Gromov–Hausdorff scaling limit at the mesoscopic critical point β=1/2

Determine the Gromov–Hausdorff scaling limit of the rescaled trees n^{-1/2} \mathcal{T}_n in the mesoscopic regime at β=1/2, where the recursive tree with limited memory \mathcal{T}_n is grown by connecting vertex n+1 uniformly among the vertices with labels in [j(n), n] with j(n)=n-\lfloor n^{1/2}\rfloor. Prove or refute the conjecture that n^{-1/2} \mathcal{T}_n converges to a compact random \mathbb{R}-tree that can be approximated by T_k—the rooted \mathbb{R}-tree with k leaves all at height 1 whose branchpoint heights X_1<\dots<X_{k-1} satisfy X_{k-1}=U_{k-1}^{1/(4k(k-1))} and, recursively, X_{\ell-1}=X_\ell U_{\ell-1}^{1/(4\ell(\ell-1))} for independent \operatorname{uniform}[0,1] random variables U_1,\dots,U_{k-1}—as k\to\infty.

Background

The paper studies random recursive trees with limited memory: at time n+1, the new vertex connects uniformly to a vertex among those with labels in [j(n), n], where j(n) increases with n. In the mesoscopic regime j(n)=n-\lfloor n^{\beta}\rfloor, the authors establish local weak limits and detailed global geometric behavior, including a phase transition at \beta=1/2. Distances scale like n^{1-\beta} and, for \beta<1/2, the globally rescaled tree converges in the Gromov–Hausdorff topology to a line segment, while for \beta>1/2 subtrees spanned by the youngest k vertices converge to k-legged stars.

At the critical value \beta=1/2, they describe the joint distribution of branchpoints in subtrees spanned by the k youngest vertices and show these subtrees have random geometry on the leading order. However, the full scaling limit of the entire tree n^{-1/2}\mathcal{T}_n in the Gromov–Hausdorff topology remains unresolved. The authors conjecture a compact random fractal limit that can be approximated by the sequence T_k formed from the subtrees spanned by n, n−1, …, n−k+1.