Jumping for diffusion in random metastable systems

The paper proves (i) a quenched CLT for random metastable systems under (P1)–(P7) by invoking the spectral CLT of Dragičević–Froyland–González-Tokman–Vaienti once uniform covering is guaranteed by (P4), and (ii) the small-ε diffusion-coefficient asymptotics lim_{ε→0} ε(Σε)^2 = 2⟨p⊙E[Ψ], (∫_0^∞ e^{t\bar G}dt)E[Ψ]⟩, with \bar G the generator of the averaged Markov jump process (equation (3) of Theorem 1.3), and illustrates it explicitly for random paired tent maps (equation (72)) . The candidate solution derives the same two assertions. For (i) it uses the same spectral method (twisted operators) as [14] in spirit; for (ii) it uses a macro–micro decomposition and an averaging argument for near-identity random matrices to reach exactly the paper’s formula. Differences are in proof strategy: the paper computes correlations via a jump-process approximation (Theorem 1.2) to obtain p_{jk}(t)=(e^{t\bar G})_{jk} and then (3) , while the candidate sketches a Duhamel/self-averaging route. Minor issues in the candidate solution include: (a) centering is assumed w.r.t. μ^ε in text whereas the paper assumes fibrewise centering w.r.t. the limiting density ϕ^0 and then works with the ε-centred observable ψ̃^ε (Definition 6.1 and (46)) ; (b) it casually claims an O(ε) convergence rate of ϕ^ε_ω→ϕ^0, while the paper uses o(1) from an averaging result ; and (c) it identifies the stable subspace as {1}^⊥ rather than {v: p^T v=0}. These do not affect the final conclusions.