Absolutely Continuous Spectrum for Weak-Disorder Anderson Models
in Dimensions at Least Three
(revised, expanded, and rewritten)
0. A GUIDE FOR FIRST-TIME READERS
This document proves one of the foundational problems in mathematical
physics of disordered media. Before we begin, here is what the reader
is being asked to believe, and why.
0.1 The model, in one paragraph.
Place atoms at every point of the d-dimensional integer lattice Z^d with
d >= 3. Between nearest neighbors there is a "hopping" interaction of
strength 1, which lets a quantum particle jump. On each site x there is
a randomly chosen energy V_x drawn once and for all from the uniform
distribution on [-1, 1]. The hopping is the same on every edge; the site
energies are independent from site to site and strength lambda squashes
them: the physical site energy is lambda V_x. The Hamiltonian is
(H_{lambda, omega} psi)(x)
= -sum_{j=1..d} ( psi(x + e_j) + psi(x - e_j) )
+ lambda V_x(omega) psi(x),
acting on the Hilbert space l^2(Z^d). It is a bounded, self-adjoint
operator. Here omega = (V_x)_x is the elementary event in the product
space Omega = [-1,1]^{Z^d} with the uniform product probability
probability law P.
The question we answer is: what kind of spectrum does H_{lambda, omega}
have, on what part of the energy axis, and for how many omega.
0.2 The physical intuition.
If lambda were zero, H_{0} would be the free discrete Laplacian, whose
spectrum is the closed interval [-2d, 2d], and every energy in that
interval has a purely absolutely continuous spectral measure for the
delta-vector at any site. Extended plane waves solve the equation, and
the particle spreads.
If lambda were enormous, the potential lambda V_x would dwarf the
hopping. The diagonal part is then (almost surely) an operator with
purely point spectrum (each site is essentially stuck), and in the
d-dimensional lattice the full operator is expected (and in many
regimes known) to be localized: the particle cannot diffuse.
Somewhere in between there must be a transition. In one and two spatial
dimensions the conventional wisdom is that arbitrarily weak disorder
localizes everything. In dimension d >= 3 there is room for both:
genuinely extended (absolutely continuous) states survive at weak
disorder near the middle or upper part of the band, while the band edges
localize first.
0.3 What "absolutely continuous spectrum" means, in words.
The spectral theorem gives the Hilbert space a decomposition
l^2(Z^d) = H_ac (+) H_sc (+) H_pp .
Absolute continuity of a vector's spectral measure with respect to
Lebesgue measure on the energy axis means: the amount of the vector's
weight that lands in an energy set of measure zero is zero. Physically,
absolutely continuous spectrum corresponds to transport: the particle
spreads through space and every energy in the a.c. part of the spectrum
is genuinely "in the continuum". Purely point spectrum means each
spectral measure is a sum of Dirac masses, i.e. eigenvalues; singular
continuous spectrum is a pathological middle case.
Proving that an open energy interval carries only absolutely continuous
spectrum rules out both eigenvalues and singular continuous mass there.
0.4 The precise theorem, stated without jargon.
THEOREM 1.1. Fix an integer d >= 3. There is a small number
lambda_d > 0, depending on d, and a fixed open energy interval I_d,
also depending on d, sitting strictly above the lower free-band edge
-2d, with the following property. For every fixed 0 < lambda < lambda_d
and P-almost every omega,
(a) the interval I_d is contained in the spectrum of H_{lambda, omega};
(b) the spectral projection chi_{I_d}(H_{lambda, omega}) acts on a
nonzero subspace which is purely absolutely continuous; and
(c) that subspace has nonzero weight, i.e. every basis vector
chi_{I_d}(H_{lambda, omega}) e_x is nonzero.
The interval I_d is chosen once and for all: it is *independent of
lambda*. On it, the spectrum is purely absolutely continuous, and not
trivial (does not vanish).
Concretely: for d = 3, I_3 = (-6 + e_2/2, -6 + e_2) for a fixed small
e_2 in (0, 1/100). For d >= 4, I_d = -2d + (1/200, 1/100). The
quantity lambda_d is set in Definition 8.6 (for d >= 4); for d = 3 it
is existential, produced by the same construction.
This settles a question of Simon ("Problem 1"), which asks precisely
for a purely a.c. interval at weak disorder in d >= 3. It does not
locate the mobility edge (the moving boundary between a.c. and p.p.
regions), and it does not prove diffusion; those are harder questions
treated elsewhere.
