Anderson models: spectral type, disorder, and dimension

Source and verification

This article explains family 261 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Random lattice Hamiltonians

The Hilbert space ${\ell }^2({\mathbb{Z}}^d)$ consists of square-summable complex functions on the integer lattice. Its coordinate vector ${\delta }_x$ represents a state concentrated at site $x$. The Anderson operator combines nearest-neighbor hopping with random site energies:

$$\begin{array}{c}(H_{h}u)(x)\\ =\sum _{|y-x{|}_1=1}u(y)+h{\omega }_{x}u(x),\end{array}$$

where the ${\omega }_x$ are independent and uniform on $[-1,1]$, and $h>0$ is the disorder strength. Here $|y-x{|}_1$ is the sum of the absolute coordinate differences, so each site has $2d$ neighbors. This bounded self-adjoint operator satisfies $\Vert H_h\Vert \le 2d+h$.

The delocalization manuscript uses negative hopping. Multiplication by $(-1{)}^{x_1+\cdots +x_d}$ reverses the hopping sign while preserving the diagonal potential, so the two conventions are unitarily equivalent. We use one positive-hopping convention throughout this introduction.

The free operator $H_0$ is diagonalized by the Fourier transform. A plane wave with frequency $k\in [-\pi ,\pi {]}^d$ has energy

$$\epsilon (k)=2\sum _{j=1}^d\cos k_j,$$

whose range is $[-2d,2d]$. Plane waves are extended states, not square-summable eigenvectors on the infinite lattice. The question is what survives after adding independent random energies at every site.

🏷️ Spectral sets and spectral types

For a bounded self-adjoint operator $H$, the spectral theorem assigns a projection $1_B(H)$ to each Borel set $B\subset \mathbb{R}$. The scalar spectral measure at site $x$ is

$${\mu }_x(B)=⟨{\delta }_x,1_B(H){\delta }_x⟩.$$

It is a probability measure. The spectrum is the closed set where $H-E$ fails to have a bounded inverse. Its spectral type describes how the measures ${\mu }_x$ distribute their mass within that set.

An absolutely continuous measure has the form $\rho (E)dE$. A pure-point measure is a countable sum of atoms. A singular continuous measure has no atoms but is concentrated on a Lebesgue-null set. A pure-point operator has a complete orthonormal eigenbasis; this does not require every point of its spectrum to be an eigenvalue.

For example, a diagonal operator whose diagonal entries form a dense sequence in an interval has a complete coordinate eigenbasis and has that entire interval as spectrum. Thus an interval spectrum does not by itself distinguish localization from delocalization. Nor does a spectral-type statement automatically establish a quantitative transport law for the evolution $e^{-itH}$.

The two manuscripts in family 261 claim the following contrast for the uniform single-site law.

Dimension	Disorder quantifier	Claimed spectral conclusion
$d=2$	Every fixed $h>0$	Pure point throughout the spectrum
Each fixed $d\ge 3$	Every fixed sufficiently small $h>0$	Purely absolutely continuous on a specified nonempty open interval

Both conclusions are almost sure. The probability-one event may depend on the fixed disorder strength. Neither paper asserts a common event for all positive strengths, a complete mobility-edge diagram, or a diffusion theorem. (OpenAI, 2026) (OpenAI, 2026b)

🏷️ The two theorem statements

Planar spectral localization, as claimed

For each fixed $h>0$, the uniform Anderson operator on ${\mathbb{Z}}^2$ almost surely has a complete orthonormal eigenbasis and

$$\sigma (H_h)=[-4-h,4+h].$$

Its eigenvalues are dense in that interval. The statement does not assert exponential eigenfunction decay, dynamical localization, or estimates uniform in energy. (OpenAI, 2026b)

The higher-dimensional paper claims a purely absolutely continuous spectral restriction with nonzero weight. There is a fixed sufficiently small $e_2\in (0,1/100)$ and, for every fixed $d\ge 3$, a positive threshold $h_d$ such that the conclusion holds for each fixed $0<h<h_d$ on

$$\begin{array}{c}{I}_3=(-6+e_2/2,-6+e_2),\\ I_d=-2d+(1/200,1/100)\\ (d\ge 4).\end{array}$$

Almost surely $I_d\subset \sigma (H_h)$ and every spectral measure restricted to $I_d$ is absolutely continuous. Both eigenvalue mass and singular continuous mass are excluded there. The interval is independent of $h$; the threshold and estimates may depend on $d$. No uniformity as $d\to \infty$ is claimed. (OpenAI, 2026a)

For $d\ge 4$, the paper specifies a threshold through an integer $n_{\ast }(d)$ and $h_d=\sqrt{3}{2}^{-n_{\ast }(d)}$. The integer is defined using infinitely many probability conditions on finite matrices. Its existence is part of the proof, and no effective numerical bound is supplied. This is not a finite certificate that these notes have executed. For $d=3$, the positive threshold is existential.

