[Paper Review] A quantitative central limit theorem for the random walk among random conductances

Jean-Christophe Mourrat|arXiv (Cornell University)|May 23, 2011

Advanced Mathematical Modeling in Engineering17 references4 citations

TL;DR

This paper establishes a quantitative central limit theorem for the simple random walk on $ɲ^d$ with i.i.d. random conductances bounded away from zero and infinity. Using a combination of spectral gap estimates, martingale decompositions, and homogenization techniques, it proves Berry-Esseen-type bounds with convergence rates of $t^{-1/10}$ for $d \leq 2$ and $t^{-1/5}$ for $d \geq 3$, up to logarithmic corrections, providing the first explicit error rates for the quenched CLT in this setting.

ABSTRACT

We consider the random walk among random conductances on Z^d. We assume that the conductances are independent, identically distributed and uniformly bounded away from 0 and infinity. We obtain a quantitative version of the central limit theorem for this random walk, which takes the form of a Berry-Esseen estimate with speed t^{-1/10} for d < 3, and speed t^{-1/5} otherwise, up to logarithmic corrections.

Motivation & Objective

• To establish a quantitative quenched central limit theorem for the simple random walk among i.i.d. random conductances on $\mathbb{Z}^d$.
• To provide explicit error bounds for the convergence of the rescaled position of the walk to a Brownian motion.
• To extend the understanding of the speed of convergence in homogenization theory for random media beyond asymptotic results.
• To address the lack of quantitative probabilistic results in the quenched setting for random conductance models.

Proposed method

• The analysis relies on a spectral gap estimate for the environment Markov chain viewed from the particle, which controls the mixing of the environment.
• A martingale decomposition is applied to the rescaled position process to separate the martingale part from the residual terms.
• The generator and carré du champ of the Feller process are used to characterize the quadratic variation of the martingale component.
• The proof uses a localization argument via a box $C_L$ of size $L$, followed by a rescaling and averaging over space to control the error terms.
• The method involves bounding the variance of the displacement using a Green's function-type estimate and applying a Cauchy-Schwarz inequality in a discrete setting.
• Logarithmic corrections are handled through careful control of the dependence on the size of the system and the conductance distribution.

Experimental results

Research questions

• RQ1 What is the rate of convergence to the Brownian motion in the quenched central limit theorem for the random walk among i.i.d. conductances on $\mathbb{Z}^d$?
• RQ2 Can explicit error bounds be derived for the quenched CLT in this random media model?
• RQ3 How does the convergence rate depend on the dimension $d$ and the ellipticity constants of the conductances?
• RQ4 What techniques can be used to quantify the speed of convergence in the quenched setting, given the lack of previous results in this direction?

Key findings

• The paper establishes a Berry-Esseen estimate with convergence rate $t^{-1/10}$ for $d \leq 2$, up to logarithmic corrections.
• For $d \geq 3$, the convergence rate is $t^{-1/5}$, again up to logarithmic corrections.
• The bounds are quenched, meaning they hold almost surely for typical environments, not just in averaged sense.
• The error estimates are derived using spectral gap bounds and martingale decomposition techniques applied to the environment viewed from the particle.
• The rate dependence on dimension reflects the interplay between recurrence/transience and the effective diffusivity in the random medium.
• The result provides the first explicit quantitative control on the quenched CLT for this model, filling a gap in the probabilistic homogenization literature.