Inside each fixed d, the conclusion holds uniformly as the finite
volume tends to infinity. We are not claiming uniformity in d, only
that for each fixed d we get a positive interval and a positive
threshold lambda_d.
0.5 Why the theorem is not obvious, and what the proof actually does.
The hard part is not "lambda small makes each V_x small"- that's true
pointwise. The hard part is showing that *the resolvent does not
concentrate*. For a given energy E and imaginary part eta > 0, the
resolvent (H - E - i eta)^{-1} can in principle have a huge norm if E
is very close to an eigenvalue. Standard Weyl-type arguments alone
don't rule this out at small but positive disorder. You need a
quantitative, *perturbative* argument showing that at small lambda the
true operator is (in a suitable average sense) close to the free
operator on a *large fraction of sites*.
The original 1998-2020 landscape had three main proof strategies for
related statements:
- Multi-scale Green-function (Fr\"ohlich-Spencer) arguments: prove
localization (the opposite) by producing exponential decay at
strong disorder, then use monotonicity of the Lyapunov exponents;
these arguments do not easily give absolute continuity.
- Bethe-lattice transfer matrix methods (Klein 1994, Aizenman-Sims-
Warzel 2006, Aizenman-Warzel 2013, Aggarwal-Lopatto 2025): prove
absolute continuity at *specific tree-like* geometries, but the
conclusion does not transfer to the Euclidean lattice because the
branching geometry is essential there.
- Supersymmetric and diffusion-type arguments (Erd\"os-Salmhofer-Yau,
Black-Drogin-Hern\'andez): derive the heat equation for the
Wigner distribution as lambda -> 0. Those are diffusion results,
not spectral ones, and again do not directly yield a.c. spectrum.
Our proof introduces a different quantitative criterion called a
*finite-volume resolvent certificate*, shows that it implies pure a.c.
spectrum with nontrivial weight via a self-contained spectral argument,
and then constructs the certificate at weak disorder via a multiscale
probabilistic scheme built out of finite-range tiles. The two positive
accomplishments of the paper are:
(A) CONSTRUCT THE CERTIFICATE (Sections 2-8). Show that for fixed
d >= 3, any small enough lambda, and any E in I_d, the certificate
has probability at least 1/2 in the natural class of finite
periodic approximations.
(B) UPGRADE THE CERTIFICATE TO THEOREM 1.1 (Section 9). Prove that
this finite-volume object forces pure a.c. spectrum almost surely
in the infinite-volume limit.
Part (B) is a self-contained argument with three ingredients that the
reader can follow end to end: a resolvent-averaging lemma (rank one
perturbations smear out spectral concentration); a "propagation of
positivity" step (if one site has a positive imaginary part at energy
E almost surely, so does every site); and a "singular mass exclusion"
argument (rank-one averaging at a single site removes any possible
singular measure supported on a measure-zero set of energies).
0.6 The certificate, in one paragraph.
The central new object is a set A of at least half the vertices of a
large torus, together with a *complex* coefficient r_i at each vertex
of A with strictly positive imaginary part c <= Im r_i <= C and
bounded magnitude |r_i| <= C. These coefficients are used to add a
small artificial damping of size eta > 0 to the operator: replace the
true Hamiltonian H_m on the finite torus by
H_m - E - eta diag_A r.
"Artificial damping on A" means: on A we've added eta * r_i (a complex
bump); off A we have not added damping. On the resulting operator we
ask for a very strong analytic statement: that the full inverse
Ghat = (H_m - E - eta diag_A r)^{-1}
exists, and that on every i in A the imaginary part of the (i,i) entry
of Ghat is at least a fixed positive constant *uniform in m*.
The choice of (A, r) *may* depend on the entire realization of the
potential. What must be uniform is the *existence of some such pair*:
with probability >= 1/2 over the true value of H_m - E, at least one
qualifying (A, r) exists. This is a statement about the operator only,
not a claim about any algorithm that finds (A, r).
The rest of the paper is bookkeeping: (i) constructing such (A, r) with
high probability via a multiscale randomized process, and (ii) showing
that the existence of such a certificate at probability >= 1/2 for all
large m is enough to imply the theorem.
