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Biased random walk on the critical curve of dynamical percolation

Assylbek Olzhabayev, Dominik Schmid

TL;DR

The paper analyzes a $\boldsymbol\lambda$-biased random walk on dynamical percolation on $\mathbb{Z}^d$, deriving a second-order expansion for the asymptotic speed and proving monotonicity on the critical curve for $d\ge 2$. The authors develop the environment seen from the walker, study a one-dimensional TARWDP projection, and construct the invariant measure $\mathbb{Q}$ via regeneration times to connect speed to time shifts; these tools yield explicit constants $C_{\mu,p}$ and $\mathcal{C}_{\mu,p,d}$ governing the $e^{-2\lambda}$ corrections. By a sequence of couplings between the RWDP and TARWDP (and reductions to 1D), they obtain precise speed expansions in both 1D and higher dimensions, and show the speed on the critical curve increases for large bias, resolving an open question. The results illuminate how update rate $\mu$, bond probability $p$, and bias $\lambda$ interplay to control trapping and transport, with potential implications for biased diffusion in dynamic random environments.

Abstract

We study biased random walks on dynamical percolation in $\mathbb{Z}^d$, which were recently introduced by Andres et al. We provide a second order expansion for the asymptotic speed and show for $d \ge 2$ that the speed of the biased random walk on the critical curve is eventually monotone increasing. Our methods are based on studying the environment seen from the walker as well as a combination of ergodicity and several couplings arguments.

Biased random walk on the critical curve of dynamical percolation

TL;DR

The paper analyzes a -biased random walk on dynamical percolation on , deriving a second-order expansion for the asymptotic speed and proving monotonicity on the critical curve for . The authors develop the environment seen from the walker, study a one-dimensional TARWDP projection, and construct the invariant measure via regeneration times to connect speed to time shifts; these tools yield explicit constants and governing the corrections. By a sequence of couplings between the RWDP and TARWDP (and reductions to 1D), they obtain precise speed expansions in both 1D and higher dimensions, and show the speed on the critical curve increases for large bias, resolving an open question. The results illuminate how update rate , bond probability , and bias interplay to control trapping and transport, with potential implications for biased diffusion in dynamic random environments.

Abstract

We study biased random walks on dynamical percolation in , which were recently introduced by Andres et al. We provide a second order expansion for the asymptotic speed and show for that the speed of the biased random walk on the critical curve is eventually monotone increasing. Our methods are based on studying the environment seen from the walker as well as a combination of ergodicity and several couplings arguments.

Paper Structure

This paper contains 25 sections, 28 theorems, 170 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Related work
  4. Overview of ideas
  5. Organization
  6. Acknowledgements
  7. Preliminaries
  8. Infected set and regeneration times
  9. Derivative regularity
  10. Monotone coupling implies speed disparity
  11. The environment process for the one-dimensional RWDP
  12. Definition and basic properties
  13. Projection of the environment process on $[-1,1]$
  14. Invariant measure for the environment process of a TARWDP
  15. Asymptotic expansion of the speed for $d=1$
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $d\geq 1$ and let $(X_t, \eta_t)_{t\geq0}$ be a $\lambda$-RWDP with parameters $\mu>0$ and $p\in (0,1)$. Then for all $\lambda>0$, there exists some $v(\lambda)=v_{\mu,p}(\lambda)$ such that almost surely The function $\lambda\mapsto v(\lambda)$ is strictly positive for all $\lambda>0$ and continuously differentiable. Moreover, for all $d\geq2$ and $\mu>0$, there exists some $\lambda_0=\lambd

Figures (3)

  • Figure 1: Visualization of the events $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, $\mathcal{D}$ used in the coupling of walkers $(X_t)_{t \geq 0}$ and $(Y_t)_{t \geq 0}$, respectively.
  • Figure 2: Visualization of the events constructed in Section \ref{['comparisoncoupling']}. Dashed arrows correspond to jumps at the respective times, and squiggly edges to updates.
  • Figure 3: The position of the walker $X$ at time $\mathcal{T}_1$. Dashed arrows show the direction of the next accepted jump.

Theorems & Definitions (48)

  • Theorem 1.1: Theorems 1.2 and 1.3 in andres2023biased
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1: Lemma 3.5 in hermon2020comparison
  • Lemma 2.2
  • Lemma 2.3: cf. Lemma 4.10 in andres2023biased
  • proof : Proof of Lemma \ref{['lemma43']}
  • proof : Proof of Theorem \ref{['moncrit']}
  • Proposition 2.4
  • ...and 38 more