[Paper Review] A central limit theorem for random walk in random environment on marked Galton-Watson trees

Gabriel Faraud|arXiv (Cornell University)|Dec 10, 2008

Stochastic processes and statistical mechanics19 references7 citations

TL;DR

This paper establishes a central limit theorem for biased random walks on marked Galton-Watson trees with i.i.d. edge weights, extending results of Peres and Zeitouni. It proves recurrence/transience criteria and shows that under moment conditions on the environment, the walk satisfies a functional central limit theorem with diffusive scaling, confirming Gaussian fluctuations in the quenched law despite quenched disorder.

ABSTRACT

In this article we focus on a general model of random walk on random marked trees. We prove a recurrence criterion, analogue to the recurrence criterion proved by R. Lyons and Robin Pemantle (1992) in a slightly different model. In the critical case, we obtain a criterion for the positive/null recurrence. Several regimes appear, as proved (in a similar model), by Y. Hu and Z. Shi (2007). We focus on the "diffusive" regime and improve their result in this case, by obtaining a functional Central Limit Theorem. Our result is also an extension of a result by Y. Peres and O. Zeitouni (2008), obtained in the setting of biased random walk in Galton-Watson trees.

Motivation & Objective

• To generalize the recurrence criterion of Lyons and Pemantle to a broader class of marked Galton-Watson trees with dependent tree structure and transition probabilities.
• To establish a functional central limit theorem for biased random walks on such random trees under moment conditions on the environment.
• To analyze the quenched asymptotic behavior of the walk, particularly the scaling limit and fluctuations, in the presence of quenched disorder.
• To extend the results of Peres and Zeitouni on biased random walks on Galton-Watson trees to a more general model with i.i.d. edge weights.

Proposed method

• Constructs a marked Galton-Watson tree using a probability measure $ q $ on $ \mathbb{N} \times (\mathbb{R}_+^*)^\mathbb{N} $, allowing dependence between tree structure and transition weights.
• Defines a random walk with transition probabilities derived from i.i.d. edge weights $ A(x) $, ensuring the walk is Markovian and respects the tree structure.
• Uses electrical network and capacitated network analogies, with conductances $ C_x = \prod_{e < z \leq x} A(z) $, to analyze hitting times and recurrence.
• Applies Azuma's inequality and martingale techniques to control deviations in the walk's position, particularly under quenched conditioning.
• Employs a coupling argument with a renewal process and a sequence of i.i.d. increments to control the walk's behavior over time.
• Applies dominated convergence and Borel-Cantelli-type arguments to show that rare events (e.g., large excursions) have vanishing probability under the annealed measure.

Experimental results

Research questions

• RQ1 Under what conditions does a biased random walk on a marked Galton-Watson tree satisfy a functional central limit theorem?
• RQ2 How does the quenched law of the walk behave asymptotically when the environment has heavy-tailed or bounded moments?
• RQ3 Can the recurrence criterion of Lyons and Pemantle be extended to models where the tree structure and transition weights are dependent?
• RQ4 What is the scaling limit of the walk's position under the annealed measure, and does it converge to a Brownian motion?
• RQ5 How do large deviations and rare events—such as long excursions or large displacements—affect the convergence to the central limit theorem?

Key findings

• The paper proves a functional central limit theorem for the biased random walk on marked Galton-Watson trees under the annealed measure, with diffusive scaling $ t^{1/2} $, confirming Gaussian fluctuations.
• It establishes that under moment conditions on $ \rho(\alpha) $, the walk satisfies $ \mathbb{P}_{\text{MT}}(\max_{s \leq t} |Y_s| \geq t^\alpha) \to 0 $ for $ \alpha < 1/2 $, indicating subdiffusive fluctuations.
• The probability of large excursions or large displacements, such as $ W_{X_s} > t^{1/4 - \varepsilon} $, vanishes as $ t \to \infty $, confirming tightness of the rescaled walk.
• The results extend Peres and Zeitouni's work by generalizing the model to include dependent tree and weight structures, while preserving the central limit behavior.
• The proof relies on controlling the number of visits to rare sets via moment bounds and coupling with i.i.d. increments, showing that such events are negligible under the annealed measure.
• The paper confirms that the quenched law of the walk, conditioned on typical environments, converges to a Brownian motion after diffusive scaling, despite quenched disorder.