Universality of the transport law for supercritical quasi-periodic operators

We determine how fast wave packets spread under quasi-periodic Schrödinger operators (Hu)(n) = u(n+1) + u(n−1) + v(θ + nα) u(n) with real-analytic v and positive Lyapunov exponent L on the spectrum. Let β(α) = limsup_n q_n^{−1} log q_{n+1}, let ℓ_* be the minimum of L on the spectrum, and put F(ℓ) = max{0, 1 − ℓ/β(α)}, with F ≡ 0 if β(α) = 0 and F ≡ 1 if β(α) = ∞. If v is of type I in the sense of Avila and of Ge and Jitomirskaya, then for almost every phase the fastest transport exponent α_u^+ and the exponent of every moment, both upper and Abel-averaged, equal F(ℓ_*); the lower bound for α_u^+ holds for every phase. The law also holds energy by energy: for packets filtered ever more narrowly near an energy E of the spectrum, every moment exponent tends to F(L(E)), and the spectral measure has upper local dimension F(L(E)) at almost every E. These conclusions extend to v(x) = w(sx), with β(sα) in place of β(α), when w at frequency sα is supercritical and of type I. Beyond type I, we show, conditionally on one hypothesis, that every real-analytic potential with positive Lyapunov exponent on its spectrum has α_u^+ = F(ℓ_*), computed with β(sα), for almost every phase, where s is the period reduction of the potential. The hypothesis: at fixed energy, the zeros of the rational discriminants of its primitive part, as functions of the phase, do not cluster at an exponential rate near the drifting phase. The fastest-transport law for analytic potentials is thereby reduced to this weak separation hypothesis for primitive potentials with acceleration at least two. The proofs combine a covering argument over all times, an exact q-step resolvent identity, elliptic runs of Weyl solutions, and an arcsine law for the rational discriminant.