[Paper Review] Quenched large deviations for random walk in a random environment

Atilla Yilmaz|ArXiv.org|Apr 1, 2008

Stochastic processes and statistical mechanics30 references3 citations

TL;DR

This paper establishes the quenched large deviation principle (LDP) for random walk in a random environment on $\mathbb{Z}^d$ with bounded jumps, using the point of view of the particle and a variational formula for the rate function. It verifies an Ansatz for the minimizer in the one-dimensional case, generalizing prior nearest-neighbor results to walks with bounded jumps.

ABSTRACT

We take the point of view of a particle performing random walk with bounded jumps on $\mathbb{Z}^d$ in a stationary and ergodic random environment. We prove the quenched large deviation principle (LDP) for the pair empirical measure of the environment Markov chain. By an appropriate contraction, we deduce the quenched LDP for the mean velocity of the particle and obtain a variational formula for the corresponding rate function. We propose an Ansatz for the minimizer of this formula. When $d=1$, we verify this Ansatz and generalize the nearest-neighbor result of Comets, Gantert and Zeitouni to walks with bounded jumps.

Motivation & Objective

• To establish the quenched large deviation principle for the mean velocity of a random walk in a random environment with bounded jumps on $\mathbb{Z}^d$.
• To derive a variational formula for the quenched rate function via contraction from the pair empirical measure of the environment Markov chain.
• To propose and verify an Ansatz for the minimizer of the variational formula in the one-dimensional case.
• To generalize the nearest-neighbor result of Comets, Gantert, and Zeitouni to walks with bounded jumps in $d=1$.
• To analyze the quenched LDP using the point of view of the particle and the Doob $h$-transform framework.

Proposed method

• Formulate the quenched LDP for the pair empirical measure of the environment Markov chain using the quenched measure $P_x^\omega$.
• Apply contraction to deduce the quenched LDP for the mean velocity, leading to a variational formula for the rate function.
• Propose an Ansatz for the minimizer of the variational formula based on the structure of the environment kernel and invariant measures.
• Use the point of view of the particle to treat the environment Markov chain as a primary stochastic process.
• Employ the Doob $h$-transform to analyze the invariant measures and transition kernels under the quenched measure.
• Verify the Ansatz in $d=1$ using moment conditions and the Garsia-Rodemich-Rumsey theorem to establish equicontinuity of rescaled functions.

Experimental results

Research questions

• RQ1 What is the quenched large deviation rate function for the mean velocity of a random walk with bounded jumps in a stationary, ergodic random environment?
• RQ2 Can a variational formula be derived for the quenched rate function through contraction from the pair empirical measure of the environment Markov chain?
• RQ3 Does the proposed Ansatz for the minimizer of the variational formula hold in the one-dimensional case with bounded jumps?
• RQ4 How does the quenched LDP generalize the nearest-neighbor result of Comets, Gantert, and Zeitouni to walks with bounded jumps?
• RQ5 What conditions ensure the lower semicontinuity or continuity of the rate function in the quenched setting?

Key findings

• The quenched LDP is established for the mean velocity of a random walk with bounded jumps in a stationary and ergodic random environment on $\mathbb{Z}^d$.
• A variational formula for the quenched rate function is derived via contraction from the quenched LDP for the pair empirical measure of the environment Markov chain.
• The Ansatz for the minimizer of the variational formula is verified in the one-dimensional case, confirming its validity under the given assumptions.
• The result generalizes the nearest-neighbor quenched LDP of Comets, Gantert, and Zeitouni to walks with bounded jumps in $d=1$.
• The rate function is shown to be finite and lower semicontinuous, though not necessarily continuous, under general conditions.
• In the periodic environment case, the rate function $\mathfrak{I}$ is continuous due to the finite-dimensionality of the state space and uniform ellipticity.