Quenched GHP scaling limit of critical percolation clusters on Galton-Watson trees

TL;DR

This work proves that critical Bernoulli percolation on a supercritical Galton–Watson tree, in the quenched setting, has a Gromov–Hausdorff–Prokhorov scaling limit given by the corresponding $α$-stable tree (CRT when $α=2$), for offspring in finite variance or in the $α$-stable regime ($α",

Abstract

We consider quenched critical percolation on a supercritical Galton--Watson tree with either finite variance or $α$-stable offspring tails for some $α\in (1,2)$. We show that the GHP scaling limit of a quenched critical percolation cluster on this tree is the corresponding $α$-stable tree, as is the case in the annealed setting. As a corollary we obtain that a simple random walk on the cluster also rescales to Brownian motion on the stable tree. Along the way, we also obtain quenched asymptotics for the tail of the cluster size, which completes earlier results obtained in Michelen (2019) and Archer-Vogel (2024).

Figures (4)

• Figure 1: A supercritical Galton-Watson tree cut at level $11$. The blue and red parts are two independent critical percolation clusters containing $\rho$ on the tree. The orange part is the intersection of the two percolation clusters.
• Figure 2: Coding functions for the given tree. We have marked two vertices on the tree along with the points corresponding to these vertices in the excursions.
• Figure 3: On the left side, we represent $\mathcal{C}^{1},\mathcal{C}^{2},\mathcal{C}^{3}$ on top and the concatenation of their contour functions on the bottom. On the right side, we represent the trees and concatenated contour function obtained by cutting at level $2$.
• Figure 4: On the left we represent the trees $\mathcal{C}^i$ for $i \in \{1,2,3,4\}$. The red part corresponds to the vertices at height larger than $n^{\varepsilon}$. On the right we represent in black the contour function $X$ of the trees $\mathcal{C}^i$. We represent in red the contour process $X^{\ge n^{\varepsilon}}$ of the red part of the trees. On the event $\mathcal{D}_n \cap \mathcal{E}_n$, the size of the black part of the left trees is bounded by $n^{1-\varepsilon/2}$. That is sufficient to see that $|k - \phi(k)| \le 4n^{1-\varepsilon/2}$. Moreover it is clear that $|\sum_{m=1}^{\Lambda_k} Y^{m}_{n^{\varepsilon}}-\Lambda^{\ge n^{\varepsilon}}_{\phi(k)}| \le Y^{\Lambda_k}_{n^{\varepsilon}}$.

Theorems & Definitions (19)

• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Corollary 1.5
• Remark 1.6
• Lemma 2.2
• Proposition 2.3
• Proposition 2.4
• Lemma 3.1
• Proposition 3.2