On some random billiards in a tube with superdiffusion

The paper proves a nonstandard CLT for the horizontal displacement S_n with normalization sqrt(n log n), variance W^2, under ν^⊗ℤ ⊗ μ, using an averaged transfer-operator approach (spectral gap in BV and continuity of the perturbed operator P_t), together with precise tail control for X(θ)=W/ tan θ; see Theorem 1.1 and the surrounding discussion of the random maps Ψ_R, invariance of μ(dθ)=½ sinθ dθ, and the Nagaev–Guivarc’h method . The candidate solution reaches the same limit law but hinges on a two-step Doeblin minorization for the annealed kernel and a Nummelin-splitting regeneration that it does not justify. In particular, the asserted uniform two-step minorization on (0,π) from geometric “one-collision events” is not established under the paper’s assumptions, and the subsequent tail estimates for block sums tacitly use independence within blocks (or a mixing bound) that is never proved. The paper’s argument is complete, while the model’s key steps are missing or require stronger assumptions (e.g., density bounded away from zero) than the paper uses .