Universality of the transport law for supercritical quasi-periodic operators

Douglas, Seth; Mghirbi, Nidhal

doi:10.5281/zenodo.23830333

Results · Transport in quasi-periodic systems

Universality of the transport law for supercritical quasi-periodic operators

Seth Douglas and Nidhal Mghirbi · version 1.0.0 · Zenodo, 31 October 2026 · 55 pages · doi:10.5281/zenodo.23830333 · PDF

In plain wordsThe law is not a quirk of one model; it governs a whole class of quasi-periodic systems, and for every analytic one with positive Lyapunov exponent the law for the fastest spreading follows from a single explicit hypothesis, open beyond type I.

The resultat three depths

In one sentenceThe exact law for how fast a wave spreads at its farthest in the Aubry–André model is not special to that model: for almost every phase it holds for every supercritical quasi-periodic operator whose real-analytic potential is of type I, with the least Lyapunov exponent in place of the strength of the potential, and for every real-analytic potential with positive Lyapunov exponent the paper proves the law for the fastest upper exponent conditionally on one explicit hypothesis, which is open beyond type I.

As the paper states it, in briefFor every real-analytic quasi-periodic potential that is supercritical and of type I, and almost every phase, the upper transport exponents follow the same law with $\log λ$ replaced by the minimum Lyapunov exponent on the spectrum, for the fastest upper exponent and for every moment. Energy by energy, for packets filtered ever more narrowly near an energy $E$ , every upper moment exponent tends to $\max (0, 1 - L (E) / β (α))$ , and the spectral measure has that upper local dimension at almost every $E$ . A potential $w (s x)$ obeys the law with $β (s α)$ when $w$ at frequency $s α$ is supercritical and of type I. For every real-analytic potential with positive Lyapunov exponent, the law for the fastest upper exponent is proved conditionally on one explicit weak-separation hypothesis, open beyond type I.

For a scientist
One paragraph

Take $(H u) (n) =$ $u (n + 1) + u (n - 1) + v (θ + n α) u (n)$ with a real-analytic potential $v$ on the circle and an irrational frequency $α$ . The companion paper proved, for the almost Mathieu operator with $λ > 1$ and almost every phase, that every moment of a packet started at one site has the upper transport exponent $\max (0, 1 - \log λ / β (α))$ . This paper shows that the same law holds when every energy of the spectrum is supercritical and of type I, in the sense of Avila’s global theory and of Ge and Jitomirskaya, with $\log λ$ replaced by the minimum $ℓ_{*}$ of the Lyapunov exponent on the spectrum: for almost every phase, the fastest upper exponent $α_{u}^{+}$ and every upper moment exponent $β^{+} (p)$ equal $\max (0, 1 - ℓ_{*} / β (α))$ , and the lower bound for $α_{u}^{+}$ holds at every phase. The law also holds energy by energy: for packets filtered ever more narrowly near an energy $E$ of the spectrum, every upper moment exponent tends to $\max (0, 1 - L (E) / β (α))$ , and the spectral measure has that upper local dimension at almost every $E$ . A potential $v (x) = w (s x)$ obeys the law with $β (s α)$ in place of $β (α)$ when $w$ at frequency $s α$ is supercritical and of type I. For every real-analytic potential with positive Lyapunov exponent on its spectrum, the law for $α_{u}^{+}$ follows, for almost every phase, from one explicit weak-separation hypothesis on the zeros of rational discriminants, which remains open beyond type I.

For an expert
The main theorems, as stated in the paper (Section 1)

Notation from the paper: $H_{v, α, θ}$ is the operator above, with $v$ real-analytic on $𝕋 = ℝ / ℤ$ , $α$ irrational and $θ \in 𝕋$ ; $Σ$ is its spectrum, which does not depend on $θ$ , and $L (E)$ its Lyapunov exponent at energy $E$ . As in the companion paper, $β (α) = \underset{n \to \infty}{lim sup} \log q_{n + 1} / q_{n}$ over the continued-fraction convergents $p_{n} / q_{n}$ of $α$ ; $P (N, T)$ and $M_{p} (T)$ are the tail probability beyond distance $N$ and the $p$ -th moment of the packet $e^{- i t H} δ_{0}$ , averaged in time with the weight $(2 / T) e^{- 2 t / T}$ ; the upper transport exponents are $β^{+} (p) = \underset{T \to \infty}{lim sup} \log M_{p} (T) / (p \log T)$ and $α_{u}^{+} = sup_{p > 0} β^{+} (p)$ , and $β_{φ}^{+} (p)$ is the exponent of the packet started at $φ$ . For $0 < β < \infty$ , $F_{β} (ℓ) = \max {0, 1 - ℓ / β}$ , with $F_{0} \equiv 0$ and $F_{\infty} \equiv 1$ .

