Erdős–Rényi laws for exponentially and polynomially mixing dynamical systems

The paper establishes an Erdős–Rényi limit law at the logarithmic window length L_n = ⌊(ln n)/I(α)⌋ for Hölder observables on systems modeled by Young towers with exponential tails, proving that max_{0≤j≤n−L_n} S_{L_n}(T^j x)/L_n → α almost surely (see Proposition 4.1 for the reduction and Theorem 5.1 for the main result, with LDP assumptions and the tower-based proof strategy) . The paper’s argument proves the upper bound from the LDP alone and the lower bound via a blocking argument on the quotient tower that leverages decay of correlations using a symbolic metric and Lipschitz approximations to indicator functions, rather than α-mixing . The model’s solution also derives the same limit: the upper bound via a union bound plus the LDP (matching the paper’s Proposition 4.1(a)), and the lower bound via separated-window testing combined with exponential α-mixing and Borel–Cantelli. This is a different proof route. Two minor issues in the model writeup are: (i) a small misalignment in the “bridging” step for the upper bound when handling n between exp(m I(α)) and exp((m+1) I(α))—which is easily fixed by applying the union bound up to n_{m+1} for window length m—and (ii) for the lower bound, one must either run the separated-window argument for both lengths m and m+1 or use boundedness of ϕ to transfer an m-exceedance to an (m+1)-exceedance. With these standard fixes, the model proof is correct. Overall, the paper’s proof is complete and correct, and the model provides a correct alternative proof under standard α-mixing that holds for Young towers with exponential tails (citing external results).