🏷️ Resolvents and single-site averaging

For $z=E+i\eta$ with $\eta >0$, define the resolvent and its diagonal entries by

$$\begin{array}{c}G(z)=(H-z{)}^{-1},\\ G_{xx}(z)=\int \frac{{\mu }_x(de)}{e-z}.\end{array}$$

The imaginary part is a Poisson smoothing of the spectral measure:

$$\begin{array}{c}\mathrm{Im}{G}_{xx}(E+i\eta )\\ =\int \frac{\eta }{(e-E{)}^2+{\eta }^2}{\mu }_x(de).\end{array}$$

After division by $\pi$, its boundary limit recovers the absolutely continuous density at almost every energy. A positive density on a large set does not rule out additional singular mass on an exceptional null set. This distinction drives the final argument in the delocalization paper.

Freeze all potentials except the value $s$ at one site. Relative to that site and its complement, write

$$H_s=\left(\begin{array}{cc}s& b^{\ast }\\ b& H^c\end{array}\right).$$

Block inversion gives

$$\begin{array}{c}{G}_{xx}^{(s)}(z)=\frac{1}{s-M(z)},\\ M(z)=z+⟨b,(H^c-z{)}^{-1}b⟩.\end{array}$$

The complement $H^c$ and the hopping vector $b$ are independent of $s$. Since $\mathrm{Im}M(z)>0$, integrating the imaginary part of $(s-M(z){)}^{-1}$ over all real $s$ gives $\pi$. Consequently, if $s$ has a bounded density $\rho$, spectral averaging yields

$$\int \rho (s){\mu }_x^{(s)}(B)ds\le \Vert \rho {\Vert }_{\infty }|B|.$$

One obtains this measure inequality by testing the Poisson identity against continuous functions and taking $\eta \downarrow 0$. It remains valid after conditioning on the complement. For the uniform law on $[-h,h]$, the density bound is $1/(2h)$. In particular, a Lebesgue-null energy set determined by the complementary potentials carries zero averaged site spectral mass. Both papers use this conditioning principle. (OpenAI, 2026a) (OpenAI, 2026b)

🏷️ Weak disorder: a finite-volume construction

The higher-dimensional argument first works on periodic boxes. Its target is a damped inverse for the true finite-volume operator, with all original random potentials already installed. With probability tending to one, there is a set $A$ containing at least half the sites and bounded complex numbers $r_i$ with imaginary parts uniformly bounded above and below by positive constants, such that

$$\begin{array}{c}\hat{G}=(H_m-E-{\eta }_m{\mathrm{diag}}_{A}r{)}^{-1},\\ \mathrm{Im}{\hat{G}}_{ii}\ge c>0(i\in A).\end{array}$$

Here ${\eta }_m\asymp 4^{-m}$ and the torus side is $L_m=4^{20m}$. The chosen damping set can depend adaptively on all potentials and auxiliary randomness. This target alone is not a claim about the undamped infinite-volume operator; the later spectral argument must remove both modifications. (OpenAI, 2026a)

The construction reveals each uniform variable through independent binary signs:

$$\begin{array}{c}h{\omega }_x=\sum _{k\ge 1}h{2}^{-k}{\epsilon }_{x,k},\\ {\epsilon }_{x,k}\in \{-1,1\}.\end{array}$$

After revealing $j$ signs, an untouched suffix is uniform on $[-h{2}^{-j},h{2}^{-j}]$ and has variance $h^2{4}^{-j}/3$. Revealing randomness gradually replaces artificial damping.

The information bookkeeping matters. At an ordinary active site, the installed value is a true revealed prefix and its suffix is untouched. A failed local trial can roll back its numerical matrix changes, but cannot erase answers already observed. Such sites are marked pending, and later trials cannot pretend their old suffixes are fresh. Temporary auxiliary digits are used only inside the construction; the final matrix again contains the original true potentials everywhere.