An energy is supercritical if $L (E) > 0$ . Its acceleration is the right derivative at $y = 0$ of $y \mapsto L (E, y) / (2 π)$ , where $L (E, y)$ is the Lyapunov exponent of the cocycle with phase complexified by $i y$ ; a supercritical energy is of type I if its acceleration is one. $G_{α}^{v}$ is an explicit set of phases: those $θ$ for which $q_{n} θ$ stays at distance at least $2 q_{n}^{- 3}$ from the turning points of the band centres of the $p_{n} / q_{n}$ -periodic approximants, for all large $n$ . The set $G_{α}$ of equation (6) consists of the $θ$ with $| \sin (2 π q_{n} θ) | \geq q_{n}^{- 2}$ for all sufficiently large $n$ . $μ_{θ}$ is the spectral measure of $δ_{0}$ , and $ℓ_{θ}$ and $ℓ^{θ}$ are the minimum and the maximum of $L$ on its support; $d_{μ}^{+} (E) = \underset{r \to 0}{lim sup} \log μ ([E - r, E + r]) / \log r$ is the upper local dimension, and $\dim_{P}^{+} (μ)$ and $\dim_{P}^{-} (μ)$ are its $μ$ -essential supremum and infimum.

For a convergent $p / q$ , $t_{q} (x, E)$ is the trace of the transfer matrix over one period of the $p / q$ -periodic approximant at the phase $x / q$ . WXS $(v, α)$ (weak separation) holds if for every $ε > 0$ and $c > 0$ , for almost every $θ$ and all large $n$ with $q_{n}^{- 1} \log | q_{n} α - p_{n} |^{- 1} \geq c$ , no energy within $1 / q_{n}$ of the spectrum has two zeros of $t_{q_{n}} (\cdot, E)$ , real or complex and counted with multiplicity, within $e^{- ε q_{n}}$ of each other with one of them within $e^{- ε q_{n}}$ of $q_{n} θ$ .

Theorem A (The transport law)

Let $v$ be real-analytic and $α$ irrational, and assume that every energy in $Σ$ is supercritical and of type I. Let $ℓ_{*} = min_{E \in Σ} L (E) > 0$ .

If $β (α) < \infty$ , then for every $θ$ in a set $G_{α}^{v}$ of full measure,
$α_{u}^{+} (H_{v, α, θ})$ $= \max {0, 1 - \frac{ℓ_{*}}{β (α)}},$
read as $0$ when $β (α) = 0$ , and $P (T^{a}, T) = O (T^{- m})$ for every $a > F_{β (α)} (ℓ_{*})$ and $m > 0$ . When $v$ is even, $G_{α}^{v}$ contains the explicit set $G_{α}$ of (6). When $β (α) = 0$ the equality holds for every phase.
If $β (α) = \infty$ , then $α_{u}^{+} = 1$ for every phase.

The lower bound $α_{u}^{+} \geq 1 - ℓ_{*} / β (α)$ holds for every phase.

Theorem B (Every moment, energy by energy)

Assume the hypotheses of Theorem A with $0 < β (α) < \infty$ , write $β = β (α)$ , and let $θ \in G_{α}^{v}$ .

