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PURE-POINT SPECTRUM FOR THE TWO-DIMENSIONAL ANDERSON MODEL
AT EVERY POSITIVE DISORDER
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A self-contained proof, written so a curious reader who knows
single-variable calculus and has seen matrix multiplication can
follow the whole arc.
This file is the whole proof, start to finish. Nothing is left
to another file. Comments in square brackets [like this] are
informal asides that you can skip on a first read without losing
the logical thread.
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READER'S NOTATION CHEAT SHEET
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We avoid LaTeX and use plain ASCII so the file is greppable.
* multiplication where ambiguous (plain juxtaposition is fine)
^ superscript (either a power, or a label like H^N or T^2)
_ subscript
== is equal to != is not equal to
<= >= less than / greater than or equal
-> arrow (function, limit)
in element of notin not in
sum_ summation prod_ product
int_ integral dd differential
forall "for every" exists "there exists"
s.t. "such that" a.s. "almost surely"
Sets: R = real numbers, C = complex numbers,
Z = integers, N = natural numbers, T = unit circle.
Vectors: l^2(Z^2) = square-summable complex functions on Z^2.
Matrices: A^T (transpose, no conjugate), A^* (adjoint)
Re A = (A + A^*)/2, Im A = (A - A^*)/(2i)
||A|| = operator norm, ||A||_HS = Hilbert-Schmidt,
||A||_1 = trace norm, tr = trace, det = determinant.
Functions: 1 = indicator function (equal to 1 on a set, 0 outside)
supp f = closure of where f != 0
dist(x, y) = shortest-coordinate distance on the torus
|x|_1 = l^1 norm (sum of |coordinates|)
|x|_infty = max norm (max of |coordinates|)
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OVERVIEW: WHAT IS PROVED AND WHY IT IS HARD
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Fix h > 0. On the infinite square lattice Z^2 put independent
random real numbers v_x, each uniform on [-h, h]. Define the
bounded self-adjoint operator H_v on the Hilbert space l^2(Z^2) by
(H_v u)(x) = sum_{y in Z^2; |y-x|_1 = 1} u(y) + v_x * u(x). (0.1)
The first term is nearest-neighbor hopping; the second is the
random "disorder" on the diagonal.
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THEOREM (main result). For each fixed h > 0, almost surely:
(a) H_v has pure-point spectral type;
(b) H_v has a complete orthonormal basis of eigenvectors;
(c) the spectrum of H_v is exactly [-4-h, 4+h];
(d) the eigenvalues are dense in [-4-h, 4+h].
The probability-one event is allowed to depend on h.
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"Pure-point spectral type" means: the closed span of the
eigenvectors is all of l^2(Z^2). It does NOT mean every point of
the spectrum is an eigenvalue; that would be false. It means the
spectral measure of the operator is carried entirely by point
masses. Equivalently: no continuous part, no singular continuous
part.
What makes this hard. In one dimension, pure-point spectrum at
any positive disorder was proved in the 1980s (Kunz-Souillard,
Carmona-Klein-Martinelli) by elementary transfer-matrix methods.
In three and higher dimensions, localization is known at large
disorder (multi-scale analysis) and at the bottom of the
spectrum (Ding-Smart for Bernoulli 2D edge; Hurtado 2026 for
broader laws, bottom interval only). In two dimensions at ALL
positive disorder and over the FULL spectrum, the result was
open; the case of arbitrarily weak disorder is particularly
puzzling because kinetic energy and disorder are of comparable
strength and neither wins trivially.
The proof below is a constructive multiscale argument that
enables an energy-by-energy bootstrap, fed finally by the
rank-one spectral averaging identity of Simon-Wolff (1986).
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THE 5 INGREDIENTS, IN ONE PARAGRAPH EACH
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(I) SCATTERING MATRICES. At a prescribed finite set of "ports"
(input/output channels) we deform the operator by adding
imaginary diagonal absorption. The resulting S-matrix is
symmetric and unitary (a standard consequence of the
Ward identity). Dirichlet energy on port data is a
convenient bookkeeping device: two systems can be
compared in that energy, and the comparison is robust
against small perturbations of the reference values on
the diagonals.
(II) A RARE RESONANT CELL. In a large square tile of side L, we
design a special probability law for the potentials so
that, with positive probability that decays only like
exp(-poly(L)), the tile supports a resonant mode of energy
E with two useful properties: (i) its transmission
amplitude from a prescribed exterior port is not tiny
relative to the boundary impedance, and (ii) the closure
error (the price of replacing the invented law by the true
product law, then closing the port) is exponentially small
in L. This gives a building block for a multiscale
thinning scheme.
(III) A HIERARCHY OF PROBABILITY SPLITS. Over the torus of side
b_N = 3^N we set up nested squares of side b_l = 3^l for
l = n, ..., N. At each tile we allow the resonant law to
replace the uniform law with tiny probability q_l p_l =
exp(-b_l^6). At each tile independently the true operator
either keeps the true uniform potentials everywhere, or it
activates (accepts) a cell port. Thinning passes peel off
these ports from bottom to top. The energy comparison on
port data is essentially monotone under thinning, up to
an error controlled by a specific additive charge.
(IV) SCREENING AND VANISHING IN 2D. At any fixed starting level
n, once the thinning scheme reaches uniform stage m >> n,
the scattering of affine coordinate data has a limiting
expected energy density d_n (per unit area). These d_n are
nondecreasing in n, bounded, and converge to a limit d.
In two dimensions the geometric fact that a logarithmic
voltage profile has arbitrarily small Dirichlet capacity
C / log R forces d = 0. The proof of d = 0 is the crux
where dimension 2 enters decisively.
(V) BOOTSTRAP TO FRACTIONAL-MOMENT DECAY, THEN SPECTRAL READOUT.
Once d = 0, the affine energy of the uniform operator
vanishes. A final buffered-barrier contraction upgrades
this to a fixed-energy fractional-moment estimate on the
true finite-torus Green function:
E | (H_N - E)^{-1}_{x y} |^{s/2} <= A exp(-a |x-y|)
for small enough fixed s > 0. Summing the square, extracting
an l^2 eigenvector limit, then feeding into the Simon-Wolff
rank-one averaging over the on-site potential v_x closes
the argument: for the true operator, E is not in the
singular continuous spectrum, and by countable intersection
almost surely the whole spectrum is pure point.
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HOW THE FILE IS ORGANIZED
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Part 1 (sections 1-2) Setup, notation, basic scattering facts.
Part 2 (sections 3-4) The rare resonant cell and its transmission.
Part 3 (sections 5-6) The hierarchy: acceptance, thinning,
energy comparison, fixed-stage limits.
Part 4 (section 7) Vanishing of the limiting affine energy
in two dimensions (the crux).
Part 5 (section 8) Bootstrap: fractional-moment decay of the
true Green function.
Part 6 (section 9) Spectral readout.
Part 7 (section 10) References.
Every numbered lemma, proposition and theorem is used in the
final argument. Proofs are complete; long estimates are broken
into steps, but no "it follows from standard results" hand-waves
that would break the chain.