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The speed of biased random walks among dynamical random conductances

Eszter Couillard

TL;DR

This work analyzes the speed of biased variable-speed random walks in dynamical random conductances on $\mathbb{Z}^d$, establishing a positive linear speed for any bias $\lambda>0$ under bounded conductances. It develops a regeneration framework via an infected-edge construction, proves an annealed CLT, and derives a precise asymptotic expansion of the speed as $\lambda\to\infty$, including a coupling-based comparison between high-dimensional and one-dimensional dynamics. The paper also proves several monotonicity results in certain regimes (notably in 1D and for large $\mu$), and demonstrates that monotonicity can fail in general, even under ellipticity. Collectively, these results extend understanding of biased random walks in dynamical environments, providing explicit speed formulas, asymptotics, and regime-dependent monotonicity phenomena with potential implications for related stochastic processes and percolation-type models.

Abstract

We study biased variable-speed random walks in dynamical random conductances. Assuming that the conductances are upper-bounded, we prove that the walk has strictly positive speed for every bias $λ>0$. We then give an explicit asymptotic formula for the speed for $λ\to + \infty$, and prove two monotonicity properties for the speed. Finally, we provide an example showing that, even for conductances that are bounded and bounded away from zero, the speed can be asymptotically decreasing in the bias.

The speed of biased random walks among dynamical random conductances

TL;DR

This work analyzes the speed of biased variable-speed random walks in dynamical random conductances on , establishing a positive linear speed for any bias under bounded conductances. It develops a regeneration framework via an infected-edge construction, proves an annealed CLT, and derives a precise asymptotic expansion of the speed as , including a coupling-based comparison between high-dimensional and one-dimensional dynamics. The paper also proves several monotonicity results in certain regimes (notably in 1D and for large ), and demonstrates that monotonicity can fail in general, even under ellipticity. Collectively, these results extend understanding of biased random walks in dynamical environments, providing explicit speed formulas, asymptotics, and regime-dependent monotonicity phenomena with potential implications for related stochastic processes and percolation-type models.

Abstract

We study biased variable-speed random walks in dynamical random conductances. Assuming that the conductances are upper-bounded, we prove that the walk has strictly positive speed for every bias . We then give an explicit asymptotic formula for the speed for , and prove two monotonicity properties for the speed. Finally, we provide an example showing that, even for conductances that are bounded and bounded away from zero, the speed can be asymptotically decreasing in the bias.

Paper Structure

This paper contains 24 sections, 28 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Definition of the model
  3. Variable speed biased random walk (VBRW) in dynamical conductances
  4. An alternative construction of the VBRW
  5. Regeneration structure
  6. Regeneration times VBRW
  7. Properties of the regeneration times
  8. VBRW on dynamical conductances has a positive speed
  9. Reversibility
  10. CLT
  11. Positivity of the speed
  12. The asymptotics of the speed of the (N-)VBRW
  13. Time-normalizing the variable speed process (NVBRW)
  14. The one-dimensional totally asymmetric NVBRW ($\lambda = +\infty$)
  15. The asymptotic speed of the one-dimensional NVBRW
  16. ...and 9 more sections

Key Result

Lemma 1

Let $\alpha,\mu >0$. Let $(A_t)_{t \geq 0 }$ be a continuous time Markov chain on $\mathbb{N}$ with the following $Q$ matrix for $i \geq 0$ Assume $A_0 = 0$ a.s. then let $\tau = \inf\{t \geq 0: A_t = 0 \text{ and } \exists 0 \leq s \leq t: \; A_s \neq 0\}$ then $\tau$ has an exponential tail. Further, for $T = \inf\{n \geq 1: A_n = 0\}$ where $(A_n)_{n \geq 0}$ is the discretization of $(A_t)_{t

Theorems & Definitions (61)

  • Lemma 1
  • Lemma 2: Exponential tail of the regeneration time
  • proof
  • Lemma 3: Existence of the speed
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Corollary 6
  • ...and 51 more