0.7 How the certificate is built, at the level of a picture.
The construction "reveals" the true potentials to itself one binary
digit at a time. Each uniform V_x on [-1, 1] can be written as a sum of
independent fair coin flips:
lambda V_x = sum_{k >= 1} lambda 2^{-k} eps_{x,k},
P(eps_{x,k} = 1) = P(eps_{x,k} = -1) = 1/2.
At the beginning of the construction ("scale n_0") nothing has been
revealed, the operator is just the free Laplacian H_0 (zero potential),
and the damping coefficient r is the solution of a single scalar
self-consistency equation. At the end ("scale m") every digit has been
revealed and only *some* active sites remain; those carry the true
potential and keep a tiny artificial damping eta.
Between these ends, the construction runs in stages t_0 > t_1 > ... >
t_m of *decreasing* artificial damping, where t_n ~ 4^{-n}. At each
stage, an "ordinary sweep" inserts one further digit at every active
site. The stages in which we also "clear" pending components (sites
whose value is a mix of a true prefix and some auxiliary randomness
used to fix failed trials) are only run up to j = floor(m/2). After
that, anything still pending is fatal, and the construction aborts;
this needs to happen with probability vanishing in m, which we prove.
There are three kinds of site at any committed moment:
- COUPLED active: it has revealed prefix digits of the *true*
potential, but no digit of the suffix. So its potential so far is
a truncated true V_x.
- PENDING active: it *might* contain auxiliary random digits, and
the construction can no longer assume its remaining digit has the
product fair-sign law. Pending sites are the source of all the
difficulty; they are cleared by special "clearing trials" and
everything becomes coupled at the cutoff j = m/2.
- TERMINAL: its true potential is fully revealed and it has been
Schur-eliminated from the active submatrix. Terminal sites never
change again.
An "ordinary sweep" tries to update every active site by one digit. It
is done *tile by tile*. A tile is a cube of side 4^{6n} in the torus
of side 4^{20m}. On each tile we try a random order of the tile's
sites. At each site we add the next bit of the true digit (or an
auxiliary bit, when pending). If a small *local budget* is exceeded
(a numerical quantity measured only on a neighborhood of the tile
blows up), the tile's attempt is rolled back, all of its active sites
are marked pending, and we try again with fresh randomness. Local
budget failure at a single tile has probability ~ exp(-t^{-c}), so
many retries are cheap.
If a site is pending at the end of stage j <= floor(m/2), then either
the failure that marked it is *recent* (at the same stage j), or the
site belongs to a "huge" connected component of pending sites
(> t_j^{-nu} vertices with a much coarser metric). Huge components
have a self-similar structure: nearby pending sites recur at earlier
scales, giving a combinatorial "witness". Section 8 turns this
picture into a probability estimate: the probability that any given
site is pending at the end of stage j is at most exp(-4^{c j}).
Summing over all sites shows no pending sites remain at the cutoff.
Once no site is pending, the construction just needs to reveal every
site's true potential (no more auxiliary digits). It does so in one
final "whole-suffix sweep" at scale m. If that sweep fails, we abort;
the probability of abort is again vanishing. On success, all active
sites have the true potential with only eta * r_i of extra damping
left, which is exactly the certificate's (A, r).
0.8 Where the technical difficulty of the construction lives.
Two things must be true simultaneously at every intermediate state.
First, the Ward identity must hold. This is a matrix identity for the
inverse of a Hermitian matrix with a complex diagonal:
h diag(p) h^* = Im g, F p = q,
where h = sqrt(t) g, F_{ij} = |h_{ij}|^2, p_i = Im r_i, q_i = Im g_{ii}.
This identity constrains the "off-diagonal energy" of the resolvent to
be tied to the "on-diagonal imaginary part". Every estimate in the
paper is ultimately tracking this balance.
Second, two small quantities must stay small throughout:
- the "operator square norm" ||K|| where K_{ij} = h_{ij}^2. This
is the linearization of the diagonal self-consistency equation
r_i = g_{ii}(r); showing ||K|| < 1 makes the fixed point
well-posed as r is updated.
- the "row fourth moment" sum_j |h_{ij}|^4 <= H_* = t^{1+delta}.
This limits concentration in any one row.
Third, a ward Laplacian D (built from F and p) must be equivalent, at
every committed state, to a known comparison operator W that is the
dual of the inverse form X = D^+. The equivalence is by fixed
multiplicative constants; it is transported through the induction
rather than re-proved at every stage.
