 of the edge the endpoint that is the ancestor of the other endpoint, which we call the descendant of the edge. The ancestor of $e$ is the endpoint of $e$ that is closer from $O$. With probability $1$, the graph $\mathcal{T}$ admits a tree structure (there is no loop) rooted at the origin $O$. For convenience, we set $\A(O)=O$.

A sequence $(X_{n})_{n\geq 0}$ of vertices of $N$ is a semi-infinite path of the RST $\mathcal{T}$ if, for any $n$, $X_{n}$ is the ancestor of $X_{n+1}$. A semi-infinite path $(X_{n})_{n\geq 0}$ has asymptotic direction $\theta\in [0,2\pi)$ if $$\lim_{n\to\infty} \frac{X_{n}}{|X_{n}|} = e^{\i\theta}$$ (by identifying $\mathbb{R}^{2}$ with the complex plane $\mathbb{C}$).\
Radial random trees in the plane have been studied in many other papers, including Pimentel [@pimentel], Bonichon and Marckert [@bonichonmarckert] or Norris and Turner [@norristurner]. The main difficulty in the case of the RST, is that the local rule used for selecting the ancestor implies complex dependences. The latter make it difficult to exhibit natural Markov processes and prevent a direct use of martingale convergence theorems or Lyapunov functions.

However, our study can rely on the following result of [@baccellibordenave Theorem 2.1] on the RST:

\[HN1\] The following properties hold almost surely:\
(i) Every semi-infinite path of $\mathcal{T}$ has an asymptotic direction;\
(ii) For every $\theta\in[0,2\pi)$, there exists at least one semi-infinite path with asymptotic direction $\theta$;\
(iii) The set of $\theta$’s in $[0,2\pi)$ such that there is more than one semi-infinite path with asymptotic direction $\theta$ is dense in $[0,2\pi)$.

This result is based on a clever method due to Howard and Newman [@howardnewman section 2.3] proving that the above properties hold for any deterministic tree that satisfies some straightness condition, which is shown to be a.s. satisfied for the RST ([@baccellibordenave Theorem 5.4]).\
Understanding finer structural properties of radial random trees, such as the asymptoti