Random skeletons in high-dimensional lattice trees

TL;DR

The paper analyzes the scaling limit of the minimal subtree connecting the root to K uniformly chosen vertices in a high-dimensional, sufficiently spread lattice tree conditioned to survive to time $ns$. It develops a general convergence framework for historical processes, verifies the required conditions for lattice trees and branching random walk, and proves joint convergence of the rescaled subtree and its historical skeleton to the corresponding skeleton of historical Brownian motion conditioned to survive, with a uniform skeleton-approximation result for large $K$. The results connect lattice-tree models to super-Brownian motion and historical Brownian motion, enabling Brownian limits for random walks on lattice trees conditioned to survive and suggesting applicability to related high-dimensional lattice models. The methodology hinges on graph spatial tree constructions, non-degenerate skeletons, and a two-condition abstract theorem (Condition fdd and Condition tau_stuff) that isolates historical convergence from branching-structure convergence. Collectively the work advances the rigorous understanding of genealogical scaling limits and spatial reach for high-dimensional branching structures, with potential extensions to oriented percolation and the voter model.

Abstract

We study the behaviour of the rescaled minimal subtree containing the origin and K random vertices selected from a random critical (sufficiently spread-out, and in dimensions d > 8) lattice tree conditioned to survive until time ns, in the limit as n goes to infinity. We prove joint weak convergence of various quantities associated with these subtrees under this sequence of conditional measures to their counterparts for historical Brownian motion. We also show that when K is sufficiently large the entire rescaled tree is close to this rescaled skeleton with high probability, uniformly in n. These two results are the key conditions used in [5] to prove that the simple random walk on sufficiently spread-out lattice trees (conditioned to survive for a long time) converges to Brownian motion on a super-Brownian motion (conditioned to survive). The main convergence result is established more generally for a sequence of historical processes converging to historical Brownian motion in the sense of finite dimensional distributions and satisfying a pair of technical conditions. The conditions are readily verified for the lattice trees mentioned above and also for critical branching random walk. We expect that it will also apply with suitable changes to other lattice models in sufficiently high dimensions such as oriented percolation and the voter model. In addition some forms of the second skeleton density result are already established in this generality.

Figures (2)

• Figure 1: A non-degenerate shape $T\in\Sigma_7$ also showing some of the corresponding edge labels. If $\ell:E(T)\to (0,\infty)$ is given then the corresponding metric space $(\bar{T},d)$ has e.g. the distance between the midpoints of $e_{9}$ and $e_{12}$ equal to $\ell(e_{9})/2+\ell(e_{11})+\ell(e_{12})/2$
• Figure 2: Left: a depiction of 3 paths ($w_2,w_3,w_1$ from top to bottom) for which the associated ${\boldsymbol \tau}$ is non-degenerate. Close parallel lines depict parts of paths that exactly coincide. Right: The corresponding shape $T({\boldsymbol \tau})$ (labels 2 and 3 can be swapped without changing the shape). Edge lengths are differences of $\tau_{\cdot,\cdot}$ (e.g. the edge adjacent to 2 has length $\ell(e_{\langle 2,2 \rangle})=\tau_{2,2}-\tau_{2,3}$) and these edge lengths define $d$ on $T({\boldsymbol \tau})$.

Theorems & Definitions (92)

• Remark 1
• Definition 2
• Remark 3
• Definition 4
• Remark 5
• Remark 6
• Definition 7
• Proposition 8
• Lemma 9
• Definition 10