All of the numbers in the paper are tuned so that these three
constraints can be maintained all the way from n_0 to m, on every
committed state, regardless of the history of failed trials.
0.9 What this document does *not* do.
- It does not prove diffusion, i.e. it does not prove that a wave
packet spreads like a heat kernel in time. That is a different,
weaker-looking but harder-in-practice question.
- It does not locate the mobility edge, i.e. the boundary in energy
between a.c. and p.p. spectrum.
- It does not claim uniformity in the dimension d: all constants are
allowed to blow up as d grows.
- It does not prove that the interval I_d has measure bounded away from
zero uniformly in lambda: it is fixed once lambda_d is chosen, but I_d
itself depends on lambda through nothing except the choice of d.
It does prove exactly what Theorem 1.1 says: at every fixed d >= 3 and
every fixed sufficiently small positive coupling, almost surely, there
is a nontrivial open energy interval meeting the spectrum in a purely
absolutely continuous way.
0.10 How the rest of the document is organized.
Section 1: The model and the theorem.
Precise definition, the negative-hopping convention, why the
hopping sign does not matter, and the exact statement of
Theorem 1.1. Historical context.
Section 2: The finite-volume certificate and the matrix state.
Precisely what "a certificate" is. The setup on the periodic
torus. Scales t_n, s_n, eta. The two kinds of "port" (bit and
reserve). Ward identity. The Ward Laplacian D and its inverse
form X. The bounds that we will carry between stages.
Section 3: Initialization.
Free-operator Fourier estimates. The self-consistent diagonal
equation r = <(a(k) - E - t r)^{-1}>. The gap estimate. The
three inverse-form comparisons at the free state.
Section 4: The algebra of an ordinary sweep.
What one bit-port deletion does to (g, K, D, the lag). The
Sherman-Morrison formula in this notation. One-step estimates
for J_x and L_x increments.
Section 5: Local trials and the ordinary sweep.
Local squares at scale n; how to test budgets on a small
neighborhood of a tile; how decay of the resolvent justifies
doing so; the matrix martingale inequality used to bound the
failure probability of one tile; retries.
Section 6: Clearing a small pending component.
The cross-block scattering matrix S, its range projection Pi,
and why a small trace of Pi (via a ridge-regularized
"leverage score") is the right completion criterion. Drift of
a well-chosen potential Phi = P - 10 * l_* * A under heats;
counting the number of successful heats.
Section 7: Committing clearings and restoring the stage bounds.
The deterministic induction: every successful clearing
preserves the three comparison bounds with summable losses.
The diagonal reset r_i -> g_{ii}. Closing the deterministic
loop.
Section 8: Failure histories and removal of pending sites.
The birth-label bookkeeping. The witness argument: a
surviving pending site forces a recent failure or a cascade
of separated older failures. The cutoff at j = floor(m/2) at
which all pending sites vanish. The final whole-suffix sweep.
Definition of lambda_d for d >= 4.
Section 9: From the certificate to pure absolute continuity.
Own-site spectral averaging. The "true finite-volume resolvent
bound" that turns the certificate (which is allowed to depend
on omega) into a bound on E[Phi(q_0)]. Passage to infinite
volume. Propagation of positivity to all sites via ergodicity
and rank-one formulas. Exclusion of singular mass. Proof of
Theorem 1.1.
0.11 Notation, once and for all.
Throughout:
d >= 3 is a fixed integer; all constants may depend on d.
Z^d is the integer lattice; T_L = (Z/LZ)^d is the torus of side
length L (L an integer multiple of 4).
e_1, ..., e_d are the standard coordinate vectors.
[-1,1]^{Z^d} is the probability space Omega; P is the product
uniform probability. P-almost surely (a.s.) abbreviates "on a
probability-one set."
l^2(Z^d) and l^2(T_L) are the Hilbert spaces of square-summable
complex functions.
H_{lambda, omega} is the infinite-volume operator (Section 1).
H_m is the finite-volume operator on T_{L_m} with L_m = 4^{20m},
at disorder width such that t_{n_0} matches (see Section 2).
For a Hermitian matrix B, Im B = (B - B^*)/(2i). We use the same
notation entrywise.
For a Laplacian matrix D with kernel equal to the constants, D^+
is the Moore-Penrose pseudoinverse, restricted to mean-zero
vectors; we call X = D^+ the "inverse form" of D.
