TOTALLY ASYMMETRIC DYNAMICAL WALKS IN RANDOM ENVIRONMENT

The paper proves: (i) a CLT for hitting times τn under Models A/A′/B (Theorem 1.1), (ii) a CLT for the scale S(zn)−n (Theorem 1.3), and crucially (iii) quenched and annealed CLTs for the position zn, where quenched needs random centerings bn(ω) and the annealed variance has an extra environment term D2 beyond the quenched variance (Theorem 1.4 and Section 9). These facts are explicit in the PDF: r n∈{1,3} via (3.2) supports the renewal viewpoint; Theorem 1.4 states quenched centering by bn(ω) and annealed centering by vn; and the annealed variance decomposes as σ2=σ2+D2 (the final equality is written in Section 9) . The model’s solution incorrectly asserts one may use the deterministic centering vn even in the quenched CLT and misses the environment-induced variance contribution D2 in the annealed limit. It also claims a universal variance formula σ2=σ2_τ/μ3 for the position process, which the paper does not state and which conflicts with the demonstrated variance decomposition in the annealed setting .