Three estimates are propagated: an entrywise complex-square control for the diagonal reset, row fourth-moment bounds that prevent resolvent mass from concentrating at a few sites, and a spatial energy norm derived from the Ward identity. In its scalar-damping form that identity is

$$\begin{array}{c}\mathrm{Im}G(E+i\eta )\\ =\eta G(E+i\eta {)}^{\ast }G(E+i\eta ).\end{array}$$

It follows by subtracting the two resolvent equations, and converts imaginary diagonal information into squared row norms. The source develops a weighted version and an associated Laplacian for its changing matrix state.

Pending components are cleared using a unitary scattering matrix. The range of the block coupling a component to the surviving sites identifies the directions through which completing that component can affect the exterior. Small projection weights make this effect controllable even if the nonzero singular values are very small. A deterministic induction then controls matrix changes and retains enough active sites; a separate probabilistic induction tracks the observed history and bounds failures. The complete clearing, reset, and history estimates are substantial source claims beyond the selected audit here.

🏷️ From the construction to pure absolute continuity

The spectral argument compares the adaptive damped inverse with the ordinary resolvent of the true torus operator. A narrow spectral projection isolates near-resonant eigenvalues. The comparison error is controlled by its rank for every qualifying damping choice, so no independence of the adaptive set $A$ is assumed.

Under the original product law, single-site averaging bounds the expected rank of a window of width $2ϵ\eta$ by $Nϵ\eta /h$, where $N$ is the number of sites. Combining that bound with the positive-probability construction yields a positive expectation for a bounded function of the true diagonal imaginary part.

The finite-to-infinite-volume comparison uses resolvent decay at positive height. Its error is bounded by a constant times

$$L_m^{d-1}{\eta }_m^{-2}{e}^{-c{\eta }_m{L}_m}.$$

Since ${\eta }_m{L}_m\asymp 4^{19m}$, the exponential dominates the prefactor. This is a limit at a fixed positive disorder strength, rather than a limit in which disorder is sent to zero before infinite volume.

At almost every fixed energy, the positive expectation gives positive probability of positive boundary imaginary part somewhere. Translation ergodicity and rank-one changes of site potentials propagate this to every site almost surely. Nonvanishing of off-diagonal boundary resolvents is another input: the first nonzero coefficient in the large-$z$ expansion counts shortest lattice paths, so the corresponding analytic function is not identically zero. A boundary uniqueness argument then rules out a positive-measure zero set.

Purity requires one more step. After removing a site, the paper constructs a full-Lebesgue-measure set of energies where $M(E+i0)$ exists finitely and has positive imaginary part. This set depends only on the complement, so every real value of the removed site has a finite diagonal boundary value there. Bounded Poisson limits imply absolute continuity on this set. Its exceptional complement is independent of the remaining site value, so conditional spectral averaging removes its spectral mass as well. Counting over sites gives the asserted pure absolutely continuous restriction. Merely establishing a positive density almost everywhere would leave this last issue unresolved. (OpenAI, 2026a)

🏷️ The planar mechanism and logarithmic capacity

The planar paper also uses damped finite matrices, but toward a different conclusion. Artificial absorbing channels, called ports, have a unitary scattering matrix $S$. For real data $f$ on the ports, the associated energy is

$${\mathcal{Q}}_S(f)=\frac{1}{2}\sum _{a,j}|S_{ja}{|}^2|f_j-f_a{|}^2.$$

Rare resonant cells provide local transmission estimates. Their conditional laws are realized through acceptance probabilities within the original product measure; they do not replace the true uniform site distribution by an unconditional Cauchy law. The errors from the bounded physical distribution must be small relative to transmission, including for weakly coupled exterior directions.

A hierarchy removes ports at increasing scales. At each fixed starting level, the torus size tends to infinity first, producing an expected affine scattering-energy density. Only afterward is the hierarchy level sent to infinity. The resulting nonnegative limit is denoted $D$ here, to distinguish it from spatial dimension.

Two-dimensional capacity supplies the decisive small quantity. For a radial voltage on $r_0<r<R$, take

$$\phi (r)=\frac{\log (R/r)}{\log (R/r_0)}.$$

It equals one on the inner circle and zero on the outer circle, and direct polar integration gives

$${\int }_{r_0<|x|<R}|\nabla \phi {|}^{2}dx=\frac{2\pi }{\log (R/r_0)}.$$

The cost tends to zero as the scale ratio grows. The manuscript uses square annuli and comparable logarithmic ramps, obtaining the same inverse-logarithmic behavior. This elementary calculation explains the dimensional mechanism, but does not itself prove localization.

