All-Disorder Localization for the One-Dimensional Anderson Model with Power-Law Hopping

Huajian Jiang, Yunfeng Shi

Nov 2, 2026arXiv:2611.02724v1

math-phmath.DSmath.PR

Scorecard· 16/16

7.0/10 impact

Rigorously closes the gap from r>8 to the physically predicted sharp threshold r>3/2 for all disorder strengths, with reusable technical tools, but confined to a niche (1D, smooth i.i.d. disorder, spectral localization only).

Abstract

We consider the one-dimensional random operator $(H_{λ,ω}u)(n) = \sum_{m\ne n}\frac{u(m)}{|n-m|^r} +λω_nu(n),$ where the random variables $ω_{n}$ are independent and uniformly distributed on $[- 1, 1]$ (i.e., \mbox{ $ω_n\stackrel{\mathrm{i.i.d.}}{\sim}\operatorname{Unif}[-1,1]$ }). For every $λ > 0$ and $r > 3 / 2$ , we prove that $H_{λ,ω}$ almost surely has pure point spectrum and admits an orthonormal eigenbasis $\{ψ_j\}$ satisfying $\sum_{n\in\bbZ}(1+|n|)^{2τ}|ψ_j(n)|^2<\infty \qquad \text{for every }j\text{ and }0\leτ<r-\frac32.$ These conclusions establish spectral localization, thereby rigorously confirming the prediction of earlier analytical and numerical studies in the physics literature, which identified $r = 3 / 2$ as the marginal exponent.

AI Impact Assessments

(1 model)

Core Contribution

The paper proves spectral localization (almost-sure pure point spectrum, with an orthonormal eigenbasis of polynomially decaying eigenfunctions) for the 1D Anderson model with hopping |n−m|^{-r}. The result holds for every disorder strength λ>0 and every r>3/2. Earlier all-disorder results required r>8 (Jakšić–Molchanov 1999). Other earlier results needed large disorder, or r>2 with energy restrictions (Aizenman 1994). The paper reaches the physically predicted threshold r=3/2, where the physics literature (Rodríguez et al., Malyshev et al., de Moura et al.) places the marginal exponent. The weighted decay exponent τ<r−3/2 also vanishes exactly at the threshold, which is a pleasing consistency check.

The technical route combines several ideas:

Barrier construction from periodic-potential spectral gaps (Jakšić–Molchanov), modified with auxiliary random marks so that the disorder between barriers stays conditionally independent.

A Schur-complement reduction onto the inter-barrier blocks.

A Laplace-transform factorization of the kernel (L+a+b)^{-r}, which gives a "spatial-cut" coupling decay of L^{-(r-3/2)} with no volume factor. This is where r>3/2 enters.

Hilbert–Schmidt weak-L¹ multi-site spectral averaging (Aizenman–Elgart–Naboko–Schenker–Stolz), with a new "column functional" Φ and mixed-Fisher-information densities.

A contractive block fractional-moment recursion, followed by Simon–Wolff rank-one arguments to get pure point spectrum with weighted ℓ² eigenvectors.

Methodological Rigor

The argument is fully rigorous and self-contained, with appendices proving the technical lemmas. I checked the architecture and found it plausible:

The key calculation is (2.15), which gives γ(K_L) ≲ L^{-α}. It follows cleanly from the Laplace factorization and a sum of (L+2n)^{-2r/3}.

The factorization calculus is well organized: subadditivity of γ at p0=2/3, the product estimate (2.17) for g_α, and the resolvent Neumann-series bound with polynomial-growth control in Lemma C.1.

The barrier selection uses Borel–Cantelli with exponentially many candidate windows, which is standard but carefully done.

The recursion closes because the single-block factors Φ(F_i)→0 as |i|→∞, so contraction holds outside a finite set. This is the mechanism that works at arbitrarily small λ.

Measurable selection of the factorizations (Appendix D) is handled explicitly, which is a sign of care.

Possible concerns are subtle. The Fisher-information bound requires smooth densities, so the paper uses an auxiliary-mark decomposition. The uniform density has to be decomposed into smooth pieces via Lemma 2.4. The extension to general single-site distributions is unclear, since the proof uses smoothness of the density. The paper is long and technical, and some steps (for example the passage between conditional and unconditional expectations) are compressed. I did not find a visible gap.

Potential Impact

This closes a gap that had stood since the 1999 Jakšić–Molchanov paper, giving a sharp threshold result for a physically important class of long-range random operators. The physics community has long used the r=3/2 prediction, and this gives a rigorous endpoint. The new tools may be reusable:

A "spatial-cut" factorization with Schatten-type column functionals.

Block fractional-moment methods for long-range hopping.

Barrier decoupling combined with fractional moments.

These could carry over to other power-law or fractional models, such as fractional Laplacians with disorder, or Hislop–Matos–Rojas-Molina-type models. Applications beyond mathematical physics are limited. The audience is mainly mathematical physicists in random Schrödinger operators, with some interest from physicists studying long-range disordered systems and many-body localization with power-law couplings.

Timeliness & Relevance

Long-range hopping and power-law interactions are of current interest in cold atoms, Rydberg arrays, trapped ions and the localization literature. The paper cites several recent preprints on the same models. The result addresses a long-standing question, but it is a specialized one: pure point spectrum, not dynamical localization. The paper does not establish exponential localization, dynamical localization, or the delocalization side for r<3/2. These remain open.

Strengths

It reaches the sharp physical threshold r=3/2 for every λ>0, a clear and checkable statement.

It introduces robust new technical machinery (the γ functional, mixed Fisher information, the contractive block recursion).

It has a clean conceptual explanation of why 3/2 appears (α=r−3/2 in the cut decay).

The proof is complete and mostly self-contained.

Limitations

It gives only spectral localization, with polynomial rather than dynamical or exponential control. It also gives no bound on eigenfunction correlators, and the weighted ℓ² statement is mild.

It is restricted to uniform (smooth-density) disorder in 1D and to positive-power kernels exactly of the form |n−m|^{-r}. This relies on the periodic gap-opening theorem and monotonicity of the symbol f_r, which does not extend easily to other kernels or higher dimensions.

The exposition is dense, with heavy notation and a very long technical chain, which limits accessibility.

It does not address the regime r≤3/2.

AI-assisted verification is disclosed, but the proofs are intricate enough that independent verification will take time.

Other Observations

No numerical experiments are needed, since it is a theorem paper. Reproducibility is therefore a matter of proof verifiability, and the explicit lemmas and appendices help. It is a solid contribution that settles a specific, well-posed open problem, but its audience is narrow.

Rating:7/ 10

Significance 6.5Rigor 8Novelty 7Clarity 6

Generated Nov 3, 2026

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Scorecard· 16/16

7.0/10 impact