the total size of the tree. We investigate this phenomenon by studying scaling limits of such trees. Using recent results concerning subexponential distributions, we study the convergence of three functions coding these trees (the Lukasiewicz path, the contour function and the height function) and show that the situation is completely different from the critical case. In particular, the height of such trees grows logarithmically in their size. We also study fluctuations around the condensation vertex.
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http://arxiv.org/abs/1205.3145
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