Localization and unique continuation for the Anderson-Bernoulli model with long-range hopping on $\mathbb{Z}$

Abstract: In this paper, we study Anderson localization near the spectral edge for the Anderson-Bernoulli model on $\mathbb{Z}$ with long-range hopping. When the hopping has a rational Laurent symbol, a quantitative version of the unique continuation principle can be proved, and localization occurs. For the unique continuation in the general case, we give some counterexamples and prove a weaker result for hopping that decays faster than exponential rate. To the best of our knowledge, this is the first localization result for the long-range Anderson model with pure Bernoulli potentials.

• The paper establishes Anderson localization near the spectral edge in the 1D Anderson-Bernoulli model with rational Laurent symbol hopping.
• It employs a quantitative unique continuation principle and multi-scale analysis to tackle the challenges posed by singular Bernoulli disorder and long-range interactions.
• The study provides sharp counterexamples that underscore the necessity of rationality in the hopping operator to guarantee the quantitative unique continuation property.

Localization and Unique Continuation in the Anderson-Bernoulli Model with Long-Range Hopping on $\mathbb{Z}$

Introduction and Context

This work rigorously analyzes spectral and dynamical localization in the one-dimensional Anderson-Bernoulli model (ABM) on $\mathbb{Z}$ with long-range hopping operators. While localization in the Anderson model with i.i.d. potentials is a well-developed subject, this regime presents challenges in two directions: the Bernoulli distribution is singular, precluding standard Wegner estimates, and the hopping operator is long-range (exponentially or even sub-exponentially decaying), invalidating transfer matrix techniques even in 1D. Prior works on Bernoulli potentials have almost exclusively treated short-range (nearest-neighbor Laplacian) hopping.

This paper addresses both difficulties simultaneously for the first time in this setting, combining a quantitative unique continuation principle (QUC) for long-range difference operators with a carefully tailored multi-scale analysis (MSA). The analysis leads to Anderson localization near the spectral edge for a class of long-range ABMs with rational Laurent symbols, extending the rigorous regime for which the Anderson-Bernoulli localization conjecture of Yeung and Oono was open.

Model Definition and Main Results

The Hamiltonian studied is of the form: $H(\omega) = T + \lambda V(\omega)$ where:

• $V(n)(\omega)$ is a sequence of i.i.d. Bernoulli random variables (taking values in $\{0,1\}$),
• $T$ is a bounded, self-adjoint, exponentially decaying convolution operator on $\ell^2(\mathbb{Z})$, specified via a kernel $f$ as $T(m,n) = f(m-n)$,
• The Fourier symbol $\hat{f}$ of the hopping is assumed to be non-constant, real-analytic on $\mathbb{Z}$0, and (by normalization) takes values in $\mathbb{Z}$1.

Core Result: If $\mathbb{Z}$2 admits a rational Laurent symbol (that is, $\mathbb{Z}$3 is a rational function), then for $\mathbb{Z}$4 there exists $\mathbb{Z}$5, depending on $\mathbb{Z}$6 and $\mathbb{Z}$7, such that Anderson localization holds almost surely on a spectral interval $\mathbb{Z}$8 at the lower edge. That is, the spectrum is pure point in $\mathbb{Z}$9 and eigenfunctions decay exponentially.

A characterizing clarification is that the class of rational Laurent symbol hopping operators $H(\omega) = T + \lambda V(\omega)$0 covers not only short-range Laplacian, but includes many physically relevant long-range operators, e.g., those with $H(\omega) = T + \lambda V(\omega)$1 for $H(\omega) = T + \lambda V(\omega)$2. Critically, this includes the exponential-hopping model in the original Yeung–Oono conjecture.

Sharp claims highlighted:

• The localization result covers the case where the hopping decays as $H(\omega) = T + \lambda V(\omega)$3, answering the open conjecture.
• Quantitative unique continuation (which underlies the analysis) fails for general exponentially decaying hopping: there exist explicit counterexamples outside $H(\omega) = T + \lambda V(\omega)$4.
• When the Laurent symbol decays faster than exponential (i.e., is entire of order zero), a weaker, dimensionally-reduced version of unique continuation holds.

Quantitative Unique Continuation for Long-Range Operators

A central aspect of the paper is the development of a quantitative unique continuation principle (QUC) for difference equations with deterministic or random bounded potentials: $H(\omega) = T + \lambda V(\omega)$5 The principle asserts that if $H(\omega) = T + \lambda V(\omega)$6 is in $H(\omega) = T + \lambda V(\omega)$7, and $H(\omega) = T + \lambda V(\omega)$8 is bounded, any solution cannot decay too fast: on any interval $H(\omega) = T + \lambda V(\omega)$9, at least $V(n)(\omega)$0 points satisfy $V(n)(\omega)$1 for some $V(n)(\omega)$2 (parameters depend only on $V(n)(\omega)$3 and $V(n)(\omega)$4).

