Accelerating Abelian Random Walks with Hyperbolic Dynamics

The uploaded paper proves two main results: an O(log n log log n) upper bound for all admissible moduli n (hyperbolic A, rk H = d, and n coprime to the H-indices a_i) and an O(log n) upper bound for almost all n, via a Fourier-factorization plus symbolic-dynamics argument using Markov partitions and a carefully controlled “bad set” W, together with an interchange inequality to sum Fourier modes. This is stated and developed explicitly in Theorem 1 and Theorem 3, with the Fourier factorization (Lemma 1), the Diaconis–Shahshahani bound, the definition of W and the uniform contraction away from W, and the block/disjointness lemmas that force a contracting letter in every block of length ≍ log n, culminating in the desired bounds . By contrast, the candidate solution introduces a scale M and a Diophantine bad set E_M with a per-step contraction ρ(M) ≤ exp(−c/M^2), then chooses M = log n and asserts a bound of the form (1/n^d)∑|p̂_t|^2 ≲ exp(−κ (t − C2 log n)/(M^2 L)) with L ≍ log M, claiming that t ≳ log n log log n suffices. This optimization is incorrect: with M = log n and L ≍ log log n, the exponent is only ≍ −t/((log n)^2 log log n), so at t ≍ log n log log n the decay is too weak to force the normalized sum to vanish. The paper’s proof instead uses a fixed-away-from-W contraction γ(η) < 1 and block length k ≍ log n to obtain the required O(log n log log n) via the interchange inequality (in particular, Lemma 6 and the S1/S2/S3 decomposition) . Hence the paper is correct, but the model’s argument is not quantitatively valid.