A note on the scaling limits of random Pólya trees

Abstract: Panagiotou and Stufler (arXiv:1502.07180v2) recently proved one important fact on their way to establish the scaling limits of random P\'{o}lya trees: a uniform random P\'{o}lya tree of size $n$ consists of a conditioned critical Galton-Watson tree $C_n$ and many small forests, where with probability tending to one as $n$ tends to infinity, any forest $F_n(v)$, that is attached to a node $v$ in $C_n$, is maximally of size $\vert F_n(v)\vert=O(\log n)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities. In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements on the bound of $\vert F_n(v)\vert$, namely $\vert F_n(v)\vert=\Theta(\log n)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given P\'{o}lya tree. Finally, we derive the limit probability that for a random node $v$ the attached forest $F_n(v)$ is of a given size.