The proof proceeds by leveraging the algebraic structure of $V(n)(\omega)$5; since $V(n)(\omega)$6 is convolutional and $V(n)(\omega)$7 is rational, $V(n)(\omega)$8 may be reduced, via an inverse convolution, to a finite-range recurrence. This enables a "cone property" akin to that satisfied by the Laplacian, adapted to this nonlocal context. The analysis shows the full QUC property holds only for rational Laurent symbols and, with additional lower bounds on $V(n)(\omega)$9, in related classes.

The methodology demonstrates that unique continuation depends crucially on the analytic continuation of the symbol: the presence of essential singularities (as in certain transcendental symbols) creates counterexamples showing failure of any form of deterministic quantitative unique continuation.

Multi-Scale Analysis in the Absence of the Wegner Estimate

Given the (singular) Bernoulli randomness, traditional approaches (relying on regularity-based Wegner bounds) are unavailable. The authors overcome this by proving a probabilistic large deviation estimate for the Green's function at initial and large scales, by leveraging the high density of free sites guaranteed by the structure of the model and its random configuration.

The MSA employs an intricate hierarchical induction, with densities of controlled "free" intervals and a refined covering structure. Key steps include:

• Elliptic analysis at the initial scale, using the algebraic reduction above, quantitative uncertainty principles, and the rational structure, culminating in off-diagonal exponential decay of the Green's function and control of the operator norm.
• Hierarchical scale coupling, maintaining sufficient density of "resonant" and "good" intervals and careful tracking of hereditary resonances.
• Wegner-type bounds on local Green's functions, proven via the developed QUC and combinatorial arguments (discrete Sperner lemma lineage).

The analysis holds uniformly over target spectral intervals near the band edge, where the random potential has a uniform effect and analytic properties of the symbol become favorable.

Counterexamples and Limitations

The authors provide explicit constructions showing the optimality of the rationality assumption for the Laurent symbol in QUC. For hopping operators whose symbols are constructed with dense zeros within a strip (using oscillatory functions like $\{0,1\}$0), the QUC fails—a large number of linearly independent, exponentially localized solutions arise. This underscores the necessity of controlling the global analytic structure of the symbol.

Moreover, for more general classes (super-exponential or faster decaying hopping), the QUC may not hold deterministically, though the authors establish weaker, dimension-dependent lower bounds (using Carleman estimates and function-theoretic arguments).

Connections with Prior and Contemporary Work

The paper generalizes and builds on the methods of [Bourgain-Kenig 2005], who clarified the link between unique continuation and Anderson localization, up to now only for short-range hopping and in the continuum via PDE estimates. The approach here is also motivated by advances on ABMs in higher dimension by [Ding-Smart 2020; Li-Zhang 2022], who extended localization results to higher-dimensional lattices with Bernoulli disorder by developing probabilistic QUCs—however, always in short-range (discrete Laplacian) settings.

In the context of long-range hopping, existing work had restricted attention to models with regular potentials (allowing for a-priori Wegner estimates), or to alloy-type (rather than Bernoulli) random variables. The current work thus provides a critical extension to the singular, Bernoulli setting for long-range one-dimensional models.

Implications and Extensions

From a theoretical standpoint, this work establishes the analytical primacy of the rationality (thus algebraic and recursive) structure embedded in the operator symbol for enabling QUC in the absence of regularity in the potential distribution. This fits within an emerging paradigm: that in Bernoulli-disordered models, spectral and dynamical localization at low energies is essentially dictated by the analytic structure of the underlying operator.

It is explicitly noted that the methodology is dimension-agnostic: the reduction to a finite-difference equation and much of the MSA procedure are not peculiar to $\{0,1\}$1. The primary obstacle to lifting these localization results to higher dimension is the absence of a suitable QUC for higher-dimensional long-range operators; already, such principles can fail for the Laplacian itself.

On the practical side, the localization result established applies to models of physical interest—such as disordered quantum wires with long-range, exponentially decaying hopping, as highlighted in the conjectures of Yeung and Oono.

The natural direction for future research is an extension of these techniques—particularly the QUC—to higher dimension and to sub-exponential and power-law hopping regimes, where the proof strategies here (and the deterministic algebraic reduction) break down.

Another relevant extension is the investigation of non-stationary disorder distributions and alloy-type models, as developed in recent years, and an exploration of minimal hypotheses on both symbol regularity and disorder structure necessary for delocalization/localization transitions.

Conclusion

This work rigorously confirms Anderson localization at the spectral edge for the one-dimensional Anderson-Bernoulli model with long-range, rational-symbol hopping, by developing and deploying a sharp deterministic unique continuation principle for difference operators. The analysis connects operator-theoretic and analytic function theory principles, and successfully adapts multi-scale analysis to a singular disorder regime which had previously resisted all attempts when combined with long-range hopping. Counterexamples provided in the paper underscore the sharpness of the conditions. The theoretical framework set forth opens directions for future research into higher-dimensional and even less regular models in the random Schrödinger operator landscape.