Limiting Behavior Of Additive Functionals On The Stable Tree

We study the shape of the normalized stable L\'{e}vy tree $\mathcal{T}$ near its root. We show that, when zooming in at the root at the proper speed with a scaling depending on the index of stability, we get the unnormalized Kesten tree. In particular the limit is described by a tree-valued Poisson point process which does not depend on the initial normalization. We apply this to study the asymptotic behavior of additive functionals of the form $$\mathbf{Z}_{\alpha,\beta}=\int_{\mathcal{T}} \mu(\mathrm{d} x) \int_0^{H(x)} \sigma_{r,x}^\alpha \mathfrak{h}_{r,x}^\beta\,\mathrm{d} r$$as $\max(\alpha,\beta) \to \infty$, where $\mu$ is the mass measure on $\mathcal{T}$, $H(x)$ is the height of $x$ and $\sigma_{r,x}$ (resp. $\mathfrak{h}_{r,x}$) is the mass (resp. height) of the subtree of $\mathcal{T}$ above level $r$ containing $x$. Such functionals arise as scaling limits of additive functionals of the size and height on conditioned Bienaym{\'e}-Galton-Watson trees.