If $D>0$, the source constructs an analytic scattering test with a nonzero central value. The small annular capacity, changes of the true potentials preserving their allowed support, and hierarchy comparisons make its boundary moments arbitrarily small. The submean inequality gives a contradiction, forcing $D=0$. Uniformity over passive exterior data and control of the number of ports are essential in this passage. (OpenAI, 2026b)

🏷️ Fixed-energy decay and the spectral conclusion

The planar bootstrap converts vanishing affine energy into an estimate for the true finite-torus Green function. For each fixed $h>0$, interior energy $E\in (-4-h,4+h)$, and $0<s<1/16$, the source claims constants $A,a>0$ and a sufficiently large starting torus size such that

$$\mathbb{E}|(H_N-E{)}_{xy}^{-1}{|}^{s/2}\le A{e}^{-a\mathrm{dist}(x,y)}.$$

The inverse exists almost surely at the specified real energy. The constants and starting scale depend on $h,E,s$; this is not a uniform-in-energy or time-evolution estimate. (OpenAI, 2026b)

The spectral consequence can be explained more explicitly. Fix a site $x$ and energy $E$. The fractional-moment estimate bounds a weighted sum of the finite-volume resolvent column. Fatou’s lemma gives an almost surely bounded subsequence, and a diagonal extraction produces a vector $u\in {\ell }^2$ satisfying $(H-E)u={\delta }_x$. Finite hopping range lets the equation pass to the limit at every fixed site. The spectral theorem then gives

$${\Gamma }_x(E)=\int \frac{{\mu }_x(de)}{(e-E{)}^2}<\infty .$$

The integrand is infinite at $e=E$. There is no contradiction with random eigenvalues: this assertion fixes $E$ before choosing the random configuration, and does not intersect over all real energies.

For a rank-one perturbation $H_a=H_0+a⟨\xi ,\cdot ⟩\xi$, write $F(z)=⟨\xi ,(H_0-z{)}^{-1}\xi ⟩$. Then

$$F_a(z)=\frac{F(z)}{1+aF(z)}.$$

At energies with finite inverse-square integral, $F(E+i0)$ is real. Where $1+aF(E+i0)=0$, the inverse vector $u$ obeys $(H_a-E)u=0$, so the possible remaining spectral mass is on eigenvalues. Spectral averaging removes the exceptional Lebesgue-null set of energies. Fubini is applied to the measurable inverse-square property of the true operator, rather than to an energy-dependent choice of auxiliary port construction. Finally one intersects over the countably many sites, whose coordinate vectors span a dense subspace. No single-site cyclicity assumption is needed.

The spectral interval follows by a separate elementary argument. The norm bound gives containment in $[-4-h,4+h]$. For any fixed box size, potential value in a dense subset of $[-h,h]$, and positive tolerance, infinitely many independent disjoint boxes have positive probability of being uniformly close to that constant. Almost surely some do. Cut-off plane waves in larger such boxes have vanishing boundary-to-volume error, and their diagonal error tends to zero with the tolerance. The approximate-eigenvector criterion fills the entire interval. Combined with the eigenbasis conclusion, it makes the eigenvalues dense there. This argument establishes the spectral set, while the preceding scattering and averaging work establishes spectral type.

🏷️ Key steps and proof outlines

Adaptive damping is removed by rank, not by independence

Here is the matrix mechanism behind the higher-dimensional spectral transfer. Let $T=(H_m-E{)}^{-1}$ for the true finite operator, and let $P_A$ project onto a qualifying adaptive damping set. After Schur complementation, the scalar-damped active inverse is $f_{\eta }(P_{A}T{P}_A)$, with

$$f_{\eta }(u)=\frac{u}{1-i\eta u}.$$

Remove the true spectral projection $Q=1_{[E-\epsilon \eta ,E+\epsilon \eta ]}(H_m)$ before compressing, and put $B_c=P_{A}T(I-Q)P_A$. The difference between the two compressed matrices has rank at most $\mathrm{rank}Q$. The resolvent identity preserves that rank after applying $f_{\eta }$, while each transformed matrix has norm at most ${\eta }^{-1}$. Thus their difference $D$ obeys

$$\Vert D{\Vert }_{\mathrm{HS}}^2\le 4{\eta }^{-2}\mathrm{rank}Q.$$

Away from the removed window, the inverse-square weight is at most $(1+{\epsilon }^{-2})$ times the ordinary Poisson weight. Combining this with the certified active-column lower bounds gives a bounded nonnegative function $\Phi$ such that

$$\frac{1}{N}\sum _i\Phi (\mathrm{Im}{G}_{ii})\ge \frac{c_{\ast }}{2}{1}_{\{\text{certificate exists}\}}-\frac{8\mathrm{rank}Q}{\eta N}.$$