For every $p > 0$ , $β^{+} (p) = F_{β} (ℓ_{*})$ . If $ℓ_{*} < β$ , then $L$ attains $ℓ_{*}$ on $supp μ_{θ}$ , and for every $γ > 0$ a fixed positive fraction of the packet lies beyond distance $e^{(b_{n} - ℓ_{*} - γ) q_{n}}$ at the times $T_{n} = | q_{n} α - p_{n} |^{- 1}$ , for all large $n$ with $b_{n} = q_{n}^{- 1} \log T_{n} \geq ℓ_{*} + 15 γ$ .
Let $E_{0} \in Σ$ , $0 < γ < ℓ_{*} / 2$ , and let $g \in C_{c}^{\infty} (ℝ)$ be real, supported in ${E : | L (E) - L (E_{0}) | < γ}$ , with $g (H) δ_{0} \neq 0$ . Then the packet started at $g (H) δ_{0}$ satisfies, for every $p > 0$ ,
$F_{β} (L (E_{0}) + γ)$ $\leq β_{g (H) δ_{0}}^{+} (p)$ $\leq F_{β} (L (E_{0}) - γ) .$
$d_{μ_{θ}}^{+} (E) = F_{β} (L (E))$ for $μ_{θ}$ -almost every $E$ . Consequently $\dim_{P}^{+} (μ_{θ}) = F_{β} (ℓ_{θ})$ and $\dim_{P}^{-} (μ_{θ}) = F_{β} (ℓ^{θ})$ .

Theorem C (Covariance)

Let $v$ be real-analytic and nonconstant, let $s$ be the greatest common divisor of the frequencies $k \neq 0$ with $\hat{v} (k) \neq 0$ , and write $v (x) = w (s x)$ . If $H_{w, s α}$ is supercritical and of type I, then for almost every $θ$ ,

α_{u}^{+} (H_{v, α, θ}) = \max {0, 1 - \frac{ℓ_{*}}{β (s α)}},

read as $0$ when $β (s α) = 0$ , and for every $θ$ if $β (α) \in {0, \infty}$ . Moreover $β (α) \leq β (s α) \leq s β (α)$ .

Theorem D (A conditional complete law)

Let $v$ be real-analytic with positive Lyapunov exponent on its spectrum, and write $v (x) = w (s x)$ as in Theorem C. If WXS $(w, s α)$ holds, then for almost every $θ$ ,

α_{u}^{+} (H_{v, α, θ}) = \max {0, 1 - \frac{ℓ_{*}}{β (s α)}},

read as $0$ when $β (s α) = 0$ , where $ℓ_{*}$ is the minimum of the Lyapunov exponent on the spectrum.

By Theorems A and C, the hypothesis of Theorem D is needed only when the primitive part $w$ has acceleration at least two at some point of the spectrum; the paper does not know whether it holds there. The almost Mathieu operator with $λ > 1$ is supercritical and of type I at every spectral energy, so Theorem A gives the companion paper’s law for $α_{u}^{+}$ at every $θ \in G_{α}$ ; the companion paper proves that case independently, by a different route.

The question

The almost Mathieu operator has a single cosine for its potential, and many of its exact results lean on that special form. Is the exact transport law of the companion paper a feature of the cosine, or a law of quasi-periodic systems?

The law across a classsupercritical, type I

For every real-analytic quasi-periodic potential that is supercritical and of type I on its spectrum, in the sense of Avila’s global theory and of Ge and Jitomirskaya, the same law holds for almost every phase, with the logarithm of the coupling replaced by the minimum of the Lyapunov exponent on the spectrum. It holds for the fastest upper exponent and for the upper exponent of every moment, and the lower bound for the fastest upper exponent holds at every phase.

It also holds energy by energy. For packets filtered ever more narrowly near an energy $E$ of the spectrum, every upper moment exponent tends to $\max (0, 1 - L (E) / β (α))$ , and the spectral measure has that upper local dimension at almost every $E$ ; for a filter of fixed width the paper gives two-sided bounds, set by the range of the Lyapunov exponent on the filter. The fastest spreading comes from the energies with the smallest Lyapunov exponent.

For the almost Mathieu operator this is the companion paper’s law. Beyond it, the law says the same thing about a whole class of systems: in the supercritical regime, whether a wave spreads, and how fast at its farthest, is set by how well the frequency is approximated by fractions, weighed against the Lyapunov exponent.

The frequency that counts

A potential of the form $v (x) = w (s x)$ repeats $s$ times around the circle, and sampling it at step $α$ is sampling $w$ at step $s α$ . The law sees this: when $w$ at frequency $s α$ is supercritical and of type I, the fastest upper exponent obeys the law for almost every phase with $β (s α)$ in place of $β (α)$ , so the arithmetic that decides transport is that of the frequency the potential actually feels. Since $β (α) \leq β (s α) \leq s β (α)$ , the two can differ. Such a $v$ is never itself of type I.

