Transport in quasi-periodic systems

Gap closed How fast does a wave packet spread in the supercritical Aubry–André model?

Exactly: one formula for the upper exponent of every moment.

1. Exact transport and spectral dimensions for the almost Mathieu operator

   Seth Douglas and Nidhal Mghirbi · version 2.0.1 · 1 November 2026 · 50 pages

   For the almost Mathieu operator with coupling $\lambda >1$, every irrational frequency $\alpha$ and almost every phase, every moment of a wave packet started at one site has the same upper transport exponent $\max (0,1-\log \lambda /\beta (\alpha ))$, where $\beta (\alpha )=\mathrm{lim\; sup}\log q_{n+1}/q_n$; the lower bound holds at every phase, and when $\beta (\alpha )>\log \lambda$ at the resonant times almost all of the packet makes the trip, uniformly in the phase. For almost every phase, both packing dimensions of every nonzero spectral measure equal the same number; transport switches on exactly at $\lambda =e^{\beta }$; and an explicit computable almost Mathieu operator has a fastest upper transport exponent of Turing degree ${\mathrm{𝟎}}^{″}$.

   • Gap closed How fast does a wave packet spread in the supercritical Aubry–André model at Liouville frequencies?
   • Answered Is the factor 2 in the packing-dimension bound near the arithmetic transition sharp?

   The paperPDFDOICode

2. Universality of the transport law for supercritical quasi-periodic operators

   Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 31 October 2026 · 55 pages

   For 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 $\log \lambda$ 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 $\max (0,1-L(E)/\beta (\alpha ))$, and the spectral measure has that upper local dimension at almost every $E$. A potential $w(sx)$ obeys the law with $\beta (s\alpha )$ when $w$ at frequency $s\alpha$ 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.

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

   The paperPDFDOICode

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

   Seth Douglas and Nidhal Mghirbi · version 2.0.0 · 2 November 2026 · 72 pages

   At 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? posed here
   • Open Does the exact worst-case law of the almost Mathieu operator hold without the parity condition? posed here

   The paperPDFDOICode

4. Band edges of the almost Mathieu operator at Liouville scales

   Seth Douglas and Nidhal Mghirbi · version 2.0.0 · 2 November 2026 · 60 pages

   For the supercritical almost Mathieu operator, write $L=\log \lambda$, $\beta =\beta (\alpha )$ and $x=2\beta /L$, and let $\rho$ be the exponential gap-decay rate, the liminf of $-\log |G_m|/(L|m|)$. Above the Liouville threshold, for finite $\beta >L/2$, infinitely many gaps are only polynomially small in their integer label, so $\rho =0$: at a convergent $p_n/q_n$, with ${\delta }_n=q_n\alpha -p_n$ and $b_n=-q_n^{-1}\log |{\delta }_n|$, the gaps with labels $m=-D{q}_{n}sgn{\delta }_n$ at the top of the spectrum have width at least $\exp (-(b_n+L/2+o(1))q_n)$, uniformly over an explicit growing range of $D$, and a separate surplus argument includes $\beta =\infty$. Below the threshold, for $0<x<1$, $\rho \le 1-x$: at $\beta$-attaining convergents, the gap with label $-D{q}_{n}sgn{\delta }_n$ has an explicit lower bound on its width for every fixed $D$. With the theorem of Liu and Shi this gives $1/C\le \rho \le 1-x<1$ for $0<x\le 2/C$, which contradicts their stated expectation of a sharper upper gap bound in that range but not their theorem. Equality $\rho =1-x$, and the line $x=1$, remain open.

   The paperPDFDOICode

• Answered Is the factor 2 in the packing-dimension bound near the arithmetic transition sharp?
• Partly answered Does the dimension bound of Jitomirskaya, Liu and Tcheremchantsev extend, with $\log \lambda$ replaced by the minimal Lyapunov exponent, to the other cases where a sharp arithmetic transition is known?