[Paper Review] Large deviations for random walks in a random environment on a strip

Jonathon Peterson|arXiv (Cornell University)|Feb 4, 2013

Stochastic processes and statistical mechanics24 references3 citations

TL;DR

This paper establishes quenched and averaged large deviation principles for random walks in a random environment on a strip of finite width $\mathbb{Z} \times [d]$, proving variational formulas that relate the quenched and averaged rate functions. The approach extends the unified framework of Comets, Gantert, and Zeitouni to higher-dimensional strips, using matrix product structures and spectral analysis to derive large deviation asymptotics for position and hitting times.

ABSTRACT

We consider a random walk in a random environment (RWRE) on the strip of finite width $\mathbb{Z} imes \{1,2,\ldots,d\}$. We prove both quenched and averaged large deviation principles for the position and the hitting times of the RWRE. Moreover, we prove a variational formula that relates the quenched and averaged rate functions, thus extending a result of Comets, Gantert, and Zeitouni for nearest-neighbor RWRE on $\mathbb{Z}$

Motivation & Objective

• To establish large deviation principles for the position $X_n/n$ and hitting times $T_n/n$ of a random walk in a random environment on a finite-width strip $\mathbb{Z} \times [d]$.
• To derive both quenched and averaged large deviation rate functions for the walk's position and hitting times under general i.i.d. environments.
• To extend the variational formula relating quenched and averaged rate functions—previously known for nearest-neighbor RWRE on $\mathbb{Z}$—to the strip model.
• To provide a unified analytical framework using matrix products and spectral theory to handle the higher-dimensional state space.

Proposed method

• Model the RWRE on the strip using i.i.d. $d \times d$ transition matrices $q_n, r_n, p_n$ for left, right, and within-level moves at each site $n \in \mathbb{Z}$.
• Define quenched and averaged laws via $P_\omega^{(x,i)}$ and $\mathbb{P}_\eta^{(x,i)} = \mathbb{E}_\eta[P_\omega^{(x,i)}]$, respectively.
• Use matrix product representations $\Phi_k(\lambda)$ to encode the moment generating functions of the walk, decomposing them into blocks $A_k(\lambda)$ and $B_k(\lambda)$.
• Establish uniform lower and upper bounds on matrix entries $A_k(\lambda), B_k(\lambda)$ for $\lambda$ below a critical value $\lambda_{\text{crit}}(\eta)$, ensuring spectral stability.
• Prove exponential decay of relative error in left and right eigenvector approximations via contraction arguments on matrix products $A_{[m,n]}(\lambda)$.
• Construct limiting vectors $\mu_k(\lambda)$ and $\nu_k(\lambda)$ for the normalized left and right eigenvectors, with explicit error bounds in $L^1$ and $L^\infty$.

Experimental results

Research questions

• RQ1 How do quenched and averaged large deviation principles for $X_n/n$ and $T_n/n$ differ in the strip model?
• RQ2 Can a variational formula be derived that relates the quenched and averaged rate functions in this higher-dimensional setting?
• RQ3 What spectral and matrix product properties ensure the existence and convergence of the relevant eigenvectors for large deviations?
• RQ4 How do the uniform bounds on matrix entries affect the stability and convergence of the large deviation rate functions?

Key findings

• The quenched and averaged large deviation principles for $X_n/n$ and $T_n/n$ are established under general i.i.d. environments on the strip.
• A variational formula is derived that expresses the averaged rate function as a minimization over quenched rate functions, extending a result of Comets, Gantert, and Zeitouni to the strip model.
• The existence of limiting eigenvectors $\mu_k(\lambda)$ and $\nu_k(\lambda)$ is proven with $L^1$ and $L^\infty$ error bounds decaying exponentially in the product length.
• Uniform bounds $c_\lambda \leq A_k(\lambda)(i,j), B_k(\lambda)(i,j) \leq 1/c_\lambda$ hold for all $k$ and $\lambda < \lambda_{\text{crit}}(\eta)$, ensuring spectral regularity.
• The convergence of normalized matrix products $\Phi_{[k,n]}(\lambda)\mathbf{1}/\|\cdot\|_1$ to a limit $\nu_k(\lambda)$ is shown with error bounds $O((1 - c_\lambda^4)^{n-k})$.