Entropy Theory for Random Walks on Lie Groups

The paper’s Theorem 1.2 and its proof are coherent: it reduces the stopped-walk problem to a clamped stopping time η′n, leverages disjoint-support additivity at small scales, and estimates H(qη′n) via a careful counting/conditioning argument, with the large deviation principle (LDP) controlling the exceptional set. By contrast, the model’s alternative proof contains a substantive gap: it tries to control the tail contribution ∑_{k>m_n} P(η_n=k) H(µ^{*k}) by bounding E[η_n 1_{η_n>m_n}] using only an LDP probability bound. This attempts to lower bound E[η_n; A_n] with A_n = {η_n ≤ m_n} by (1−ε′)L_n P(A_n), which is unjustified because A_n imposes only an upper bound on η_n. Without a two-sided truncation (or additional moment assumptions), the proposed bound on the tail expectation—and hence the entropy truncation step—does not follow. The paper’s argument avoids this pitfall by working directly with η′n and precise entropy inequalities.