[Paper Review] Some properties of the rate function of quenched large deviations for random walk in random environment

Alexis Devulder|arXiv (Cornell University)|Feb 15, 2005

Stochastic processes and statistical mechanics9 references3 citations

TL;DR

This paper establishes that the quenched large deviation rate function for one-dimensional random walk in a random environment exhibits infinite second derivative at zero, confirming a conjecture by Greven and den Hollander. Using hitting time analysis and asymptotic estimates, it further proves that the rate function for Brox-type diffusions in drifted Brownian potentials is strictly smaller than that of a Brownian motion with the same speed, providing a continuous analogue to the RWRE problem.

ABSTRACT

In this paper, we are interested in some questions of Greven and den Hollander about the rate function $I\_η^q$ of quenched large deviations for random walk in random environment. By studying the hitting times of RWRE, we prove that in the recurrent case, $\lim\_{θ o 0^+}(I\_η^q)''(θ)=+\infty$, which gives an affirmative answer to a conjecture of Greven and den Hollander. We also establish a comparison result between the rate function of quenched large deviations for a diffusion in a drifted Brownian potential, and the rate function for a drifted Brownian motion with the same speed.

Motivation & Objective

• To resolve Open Problem 2 of Greven and den Hollander concerning the behavior of the quenched large deviation rate function near zero in recurrent random walks in random environments.
• To analyze the asymptotic behavior of hitting times for RWRE to establish properties of the rate function.
• To investigate whether the quenched rate function for RWRE with positive speed is smaller than that of a simple random walk with the same speed, as conjectured by Greven and den Hollander.
• To provide a continuous analogue by comparing rate functions of Brox-type diffusions and Brownian motion with identical speed.

Proposed method

• Analyzes the hitting time τ₁ of the random walk in random environment using moment generating function estimates.
• Derives asymptotic behavior of E[τ₁^α e^{-rτ₁}] as r → 0⁺, showing polynomial growth with exponent α.
• Applies large deviation theory to the quenched law Pω to establish the existence and convexity of the rate function Iηq.
• Uses the connection between Brox diffusions and RWRE via scaling limits and stochastic differential equations.
• Employs modified Bessel functions and asymptotic expansions to analyze the cumulant generating function Γκ(λ) of the diffusion.
• Compares the cumulant generating function of the diffusion with that of a Brownian motion with the same speed using differential equations and series expansions.

Experimental results

Research questions

• RQ1 Does the quenched rate function Iηq for recurrent RWRE satisfy (Iηq)''(θ) → +∞ as θ → 0⁺?
• RQ2 Can the rate function of a Brox-type diffusion with drift be shown to be strictly smaller than that of a Brownian motion with the same speed?
• RQ3 Is there a continuous analogue to the conjecture that quenched RWRE rate functions are smaller than those of simple random walks with matching speed?
• RQ4 What is the precise asymptotic behavior of the moment generating function of hitting times in the recurrent RWRE regime?

Key findings

• The second derivative of the quenched rate function Iηq diverges to infinity as θ → 0⁺, confirming the conjecture of Greven and den Hollander.
• For any α > 0, the moment generating function of the hitting time τ₁ satisfies E[τ₁^α e^{-rτ₁}] = (1/r)^{α + o(1)} as r → 0⁺.
• The cumulant generating function Γκ(λ) of the Brox diffusion satisfies Γκ(λ) < φvκ(λ) for all λ > 0, where φvκ is the cumulant generating function of Brownian motion with speed vκ.
• The rate function Jκ(x) for the diffusion satisfies Jκ(x) < (1/2)(x - vκ)^2 for all x > vκ, proving strict dominance over the Brownian motion rate function.
• Asymptotic expansion of Γκ(λ) yields Γκ(λ) = √(2λ) - (1/4)(κ - 1/2) + O(1/√λ) as λ → ∞.
• The comparison result holds for all κ ≥ 0, with equality only in the case κ = 1/2, where the diffusion reduces to a Brownian motion with drift.