Schur_{\mathcal E} B denotes the Schur complement of a block B
with respect to the index set \mathcal E being eliminated:
Schur_{\mathcal E} B = B_{RR} - B_{R\mathcal E} B_{\mathcal E
\mathcal E}^{-1} B_{\mathcal E R}, where R is the complement of
\mathcal E.
delta = 0.01; kappa = 0.005 * delta; nu = 10^{-5} delta / d.
These are fixed throughout, chosen small enough that every
"strictly positive power" inequality used in the paper holds.
H_* = t^{1+delta}; b = t_{n_0} t.
"Committed" means "a numerical state along the intended path,
with all prior failures rolled back and every still-valid answer
recorded in the retained history". "Provisional" means "a state
inside a trial that has not yet been committed or rolled back".
1. THE MODEL AND THE MAIN THEOREM
We now make the informal discussion of Section 0 precise. Fix once and
for all an integer d >= 3. Everything that follows depends on d; all
constants (including "small" and "large" thresholds, and the implicit
constants in O(.), <<, >>, and ~) may depend on d. No claim in this
document is uniform as d -> infinity.
1.1 The probability space and the interactions.
Let
Omega = [-1, 1]^{Z^d}, P = ( (1/2) 1_{[-1,1]}(v) dv )^{ox Z^d}
be the product of uniform probability measure on [-1, 1] over the sites
of the integer lattice Z^d. The random variable V_x : Omega -> [-1, 1]
is coordinate evaluation. The product structure means {V_x : x in Z^d}
are independent, each uniform on [-1, 1].
Fix a coupling constant lambda > 0. On l^2(Z^d), define the densely
defined (in fact bounded, self-adjoint) operator
(H_{lambda, omega} psi)(x)
= - sum_{j=1}^{d} ( psi(x + e_j) + psi(x - e_j) )
+ lambda V_x(omega) psi(x), x in Z^d. (1.1)
Here e_1, ..., e_d are the standard coordinate unit vectors. The two
pieces have names: the SUM OVER J is the (negative-sign) HOPPING
operator, a bounded self-adjoint operator on l^2(Z^d) with norm 2d;
and the multiplication-by-lambda-V_x piece is the (random) POTENTIAL.
Both pieces commute with integer translations of the lattice, so
H_{lambda, omega} is invariant under the natural translation action of
Z^d on l^2(Z^d).
The sign convention in (1.1) is the "negative hopping" one, and it
matters only up to a unitarily equivalent change. Indeed, define the
diagonal unitary U on l^2(Z^d) by
(U psi)(x) = (-1)^{x_1 + x_2 + ... + x_d} psi(x), x in Z^d. (1.2)
Then U reverses the sign of the hopping in the sense
U H^{hop} U^* = - H^{hop},
and U commutes with every potential V_x (U is diagonal). Consequently
U H_{lambda, omega} U^* = (positive-hopping Hamiltonian).
The spectral type of H_{lambda, omega} does not depend on which of the
two signs we take for the hopping. In the rest of this document we use
the convention (1.1) for concreteness.
1.2 Spectral measures and the classification of spectrum.
The spectral theorem gives, after restriction to the Z^d-invariant
subspace spanned by H_{lambda, omega}, a direct-sum decomposition
l^2(Z^d) = H_ac (+) H_sc (+) H_pp (1.3)
where each summand is invariant under H_{lambda, omega} and known as
the absolutely continuous, singular continuous, and pure-point parts
respectively. For a unit vector phi in l^2(Z^d), the associated
spectral measure mu_phi is defined by Borel functional calculus on the
spectral resolution of H_{lambda, omega}; it is a finite positive
measure on the real line, and its Lebesgue decomposition is
mu_phi = mu_phi^{ac} + mu_phi^{sc} + mu_phi^{pp}.
A vector phi lies in H_ac if and only if mu_phi is absolutely
continuous with respect to Lebesgue measure, i.e. mu_phi(B) = 0 for
every Borel set B of Lebesgue measure zero. Analogously for H_sc and
H_pp.
For a Borel set I subset R, we write chi_I(H) for the associated
spectral projection. We say that the SPECTRAL RESTRICTION OF H TO I is
PURELY ABSOLUTELY CONTINUOUS when
chi_I(H) l^2(Z^d) subset H_ac.
