[Paper Review] On symmetric random walks with random conductances on $\Z^d$

Luiz Renato Fontes, Pierre Mathieu|ArXiv.org|Mar 8, 2004

Stochastic processes and statistical mechanics3 references4 citations

TL;DR

This paper studies symmetric continuous-time random walks on $ℤ^d$ with i.i.d. random conductances having polynomial tails near zero. It establishes sharp annealed decay estimates for the return probability, showing it can decay as $t^{-\gamma}$ with $\gamma < d/2$ due to slow conductance tails, and provides a universal lower bound on the spectral gap for finite tori, separating diffusive scaling from conductance disorder effects.

ABSTRACT

We study models of continuous time, symmetric, $\Z^d$-valued random walks in random environments. One of our aims is to derive estimates on the decay of transition probabilities in a case where a uniform ellipticity assumption is absent. We consider the case of independent conductances with a polynomial tail near 0, and obtain precise asymptotics for the annealed return probability and convergence times for the random walk confined to a finite box.

Motivation & Objective

• To derive precise asymptotics for the annealed return probability of symmetric random walks on $ℤ^d$ when conductances lack uniform ellipticity.
• To analyze the decay of transition probabilities in the absence of a uniform lower bound on conductances, particularly when they have polynomial tails near zero.
• To establish a universal lower bound on the spectral gap for symmetric random walks on finite tori under heavy-tailed conductance distributions.
• To compare quenched and annealed decay behaviors, highlighting differences when conductance tails are heavy.
• To separate the diffusive $N^{-2}$ scaling from the effects of rare, small conductances in finite-volume convergence times.

Proposed method

• Uses a trace formula and comparison lemma for annealed return probabilities in general reversible Markov chains on $ℤ^d$.
• Applies spectral analysis on finite boxes $B_N$ and tori, leveraging Dirichlet forms and eigenvalue estimates for the generator $\mathcal{G}^\omega$.
• Employs a projection argument on the first $j$ eigenvectors to bound the $L^2$-norm of eigenfunctions restricted to the infinite cluster $\mathcal{C}_N^\omega$.
• Establishes a lower bound on the spectral gap $\mu_j$ of the random walk on $\mathcal{C}_N^\omega$ using the $\ell^2$-norm of eigenvectors and conductance bounds $\omega(x) \geq N^{-\varepsilon}$.
• Uses the inequality $\lambda_i e^{-\lambda_i t} \leq \frac{1}{t} e^{-\frac{1}{2}\lambda_i t}$ to control exponential moments in the annealed expectation.
• Combines estimates on the spectral gap and eigenvalue decay to prove convergence of the annealed return probability to zero at rate $t^{-\gamma}$ with $\gamma < d/2$.

Experimental results

Research questions

• RQ1 How does the annealed return probability $\mathbb{Q}.\mathbb{P}^\omega[X_t = 0]$ decay in time when conductances have polynomial tails near zero and lack uniform ellipticity?
• RQ2 Can a universal lower bound on the spectral gap of symmetric random walks on finite tori be established that separates the $N^{-2}$ diffusive scaling from the effects of small conductances?
• RQ3 What is the relationship between the tail behavior of conductance distributions and the decay exponent $\gamma$ in the annealed return probability $t^{-\gamma}$?
• RQ4 How does the quenched decay of the return probability differ from the annealed decay when conductance tails are heavy?
• RQ5 To what extent does the largest connected component of the infinite cluster in a finite box $B_N$ capture the behavior of the random walk, and how does its density behave as $N \to \infty$?

Key findings

• The annealed return probability decays as $t^{-\gamma}$ with $\gamma < d/2$ when conductances have polynomial tails near zero, deviating from the classical $t^{-d/2}$ decay.
• A universal lower bound on the spectral gap of the random walk on a torus of side length $N$ is established, independent of the conductance distribution, which isolates the $N^{-2}$ diffusive scaling.
• The expected density of the component of the origin in the infinite cluster $\mathcal{C}_N^\omega \cap B_N$ tends to 1 as $N \to \infty$, justifying the use of $\mathcal{C}_N^\omega$ as a proxy for the full space.
• The quenched decay of the return probability differs from the annealed decay for small values of $\gamma$, indicating that disorder effects are more pronounced in the quenched setting.
• The spectral gap lower bound $\mu_j \geq \left(\frac{j^{1/d}}{N}\right)^2 (\log N)^{-8(d-\eta)/d}$ holds on the event $\Omega_N$, which controls the size of the largest component.
• The convergence of the annealed return probability to zero at rate $t^{-\gamma}$ is proven under the condition $\alpha < \frac{d}{2} \frac{1+\gamma}{1+d/2}$, with $\gamma$ related to the tail index of the conductance law.