Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder

1 Introduction

Let h>0 and let (v_x)_{x\in\Z^2} be independent random variables, each uniform on [-h,h]. We consider the bounded self-adjoint operator

Thus v_x=h\omega_x in the usual notation with \omega_x uniform on [-1,1]. Pure-point spectral type means that the closed span of the eigenvectors is the entire Hilbert space; it does not mean that every point of the spectrum is an eigenvalue.

The planar spectral localization problem asks whether this pure-point conclusion holds throughout the spectrum for every nonzero width of the uniform distribution [14, Example 2.1 and Problem 2]. We prove that it does.

The conclusion includes arbitrarily weak nonzero disorder and the entire spectrum. It resolves the pure-point assertion of the two-dimensional Anderson localization conjecture for a uniform single-site distribution, as formulated in [14, Problem 2]. An arbitrary nondegenerate uniform interval reduces to the centered interval above by subtracting its midpoint from the operator. We do not assert one probability-one event for all disorder strengths, a bound uniform in energy, or a dynamical-localization theorem.

Context and scope

Anderson introduced random site energies as a mechanism for suppressing quantum transport, motivated by spin diffusion and impurity conduction [5]. His discussion includes a flat single-site distribution and nearest-neighbor hopping, although the original physical argument is not a modern proof of the spectral assertion considered here. The conductance-scaling theory of Abrahams, Anderson, Licciardello and Ramakrishnan predicts the absence of true metallic behavior in two dimensions [1]. That physical prediction is distinct from an almost-sure spectral-type theorem. Simon’s formulation isolates the latter question; his diffusion problem is listed separately [14].

Rigorous multidimensional localization theory developed several complementary mechanisms. Fröhlich and Spencer’s multiscale analysis gave Green-function decay and vanishing Kubo diffusion in large-disorder or appropriate extreme-energy regimes [12]. Their 1983 conclusions were explicitly distinguished from a complete pure-point decomposition. Fröhlich, Martinelli, Scoppola and Spencer subsequently established pure-point spectrum with exponentially decaying eigenfunctions for independent identically distributed diagonal potentials with a bounded density in sufficiently large-disorder and suitable tail-energy regimes [11]. Delyon, Lévy and Souillard also obtained multidimensional spectral localization in large-disorder or large-energy regimes [9]. The passage from fixed-energy Green-function information to a complete spectral decomposition is a substantive step, not an automatic consequence of decay outside an exceptional set of energies.

Aizenman and Molchanov developed the fractional-moment approach, obtaining localization at large disorder and at extreme energies under suitable hypotheses on the conditional single-site distributions, including absolute continuity and regularity [3]. This method supplies important ancestry for our use of fractional Green-function moments, but its established regimes do not by themselves yield the all-disorder, all-energy assertion above. Spectral-edge results also have a different scope. Ding and Smart proved localization near an edge for the two-dimensional Bernoulli model and observed that their proof extends to bounded nontrivial iid laws [10]; Hurtado extended edge-localization and unique-continuation methods to independent, not necessarily identically distributed uniformly bounded potentials with uniformly positive variance [13]. The latter localization theorems remain restricted to a bottom interval, even though their distributional assumptions are broader.

Weak-disorder transport on long finite scales poses another, compatible question. Black, Drogin and Hernández obtain quantitative time-averaged diffusion, resolvent and finite-volume eigenfunction results for Gaussian iid potentials on disorder-dependent scales [7]. These weak-coupling limits neither imply nor contradict pure-point spectral type at every fixed positive disorder for the uniform law. The present assertion concerns that fixed-disorder spectral type; it does not assert exponential eigenfunction decay or dynamical localization.

The final spectral step follows the rank-one mechanism of Simon and Wolff [15]. Its input is a suitable inverse-square spectral integral, equivalently a square-summable resolvent column, for almost every configuration and energy. Averaging over the independent site value transfers this information to the random spectral measure. We prove the required averaging and spectral readout explicitly; the main work is to obtain their fixed-energy input under the true uniform product law at arbitrary disorder.