Every qualifying choice implies the same true-operator inequality. Consequently the averaging step needs neither an independent damping set nor a measurable selection of one. Single-site averaging under the original product law controls the rank term; choosing $\epsilon$ small leaves a positive lower bound. Compression never has to commute with the spectral projection. (OpenAI, 2026a)

Planar barriers must remain likely after weighting by transmission

In the planar bootstrap, let $V$ denote a scattering-block norm measuring transmission from a filled annulus to a remote target. A rare bad-barrier estimate under the unweighted law is insufficient if $\mathbb{E}{V}^s$ is itself tiny. The manuscript therefore uses the probability measure

$$d\nu =\frac{V^s}{\mathbb{E}{V}^s}d\mathbb{P}$$

when the denominator is positive. Schur complementation at one site writes the transmission block as a fixed matrix numerator divided by a scalar dissipative denominator. On a central interval the numerator is affine. Its maximum controls it from below on a subinterval of fixed relative length, while integrability of $|v-a{|}^{-q}$ for $q<1$ controls the denominator. This proves a conditional reverse moment estimate

$$({\mathbb{E}}_i(V^s{)}^2{)}^{1/2}\le K_s{\mathbb{E}}_i{V}^s.$$

Cauchy–Schwarz now bounds a one-site bad event under $\nu$ by $K_s$ times the square root of its original probability, uniformly in every frozen exterior. Iterating conditional estimates along distinct sites and counting simple lattice paths gives exponentially unlikely barrier failure under both laws. Independence under $\nu$ is never asserted.

Resampling separates the return factor without conditioning on success

Buffered inner and outer barriers permit two fresh configurations: one retains the original exterior and resamples the interior; the other retains the original interior and resamples the exterior. These two configurations are independent under the enlarged, unconditioned product law. Cutoff commutator estimates and the Ward identity compare the original transmission $V$ and return $U$ with their respective fresh versions on the good-barrier event $\mathcal{B}$. Dropping that indicator before factorization gives

$$\mathbb{E}[1_{\mathcal{B}}{V}^s{U}^s]\le 2^{2s}\mathbb{E}{V}^s(\mathbb{E}{U}^2{)}^{s/2}.$$

The tilted estimate pays for the complement. Vanishing hierarchy energy makes the return term small; repeated buffered cutoffs attenuate direct crossing exponentially. Choose the barrier-strength parameter first, then the hierarchy levels and torus size in the source’s order. The resulting one-step multiplier can be made arbitrarily small, and a supremum over deterministic masks satisfies $F_N(r)\le \frac{1}{2}{F}_N(r-4R)$. Iteration yields exponential fractional-moment decay at the fixed energy. The earlier capacity argument supplies the vanishing-energy input; this bootstrap explains how that input becomes a resolvent estimate. (OpenAI, 2026b)

🏷️ Connections and verification limits

Random regular spectra includes weak Anderson disorder on random regular graphs. Its finite-graph eigenvalue statistics and infinite-tree density belong to a different geometry from the Euclidean lattice treated here. Simple Lebesgue spectrum uses the same absolutely continuous spectral language for Koopman operators, with a different operator and construction.

The audit covered both theorem statements, introductions, the finite-volume targets, and selected spectral-transfer and planar capacity arguments. The complete adaptive clearing, rare-cell transmission, hierarchy, and bootstrap proofs were not independently verified. No finite random-matrix simulation would by itself verify a fixed-disorder infinite-volume spectral-type theorem, and none is presented as such here.

🏷️ Manuscripts and verification scope

• Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three — pinned manuscript and companion source.
• Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. Only the planar manuscript is listed. The selected result constructs its bounded self-adjoint Anderson operator and identifies the almost-sure spectral set [-4-h,4+h] for each positive uniform disorder strength. The scope explicitly does not assert the pure-point spectral type claimed in that manuscript. The higher-dimensional delocalization companion is not listed. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three.

🐻  OpenAI 2026b. Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder.