Feb 14  Wed  Robin Stephenson (Sheffield)  Probability seminar  
16:00  Where do trees grow leaves?  
Hicks Seminar Room J11  
Abstract: We study a model of random binary trees grown ``by the leaves" in the style of Luczak and Winkler (2004). If $\tau_n$ is a uniform plane binary tree of size $n$, Luczak and Winkler, and later explicitly Caraceni and Stauffer, constructed a measure $\nu_{\tau_n}$ such that the tree obtained by adding a cherry on a leaf sampled according to $\nu_{\tau_n}$ is still uniformly distributed on the set of all plane binary trees with size $n+1$. It turns out that the measure $\nu_{\tau_n}$, which we call the leafgrowth measure, is noticeably different from the uniform measure on the leaves of the tree $\tau_n$. In fact we prove that as $n \to \infty$, with high probability it is almost entirely supported by a subset of only $n^{3 ( 2  \sqrt{3})+o(1)} \approx n^{0.8038...}$ leaves. In the continuous setting, we construct the scaling limit of this measure, which is a probability measure on the Brownian Continuum Random Tree supported by a fractal set of dimension $ 6 (2  \sqrt{3})$. We also compute the full (discrete) multifractal spectrum. This work is a first step towards understanding the diffusion limit of the discrete leafgrowth procedure. 


