Worst-case transport and light cones for quasi-periodic operators

As the paper states it, in briefAt the worst phase rather than a typical one. For every real-analytic quasi-periodic potential with positive Lyapunov exponent on the spectrum and every frequency with $0<\beta (\alpha )<\infty$, uniformly in the phase, the fastest upper transport exponent is at most the maximum over the spectrum of $1-\ell /(2\omega \beta )$ when $\beta \ge \ell$, where $\ell$ is the Lyapunov exponent and $\omega$ the acceleration, with an explicit formula when $\beta <\ell$; so a finite $\beta$ keeps transport below ballistic at every phase, and the factor $2\omega$ is attained. For the almost Mathieu operator, when ${\beta }_e=\beta$ is finite and greater than $\log \lambda /2$, where ${\beta }_e$ is the limsup along even denominators, the worst exponent is exactly $1-\log \lambda /(2\beta )$: it is attained at the phase $1/4$ and on a dense $G_{\delta }$ set of phases, while almost every phase has $\max (0,1-\log \lambda /\beta )$. In the Aubry–André XY spin chain, for the specified observables, the light cone at a fixed site follows the typical law at almost every phase, and the light cone uniform over sites follows the worst-case law; an explicit computable chain has two different light-cone exponents, both of Turing degree ${\mathrm{𝟎}}^{″}$. The Lyapunov exponent of a computable analytic potential is computable at every computable irrational frequency, and at one explicit frequency the transport exponent has degree exactly ${\mathrm{𝟎}}^{″}$ for every computable even potential in an explicit open neighbourhood of $4\cos 2\pi x$.

• Partly proved How fast can a wave packet spread at the worst phase, rather than a typical one?
• Proved Is the worst phase rare?
• Proved How fast do signals travel in the quasi-periodic XY spin chain?
• Proved Where does the uncomputability of transport come from?
• Open Is the worst-phase factor two, rather than twice the acceleration, for primitive potentials? left open
• Open Does the exact worst-case law of the almost Mathieu operator hold without the parity condition? left open

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Label checked 1 November 2026.

Novelty

Extends named prior work: the endpoint $\beta (\alpha )=\infty$ is due to Jitomirskaya and Zhang, and $\beta (\alpha )=0$ to Damanik and Tcheremchantsev and to Jitomirskaya and Mavi. It credits the link between the acceleration and zeros of determinants to Han and Schlag, the closed central gap to Bellissard and Simon and to Lamoureux and Mingo, the reduction of XY dynamics to one-body dynamics to Lieb, Schultz and Mattis and to Hamza, Sims and Stolz, the single-site OTOC formula to Riddell and Sørensen, and earlier uncomputability in physics to Cubitt, Perez-Garcia and Wolf, to Bausch, Cubitt and Watson, and to Purcell, Li and Cubitt. The computability of the Lyapunov exponent is an algorithmic consequence of Avila’s global theory, for which it makes no priority claim.

PDF

papers/worst-case-transport-v2.0.0.pdf, version 2.0.0, 1,201,379 bytes, 72 pages.

SHA-256 8da653613f3cab77be285e37ce8ed023c3ed5c72a6eb215824fdf68e36d19f54

DOI

10.5281/zenodo.23875774; all versions 10.5281/zenodo.23845227

Code

github.com/Apsiape/worst-case-transport-light-cones

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All corrections, and what counts as one.

@misc{douglas2026worstcasetransport,
  author  = {Douglas, Seth and Mghirbi, Nidhal},
  title   = {Worst-case transport and light cones for quasi-periodic operators},
  year    = {2026},
  version = {2.0.0},
  doi     = {10.5281/zenodo.23875774},
  url     = {https://doi.org/10.5281/zenodo.23875774}
}

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