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Scaling limit for the random walk on critical lattice trees

Gérard Ben Arous, Manuel Cabezas, Alexander Fribergh

TL;DR

The paper establishes a scaling limit for the simple random walk on critical lattice trees in dimensions $d\ge 8$, showing convergence to the Brownian motion on the Integrated Super-Brownian Excursion (BISE). It develops a lace-expansion–based framework and proves a new convergence theorem that reduces the required conditions to a trio by leveraging a uniform edge-volume sampling scheme for spanning points, then applies the results to lattice trees. The approach connects discrete random-graph diffusion to continuum objects (CRT, ISE, BISE) via graph-skeletons and empirical measures, illustrating universality of anomalous diffusion on high-dimensional critical graphs. The findings illuminate how random walks on critical geometries converge to a universal diffusion process, with time scaling governed by the limiting volume and edge-structure parameters.

Abstract

We prove a scaling limit theorem for the simple random walk on critical lattice trees in $\mathbb{Z}^d$, for $d\geq 8$. The scaling limit is the Brownian motion on the Integrated Super-Brownian Excursion (BISE) which is the same one that we have identified earlier for other simpler models of anomalous diffusion on critical graphs in large enough dimension. The proof of this theorem is based on a combination of the tools of lace-expansion (contained in the articles \cite{CFHP} and \cite{CFHP2}), and a new and general convergence theorem.

Scaling limit for the random walk on critical lattice trees

TL;DR

The paper establishes a scaling limit for the simple random walk on critical lattice trees in dimensions , showing convergence to the Brownian motion on the Integrated Super-Brownian Excursion (BISE). It develops a lace-expansion–based framework and proves a new convergence theorem that reduces the required conditions to a trio by leveraging a uniform edge-volume sampling scheme for spanning points, then applies the results to lattice trees. The approach connects discrete random-graph diffusion to continuum objects (CRT, ISE, BISE) via graph-skeletons and empirical measures, illustrating universality of anomalous diffusion on high-dimensional critical graphs. The findings illuminate how random walks on critical geometries converge to a universal diffusion process, with time scaling governed by the limiting volume and edge-structure parameters.

Abstract

We prove a scaling limit theorem for the simple random walk on critical lattice trees in , for . The scaling limit is the Brownian motion on the Integrated Super-Brownian Excursion (BISE) which is the same one that we have identified earlier for other simpler models of anomalous diffusion on critical graphs in large enough dimension. The proof of this theorem is based on a combination of the tools of lace-expansion (contained in the articles \cite{CFHP} and \cite{CFHP2}), and a new and general convergence theorem.

Paper Structure

This paper contains 39 sections, 19 theorems, 117 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Critical lattice trees
  3. Simple conditions for the convergence towards the Brownian motion on the ISE
  4. Organization of the paper
  5. Notations
  6. Brownian motion on super-brownian motions
  7. The continuum random tree (CRT) and random trees conditioned on height
  8. The integrated super-Brownian excursion (ISE) and its counterpart conditioned on height
  9. The Brownian motion on the ISE: $B^{\text{ISE}}$
  10. Graph spatial trees and approximations of the ISE and the $B^{\text{ISE}}$
  11. Graph spatial trees
  12. The $K$-CRT, the $K$-ISE and the $B^{K\text{-ISE}}$ and their counterparts conditioned on height
  13. The skeleton of a graph
  14. Approximating a tree-like graph by a graph spatial tree
  15. Adding a measure associated to the volume of the graph
  16. ...and 24 more sections

Key Result

Theorem 1.1

Fix $d>8$. There exists $L_0(d)$ such that for all $L\geq L_0$ there exists constants $C_1, C_2 \in (0,\infty)$ such that, for each $h>0$, where the convergence is annealed (under $\mathbb{P}^h_{ n}$) and occurs with the topology of uniform convergence over compact sets of time. The limiting process $B^{h\text{-ISE}}$ is the Brownian motion on the integrated super-Brownian excursion conditioned t

Figures (3)

  • Figure 1: The transformation a graph $G$ is turned into $G(4)$.
  • Figure 2: Notations associated to the construction of the atoms of the measure $\hat{\mu}^{(n,K,K_1,K_2)}$
  • Figure 3: A dense set for $K'$ large enough and an example of $\vec{x}_{n,K,K'}$

Theorems & Definitions (50)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.5
  • ...and 40 more