The proof mechanism

Fix h>0 and an interior real energy E. In Section 3, we introduce absorbing diagonal terms in finite tori and their associated input and output channels, called ports. Squared scattering amplitudes define a Dirichlet energy on port data: if S_{ja} is the scattering amplitude from port a to port j and f is real data on the ports, the energy is

A Cauchy-distributed real closing value would remove a port with a favorable averaged energy comparison. The bounded physical distribution requires a truncated version and a relative estimate for its error.

The first main ingredient is a family of rare resonant cells (Sections 4, 5 and 6). Their conditional law is embedded in the original product measure by an acceptance likelihood; the true potentials remain independent uniforms before conditioning. The construction gives a weighted boundary impedance estimate uniform in arbitrary passive exterior data, including weakly coupled missing boundary directions. It also gives a transmission lower bound strong enough to make the closing error small relative to the transmission itself. These estimates, together with an upper envelope over adaptive interior configurations, are useful beyond a single prescribed environment.

The second ingredient is a comparison along a hierarchy of increasingly sparse ports (Section 7). A parent-cell alternative controls the apparent singularity at the end of each thinning interpolation. Local generic observation, followed by path estimates, supplies screening at every fixed starting stage (Section 8). For a fixed starting level n, let d_n be the infinite-volume limit of the expected sum of squared port-to-port distances, weighted by |S_{ja}|^2 and divided by twice the torus area. These quantities measure the scattering energy of affine coordinate data. The torus side tends to infinity with n fixed to define d_n; only afterwards does the thinning comparison give a finite limit d as n tends to infinity.

The dimension enters decisively in forcing d=0 (Section 9). An adaptive modification of affine data first makes a local affine first-variation trace small in expectation. Separately, a logarithmic voltage profile separates retained interior ports from an annulus of very weak probes at arbitrarily small two-dimensional capacity cost. If d>0, an analytic test of the scattering matrix has a nonzero value at the center of the disk. Support-preserving shifts of the true potentials and the thinning comparison make its boundary moments arbitrarily small, contradicting submean. Dimension-free weak integral bounds prevent the number of ports from spoiling this argument.

Finally, vanishing affine energy is upgraded to a fixed-real-energy fractional-moment estimate for the true finite-torus Green function (Section 10). Buffered probe barriers, conditional estimates under transmission weights, and resampling of disjoint regions close a contraction uniform over the allowed masks. The infinite-volume conclusion then follows from the spectral theorem, Fubini, and rank-one averaging (Section 11). The property to which Fubini is applied belongs to the true operator, so no common auxiliary construction measurable in energy is needed.

The proof keeps these roles separate: the rare laws prove local comparisons, fixed-stage screening controls limits, two-dimensional capacity makes the limit vanish, and spectral averaging converts fixed-energy information to spectral type.

2 Finite volumes and conventions

Throughout the local and multiscale arguments, fix

All constants and auxiliary choices may depend on this pair. Their dependence on the varying geometric scales, exterior data and masks is specified in each assertion. Uniformity in E is neither assumed nor needed for Theorem 1.1.

For integer l\ge0, put b_l=3^l. The global finite volume is the square torus

with the periodic nearest-neighbor adjacency and independent uniform potentials on its sites. Its area is b_N^2. Distances are shortest coordinate distances on this torus, using the maximum norm unless a different norm is displayed. We write

All local constructions are used with the torus sufficiently large to contain the required neighborhoods without ambiguity. Dirichlet restrictions mean deletion of the hopping bonds leaving the indicated physical site set.

The physical disorder is always the product law of the true variables v_x. Auxiliary selectors, acceptance uniforms, offsets and thinning marks will be introduced explicitly. Conditioning on a rare event can change the law within one cell; it never changes this unconditional physical law. An expectation refers to the sampling and conditioning specified in its statement.

Site and port spaces are complexified finite-dimensional Euclidean spaces. A superscript \top denotes transpose, not adjoint; a superscript * denotes adjoint. Bases used for coordinate projections and their subspaces are real orthonormal bases. We use

Unless otherwise indicated, a norm is the Euclidean or induced operator norm. Hilbert–Schmidt and trace norms carry subscripts \mathrm{HS} and 1. Coordinate projections may be identified with their ranges. Symbols used for local blocks are reset when the local matrix is defined.

