Limit of trees with fixed degree sequence

We show, under natural conditions, that uniform rooted trees with fixed degree sequence converge after renormalization toward inhomogeneous continuum random trees (ICRT). We also provide a sharp upper-bound for the tail of their heights. We also extend our results to P-trees, ICRT, and trees with random degree sequence. In passing we confirm a conjecture of Aldous, Miermont, and Pitman stating that Lévy trees are ICRT with random parameters.

💡 Research Summary

The paper studies uniform rooted trees with a prescribed degree sequence—so‑called D‑trees—where the degree sequence D = (d₁,…,dₙ) satisfies ∑dᵢ = n − 1. The authors investigate the scaling limits of these trees when the number of vertices n tends to infinity. Their main contribution is to show that, under natural regularity conditions on the degree sequence, the rescaled metric space (Tₙ, (σₙ/n)·dₙ) converges to an inhomogeneous continuum random tree (ICRT) introduced by Aldous, Camarri and Pitman. The ICRT is parametrised by a non‑increasing sequence Θ = (θ₀,θ₁,θ₂,…) with ∑θᵢ² = 1; the parameters are obtained as limits of the normalised degrees dᵢ/σₙ, where σₙ = n·∑_{i=1}ⁿ dᵢ(dᵢ − 1) measures the overall degree variance.

The authors first present a “stick‑breaking” construction for D‑trees due to Foata and Fuchs. In this construction the tree is built by repeatedly attaching a new vertex to a uniformly chosen “available slot” among the remaining half‑edges. By encoding the times at which each vertex first appears (Xᵢ), the times of the first repetitions (Yᵢ) and the locations of the repetitions (Zᵢ), the whole geometry of the tree can be described by a triple of point processes. They then introduce an analogous construction for ICRT, based on a Poisson point process on the half‑plane with intensity dy × dμ, where μ is a mixture of a Lebesgue part (θ₀²dx) and atomic parts at the exponential variables Xᵢ∼Exp(θᵢ).

A key technical result (Proposition 2.3) proves joint convergence of the rescaled triples (Xᵢ·σₙ/n, Yᵢ·σₙ/n, Zᵢ·σₙ/n) to the corresponding ICRT triples (Xᵢ, Yᵢ, Zᵢ). This is achieved by coupling the discrete Foata‑Fuchs algorithm with a continuous version that replaces the uniform choices of slots by independent exponential waiting times and then shows that the resulting point processes converge to the Poisson construction of the ICRT. The coupling argument is delicate: it uses a Skorokhod representation, a careful bookkeeping of the “available slots”, and the fact that the exponential variables have the same mean as the original discrete waiting times after scaling.

With this joint convergence in hand, the authors establish two types of metric‑space convergence. Theorem 1.1 shows weak convergence in the Gromov–Prokhorov (GP) topology: for any probability measure pₙ on the vertex set that becomes asymptotically negligible (for instance the uniform measure on leaves), the measured metric space (Tₙ, (σₙ/n)·dₙ, pₙ) converges to (T_Θ, d_Θ, p_Θ). This result captures the distribution of distances between a finite number of random vertices.

To obtain convergence of the whole tree, including global quantities such as height and diameter, a stronger topology is required. The authors introduce a function ψ_D(l) = l ∑_{i=1}ⁿ dᵢ(1 − e^{−dᵢ l/σ_D}) and impose Assumption 2, namely that ∫₀^∞ ψ_D(l) dl < ∞ uniformly in n. This condition is essentially the compactness criterion for ICRT and mirrors the integrability condition for Lévy trees. Under this assumption, Theorem 1.2 proves weak convergence in the Gromov–Hausdorff–Prokhorov (GHP) topology, which implies convergence of the entire metric space together with its natural probability measure.

A further major contribution is an explicit upper bound for the tail of the tree height. Theorem 1.3 shows that there exist constants c, C > 0 such that for any D‑tree, \