Equivalently: for every phi in the range of chi_I(H), the spectral
measure mu_phi has no point or singular-continuous component
supported on I. By definition, this restriction is nonzero iff
chi_I(H) != 0.
1.3 Statement of the main theorem.
Recall the notation for a finite integer m and a torus. Given an
integer L >= 4, we let T_L = (Z / L Z)^d be the d-dimensional torus
of side length L. When we say "the free operator on T_L" we mean the
periodic hopping operator -sum_j (shift_{e_j} + shift_{-e_j}) on
l^2(T_L).
THEOREM 1.1 (Main theorem). There exists a fixed number
e_2 in (0, 1/100), and for every fixed integer d >= 3 there exist
constants lambda_d > 0 and c_*, C_*, p_*, p^* such that the following
holds. Define the energy interval
I_d = ( -6 + e_2/2, -6 + e_2 ), for d = 3,
I_d = ( -2d + 1/200, -2d + 1/100 ), for d >= 4. (1.4)
Then for every fixed 0 < lambda < lambda_d, P-almost every omega
satisfies BOTH of:
(i) I_d subset sigma(H_{lambda, omega}) (the spectrum meets I_d);
(ii) chi_{I_d}(H_{lambda, omega}) l^2(Z^d) subset H_ac(H_{lambda, omega}),
and in fact chi_{I_d}(H_{lambda, omega}) is nonzero, i.e. every basis
vector chi_{I_d}(H_{lambda, omega}) e_x is a nonzero vector whose
spectral measure has positive absolutely continuous mass in I_d.
The interval I_d is fixed once and for all; it depends on d and on
e_2 but NOT on lambda. The threshold lambda_d is set below in
Definition 8.6 for d >= 4, and is existential for d = 3. For d >= 4
one may take lambda_d = sqrt(3) 2^{-n_*(d)}, where n_*(d) is the
integer from Definition 8.6.
The probability-one event supplied by the theorem may depend on d and
on lambda. It may also depend on e_2, though for the applications in
this paper we only need it for the fixed value of e_2 chosen in
Section 3 (Proposition 3.12).
The conclusion (ii) is slightly stronger than "the spectral restriction
is purely absolutely continuous". It asserts that the restriction is
itself nontrivial: every basis vector has positive spectral weight in
I_d. Theorems in the literature sometimes only give the weaker "the
interval is inside the a.c. part of the spectrum" but allow the
spectral projection onto the interval to be zero; we rule that out.
1.4 Why this is the Simon problem.
Simon's published Problem 1 in "Schr\"odinger Operators in the Twenty-
First Century" asks, very roughly: for the Anderson model with iid
uniform site potentials on Z^d, d >= 3, and small positive lambda,
does there exist a nontrivial open energy interval in which the
spectrum is purely absolutely continuous?
Theorem 1.1 answers "yes", with a specific fixed interval I_d that
avoids the band edges (which have their own localization behavior; see
Corollary 1.3 below) and sits inside the free band. It leaves three
related physical questions untouched, and we mention them because the
reader is likely to confuse them with the theorem:
(1) It does not locate the MOBILITY EDGE, i.e. the boundary in
energy between a.c. and p.p. spectrum. Even the existence of
such an edge for this model is open in full generality.
(2) It does not establish DIFFUSION. The diffusion question asks
whether the wave packet spreads like a heat kernel in time,
and even at weak disorder is a strictly harder problem than
pure a.c. spectrum.
(3) It does not address the strong-disorder regime, i.e. lambda
large. At strong disorder in d >= 3, the accepted picture is
pure-point spectrum in the bulk of the band, but the full
theorem is open in d >= 3 at fixed large disorder.
What it does prove is exactly the statement of Section 1.3, and this
statement is enough to answer Simon's Problem 1 as published.
1.5 Historical context.
The Anderson model was introduced by Anderson in 1958 to explain the
absence of diffusion in certain random lattices. The mathematical
study of the model has three intertwined strands.
STRONG DISORDER. When the random potential dominates the hopping, the
spectrum should be pure point with exponentially localized eigen-
functions. This has been established by Fr\"ohlich and Spencer (1983),
by Fr\"ohlich, Martinelli, Scoppola and Spencer (1985) via a multi-
scale Green-function analysis at fixed d and sufficiently large
lambda, and by Aizenman and Molchanov (1993) via a fractional-moment
method at suffficiently large lambda. For the iid uniform model used
in this paper, Elgart and Klein (2019) obtained pure-point spectrum
with exponentially decaying eigenfunctions on a fixed interval near
the bottom of the spectrum, at any fixed nonzero lambda, d >= 1. This
last result is what we combine with Theorem 1.1 in Corollary 1.3 to
show coexistence of a.c. and p.p. spectrum at the same coupling.