The open frontierweak separation

For every real-analytic potential with positive Lyapunov exponent on its spectrum, type I or not, the paper shows that the law for the fastest upper exponent $α_{u}^{+}$ follows, for almost every phase, from one explicit hypothesis, weak separation: at a fixed energy, the zeros of the rational discriminants of the potential’s primitive part, as functions of the phase, do not cluster at an exponential rate near the drifting phase. The hypothesis is needed only beyond type I, for primitive potentials with acceleration at least two, and there it is open. A proof of it would give the law for the fastest upper exponent for every real-analytic quasi-periodic operator with positive Lyapunov exponent on its spectrum. Without any hypothesis, a third paper, Worst-case transport and light cones for quasi-periodic operators, proves an upper law at every phase for every such potential, in which the worst phase divides the localization strength by at most twice the acceleration.

For the two-harmonic family $2 λ \cos 2 π x + 2 μ \cos 4 π x$ the paper reduces the hypothesis further, to a large-deviation estimate for one explicit polynomial, which it also leaves open. The law could also hold where weak separation fails; the paper’s methods do not decide this. The question is listed as question 50.

Questionson the index

Partly answered Does the dimension bound of Jitomirskaya, Liu and Tcheremchantsev extend, with $\log λ$ replaced by the minimal Lyapunov exponent, to the other cases where a sharp arithmetic transition is known? posed by others
Gap partly closed Is this transport law universal?
Open Does the weak-separation hypothesis hold beyond type I? left open

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Novelty and credit

Novelty: Extends named prior work: Avila’s global theory, Ge and Jitomirskaya’s type I class and the companion paper’s law for the almost Mathieu operator, credited; for general supercritical analytic potentials it credits the case $β (α) = 0$ to Jitomirskaya and Mavi.

The classification of energies as subcritical, critical and supercritical, and the acceleration, come from Avila’s global theory of one-frequency Schrödinger operators; the type I class, and sharp arithmetic transitions for it, are Ge and Jitomirskaya’s. For the almost Mathieu operator, the arithmetic spectral transition is Avila, You and Zhou’s; the case $β (α) = 0$ of the transport law is Damanik and Tcheremchantsev’s and the case $β (α) = \infty$ Jitomirskaya and Zhang’s; the exact value in between is determined in the companion paper. For general supercritical analytic potentials, the endpoint $β (α) = 0$ was observed by Jitomirskaya and Mavi.

Near the transition, the paper credits lower bounds of the form $1 - C \log λ / β$ to Jitomirskaya and Zhang and to Haeming, who also proved lower bounds for every moment, at every phase and along sequences of times; upper bounds on fractal dimensions of spectral measures to Jitomirskaya, Liu and Tcheremchantsev, which Levi extended, with the same factor two, to potentials that are monotone on the period; exponential dynamical localization in phase expectation for small $β (α)$ to Jitomirskaya and Krüger; and refined growth rates for weakly Liouville frequencies to Bradshaw, de Jong, Liu, Wang, Wang and Yang.

The structural inputs are Avila’s global theory, Ge and Jitomirskaya’s type I class, the finite harmonic structure of rational discriminants of Jitomirskaya and Marx and of Shamis, and, for the spectral consequences, the inequality of Barbaroux, Germinet and Tcheremchantsev between transport and generalized dimensions. The proofs also use the complex-energy method of Damanik and Tcheremchantsev and the periodic magic formula of Damanik, Killip and Simon.

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Every archived version, newest first. Dates are Zenodo publication dates. All versions: 10.5281/zenodo.23830332, which always resolves to the newest.
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Cite

@misc{douglas2026universality,
  author  = {Douglas, Seth and Mghirbi, Nidhal},
  title   = {Universality of the transport law for supercritical quasi-periodic operators},
  year    = {2026},
  version = {1.0.0},
  doi     = {10.5281/zenodo.23830333},
  url     = {https://doi.org/10.5281/zenodo.23830333},
  note    = {Preprint}
}

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