Law of large numbers for a finite-range random walk in a dynamic random environment with nonuniform mixing

TL;DR

The paper addresses the law of large numbers for a finite-range random walk in a dynamic random environment with time correlations decaying polynomially. It extends the multi-scale renormalization framework from prior work to finite-range jumps, compensating for the loss of a monotonicity property with a uniform ellipticity assumption and a coalescence-like coupling, to prove a.s. convergence of the speed $\nu$ in both discrete and continuous time. The authors introduce limiting speeds $v_-$ and $v_+$, prove deviation bounds, and then show $v_- = v_+$ via traps and threats to obtain the LLN with a polynomial rate of convergence; the approach applies to a broad class of environments, including East-like models. This provides a robust, general method for LLNs in dynamic random environments with polynomial mixing, expanding the scope beyond nearest-neighbor or rapidly mixing cases and enabling applications in statistical mechanics and related fields.

Abstract

In this paper, we study random walks evolving on Z in a dynamic random environment that we assume to have time correlations that decrease polynomially fast. We show a law of large numbers by generalizing methods already used for the nearest-neighbor framework to the finite-range one. This requires some new ideas to get around the absence of a monotonicity property that was crucial in the proof for the nearest-neighbour case. Our proof works both in discrete and continuous time.

Figures (5)

• Figure 1: Illustration of the mixing property. Events describing respectively the two sample paths drawn inside the boxes can be decoupled using Fact \ref{['D:p:mixing']}.
• Figure 2: Illustration of $A_{H,w}(v)$. Starting from the point $y\in I_H(w)$, the random walk has an average speed at least $v$ between times $0$ and $H$.
• Figure 3: The sequence of speeds $(v_k)_{k\geqslant k_{\ref{['D:k:first_scale']}}}$.
• Figure 4: Illustration of $w$ being $k$-trapped -- here $v_0=0$. At least a third of the random walks starting in $J_k(w)$ reach height $\pi_2(w)+h_k$ with a nonpositive average speed. When $R\geqslant 2$, sample paths can jump over each other as in the drawing.
• Figure 5: Along the way, the random walk starting at $y$ passes near threatened points represented by the big filled dots. These points are likely to be responsible for a delay of our random walk. Here, the second threatened point encountered comes with a trap that causes a delay. Events $\mathrm{Trap}_{j}$, $\mathrm{Bar}_{j}$ and $\mathrm{Del}_j$ occur (for some $j\in\llbracket 2r,3r-1\rrbracket$), in fact here our random walk coalesces with $Z^{Y_j}$.

Theorems & Definitions (47)

• Definition 2.1: discrete-time setting
• Definition 2.2: continuous-time setting
• Remark 2.3
• Definition 2.5
• Theorem 2.7: LLN in discrete time
• Theorem 2.8: LLN in continuous time
• Definition 2.10
• Definition 2.11
• Remark 3.1
• Remark 3.2