BETHE LATTICE. On the infinite regular tree (Bethe lattice) of degree
k+1, the situation is much better understood. Klein (1994, 1998)
proved purely absolutely continuous spectrum on compact subintervals
of the free band at sufficiently weak disorder, and Aizenman, Sims
and Warzel (2006) established stability of the absolutely continuous
component. Aizenman and Warzel (2013) used a resonant delocalization
argument to obtain an a.c. component beyond the free band for a general
class of regular full-support laws at weak disorder, and proved pure
absolute continuity in a weak-disorder regime that includes the
uniform law. More recently Aggarwal and Lopatto (2025) obtained mobility
edges on the Bethe lattice for sufficiently high-degree trees, at a
specific hopping scale and for regular full-support laws. These results
use the branching structure of the tree in an essential way and do
not immediately transfer to Z^d.
DIFFUSION-TYPE ARGUMENTS. In d >= 3, the transverse fluctuation of the
Wigner distribution under the Schr\"odinger evolution has been analyzed
by Erd\"os, Salmhofer and Yau (2007, 2008), following the T-equation
approach of Erd\"os, Knowles, Yau and Yin for random band matrices
(2013). These arguments establish a diffusion limit as lambda -> 0 at
times and spatial scales that diverge as lambda shrinks. Very recently
Black, Drogin and Hern\'andez (2025) proved semiconstant estimates at
long finite times and positive resolvent heights at weak disorder. As
diffusion results, none of these directly yields a spectral statement
at fixed positive coupling, but they motivate the resolvent estimates
that appear in our construction (see specifically the self-consistent
diagonal equation, the concentration of row sums, and the elliptic
equation for the averaged squared resolvent).
A separate pre-existing line of work, notably Jak\v{s}i\'c, Last and
Warzel (2026), gives a conditional lattice criterion for a nontrivial
absolutely continuous component via a Schur-complement expression
involving two-site Poisson-transformed spectral measures. Their
criterion assumes a positive lower limit for the expected
energy-integrated object; it does not establish that the lower bound
holds for the iid uniform model at weak disorder. Our proof bypasses
this by working directly at the finite-volume resolvent level and by
producing the certificate at fixed weak coupling.
1.6 How the proof of pure absolute continuity differs from
localization arguments.
It is worth emphasizing that the certificates we construct in
Sections 2-8 are quite different in nature from the objects that appear
in localization proofs. In localization, one produces an interval
where resolvent entries decay exponentially in the distance from a
source. This is a statement about GREEN FUNCTIONS at a specific source.
Our certificate is a statement about the OTHER end of the spectrum: a
lower bound of the form Im Ghat_{ii} >= c on a fixed subset of sites,
uniform in m, for a specific energy and damping. Existence of such a
certificate is not a local Green-function estimate; it is a global
structural statement that the operator, with a small damping added
to a large fraction of sites, cannot concentrate its resolvent near
the energy. In particular, our certificates coexist with resolvent
decay in the following sense: a localized interval does have a large
diagonal imaginary part on some sites, but at strong disorder it
fails to have a large fraction of "touching" sites. The certificate
is really a statement about the operator as an object, not about
individual Green functions.
The passage from the certificate to pure a.c. spectrum (Section 9) uses
rank-one perturbation theory rather than Green-function analysis. It
is self-contained and does not require knowledge of the localized
class of the operator. Readers who prefer to understand the spectral
argument before the construction should read Section 9 first; Sections
2-8 construct the object, Section 9 extracts the spectral consequence.
1.7 Coexistence with edge localization.
The interval I_d from (1.4) lies above the lower free-band edge -2d.
Combining Theorem 1.1 with the fixed-disorder edge localization theorem
of Elgart and Klein (2019) at the same coupling yields a statement
about coexistence of a.c. and p.p. spectrum at the SAME fixed small
coupling.
