[Paper Review] Annealed invariance principle for random walks on random graphs generated by point processes in $\mathbb{R}^d$

Arnaud Rousselle|arXiv (Cornell University)|Jun 18, 2015

Stochastic processes and statistical mechanics28 references3 citations

TL;DR

This paper establishes an annealed invariance principle for continuous-time simple random walks on random geometric graphs—such as Delaunay triangulations, Gabriel graphs, and creek-crossing graphs—constructed from point processes in $\mathbb{R}^d$. Under mild moment and ergodicity conditions on the underlying point process, the rescaled walk converges to a Brownian motion under the annealed measure, leveraging the environment-marked particle process and Markovian embedding techniques.

ABSTRACT

We consider simple random walks on random graphs embedded in $\mathbb{R}^d$ and generated by point processes such as Delaunay triangulations, Gabriel graphs and the creek-crossing graphs. Under suitable assumptions on the point process, we show an annealed invariance principle for these random walks. These results hold for a large variety of point processes including Poisson point processes, Matérn cluster and Matérn hardcore processes which have respectively clustering and repulsiveness properties. The proof relies on the use the process of the environment seen from the particle. It allows to reconstruct the original process as an additive functional of a Markovian process under the annealed measure.

Motivation & Objective

• To establish an annealed invariance principle for continuous-time simple random walks on random geometric graphs in $\mathbb{R}^d$.
• To extend the invariance principle to geometric graphs generated by diverse point processes, including those with clustering (Matérn cluster) or repulsion (Matérn hardcore).
• To unify the analysis of random walks on geometric graphs via the environment seen from the particle framework.
• To validate the scaling limit for a broad class of point processes, including stationary, ergodic, and aperiodic ones.
• To provide a general framework for proving diffusive limits under the annealed measure, applicable beyond lattice-based models.

Proposed method

• Uses the environment seen from the particle to construct a Markovian process that reconstructs the original walk as an additive functional.
• Applies the annealed measure as a semi-direct product measure over the point process realizations and the walk paths.
• Employs the generator $\mathcal{L}^{\xi^0}f(y) = \sum_{z \in \xi^0} \mathbf{1}_{\{\{y,z\} \in E_{G(\xi^0)}\}} (f(z) - f(y))$ to define the continuous-time walk with exponential holding times.
• Relies on the Feller-Meyer-Meyer-Weinberger (FMW) approach and the martingale functional central limit theorem under the annealed measure.
• Establishes moment conditions on the degree distribution to ensure non-explosion and tightness of the rescaled process.
• Uses the Palm version $\mathcal{P}_0$ to condition on the origin being in the point process, enabling stationarity and ergodicity arguments.

Experimental results

Research questions

• RQ1 Does the annealed scaling limit of a continuous-time simple random walk on a random geometric graph converge to a Brownian motion?
• RQ2 Can the invariance principle be extended to geometric graphs derived from point processes with clustering or repulsion, such as Matérn cluster or hardcore processes?
• RQ3 Under what conditions on the point process does the environment seen from the particle yield a Markov process that allows for a functional central limit theorem?
• RQ4 How do assumptions like stationarity, ergodicity, aperiodicity, and finite-degree moments affect the validity of the annealed invariance principle?
• RQ5 Can the framework handle multiple geometric graph constructions (Delaunay, Gabriel, creek-crossing) under a common analytical umbrella?

Key findings

• The annealed invariance principle holds for random walks on Delaunay triangulations, Gabriel graphs, and creek-crossing graphs under suitable moment and ergodicity conditions.
• The rescaled random walk converges weakly to a Brownian motion in $\mathbb{R}^d$ under the annealed measure, with the limiting diffusion matrix being non-degenerate under mild assumptions.
• The result applies to Poisson point processes, Matérn cluster processes, and Matérn hardcore processes, all of which satisfy the required moment and structural conditions.
• The second factorial moment measure of the point process is absolutely continuous with respect to the $(2d)$-dimensional Lebesgue measure, ensuring sufficient mixing for the central limit theorem.
• The degree of the origin has polynomial moments of all orders under $\mathcal{P}_0$, which guarantees the non-explosion of the continuous-time walk.
• The proof technique via the environment seen from the particle allows the original process to be represented as an additive functional of a Markov process under the annealed measure, enabling the use of martingale CLT tools.