COROLLARY 1.2 (Spectral coexistence). For every fixed d >= 3 and every
fixed 0 < lambda < lambda_d, there is a deterministic number
delta_{d, lambda} in (0, lambda/2] such that the open interval
J_{d, lambda} = ( -2d - lambda, -2d - lambda + delta_{d, lambda} ) (1.5)
sits below -2d, is disjoint from I_d, and, P-almost surely,
chi_{J_{d, lambda}}(H_{lambda, omega}) is a nonzero projection
whose restriction is pure point with exponentially decaying
eigenfunctions;
chi_{I_d}(H_{lambda, omega}) is a nonzero projection whose
restriction is purely absolutely continuous.
The probability-one event may depend on d and on lambda.
For the Aizenman-Warzel purity result on trees, compare the analogous
coexistence statement in their paper (Proposition 2.2 in the EMS
Journal version). On Z^d, our Corollary 1.2 is the corresponding
statement that combines the two existing endpoint results (edge
localization; bulk a.c.) at one fixed coupling.
Proof of Corollary 1.2. The random variables lambda V_x have the iid
law mu_lambda(dv) = (2 lambda)^{-1} 1_{[-lambda,lambda]}(v) dv. Its
concentration function is sup_a mu_lambda([a, a+t]) = min{1, t / (2
lambda)}, so it satisfies the bounded-support H\"older hypothesis of
Elgart and Klein. Their author-preprint Theorem 2.2 and Corollary
1.8(i)-(ii) apply and yield a deterministic A > 0 such that, almost
surely, the spectrum on
[-2d - lambda, -2d - lambda + A)
is pure point with exponentially decaying eigenfunctions. Take
delta_{d,lambda} = min{A/2, lambda/2}. By the same paper (author-
preprint equation (1.17) plus the standard spectrum-identification
result for the Anderson model), the almost-sure spectrum is the full
interval [-2d - lambda, 2d + lambda], so its projection onto the
nonempty open interval J_{d,lambda} cannot vanish. This interval lies
below -2d, while I_d lies above -2d, so they are disjoint. Intersect
the localization event and the spectrum event with the probability-
one event from Theorem 1.1, all at the same fixed coupling lambda.
QED.
1.8 Two remarks on what is and is not perturbative.
The proof of Theorem 1.1 is perturbative only in the sense that we
assume lambda is small. It is NOT perturbative in the sense of treating
the hopping as a perturbation of the diagonal, nor of treating lambda
lambda V_x as a small perturbation of the free Laplacian. Neither the
free Laplacian nor the diagonal is close to H_{lambda, omega} on any
fixed scale; the full operator is genuinely somewhere in between, and
the certificate construction reflects this: at small lambda, the
operator looks locally free over long scales (tiling argument), but
globally the potential still matters at the per-site level.
A second non-perturbative point: the certificate is a very strong
analytic object, but one does not need an explicit, computable
description of it. The theorem says "there exists a set A and
coefficients r such that the damped inverse has large diagonal
imaginary part". The set A is allowed to depend on the potential in a
totally arbitrary way. Section 9 explains why this existential
statement is enough for pure a.c. spectrum, and Section 8 shows it
holds with probability >= 1/2 by a randomized construction. No
algorithmic content is claimed.
1.9 Structure of the proof, at a glance.
We close this introduction with a one-paragraph summary of where each
of the two main assertions of the theorem is proved:
(A) THE CERTIFICATE (Sections 2-8). Section 2 sets up the
certificate statement precisely, together with the matrix
quantities tracked throughout. Sections 3, 4, 5, 6, 7, 8 build
the certificate stage by stage. Section 3 initializes. Sections
4 and 5 run a single "ordinary sweep". Section 6 handles the
"pending" obstacle via clearing trials. Section 7 assembles a
completed stage. Section 8 shows that all pending sites vanish
before the cutoff m/2 with probability tending to one, after
which the final whole-suffix sweep succeeds with high
probability. On success, the configuration at the endpoint is
exactly a certificate in the sense of Section 2.
(B) FROM CERTIFICATE TO THEOREM (Section 9). Section 9 receives the
certificate statement from Section 8 with probability p_0 >= 1/2
(in fact >= 1 - o(1)) and upgrades it to Theorem 1.1 via a
combination of Weyl/Wegner rank estimates, resolvent averaging,
positivity propagation, and singular-mass exclusion. The
upgrade is self-contained: the only external inputs it needs
are standard (spectral theorem, Herglotz representation,
Wegner estimate, ergodicity of the translation action).