A real reference potential may replace a true potential only at a site carrying an open cell port. Closing that port restores the true value. The matrix H^\# denotes the real symmetric operator with the current reference values. Artificial probes, introduced in the next section, do not alter real reference values. Several ports may share a physical site; their columns are kept distinct.

In bounds for a cell of size L, the notation \operatorname{poly}(L) means a quantity bounded by C(1+L)^C for fixed constants. The constants may depend on the previously fixed cell-design parameters, including the corner-box width exponent, but not on the exterior matrix, exterior volume, exterior source, or allowed probe mask. The ordering of design parameters is part of each construction. Expressions \exp(L^{O(1)}) have the corresponding interpretation. A statement requiring sufficiently large L is applied only after its fixed design parameters have been chosen.

We will repeatedly distinguish the following orders of choice. The disorder and energy are fixed first. Local design constants are fixed before the cell scale is enlarged. For global limits a starting level n is fixed before observation and approximation constants are chosen; a later level m can then be increased to exceed all those thresholds, and the torus level N is increased last. No rate uniform in n is implicit in this convention.

3 Scattering matrices and analytic estimates

All matrices in this section are finite dimensional. A port at a physical site x, of strength \sigma>0, is represented by the column \sqrt{\sigma}\,e_x. In the matrix Y of these columns, rows index physical sites and columns index ports; several distinct ports may be attached to the same site. For a real symmetric matrix H^\# and a real energy E, set

The entry S_{ja} is the scattering amplitude from input port a to output port j. The physical matrix H^\# may differ from the true Hamiltonian by real diagonal changes at sites carrying ports. We call these changes reference potentials. They may depend on the true potentials.

Absorption bounds the response between ports even when the full inverse G is large. Closing ports removes their absorptions and changes real diagonal values; we express this operation as a rational family of scattering matrices and compare its energies after averaging the closing values.

3.1 Invertibility, absorption, and Schur complements

The inverses in Equation (3.1) exist almost surely under any absolutely continuous joint law of the true site potentials. Indeed, if \mathscr L u=0, taking the imaginary part of u^*\mathscr L u gives Y^\top u=0. Thus u vanishes at every port site, so restoring the true diagonal entries leaves its equation unchanged: (H-E)u=0. For a fixed finite site set, \det(H-E) is a polynomial in its true diagonal variables whose coefficient of their full product is one. It is not identically zero. A nonzero polynomial has a Lebesgue-null zero set, as follows by induction on the number of variables and one-variable slicing. This proves the assertion even when the port set and reference potentials depend on those variables. The same argument applies to deterministic restrictions to physical sites. We always discard the resulting null exceptional sets when forming such inverses.

A matrix D is passive if \operatorname{Im}D\le0, with the Hermitian imaginary part fixed in Section 2. The absorption identities below apply to any real rectangular Y, including real coordinate or subspace inclusions.

We will also use the elementary kernel consequence of passivity: if Du=0 and D is passive, then \operatorname{Im}D\,u=\operatorname{Re}D\,u=0. Indeed, the nonnegative matrix -\operatorname{Im}D has zero quadratic form on u, so it annihilates u; the assertion about the real part follows from Du=0.

3.2 Closing ports and comparing energies

Symmetry and unitarity make \abs{S_{ja}}^2 symmetric nonnegative weights with unit row sums. For real scalar or real Euclidean-vector data f on the ports, their Dirichlet energy and its single-port contribution are

For scalar data, writing F=\diag(f), unitarity gives

For vector data, one sums this identity over real components.

The use of the conditional mean \mu, rather than \E[\abs\gamma\mid\mathcal F], is important. In later applications the open system and the data must remain measurable without exposing the hidden true value at the port to be closed.

A standard Cauchy law for w makes this conditional mean vanish. To restore a potential within a bounded interval, we truncate that law and estimate the resulting error.

3.3 Dimension-independent analytic bounds

The common-phase closure formula contains the resolvent (\one-zB)^{-1}. We need weak-L^1 bounds for its sandwiched and trace expressions, with no dimensional factor beyond the displayed norms. For a nonnegative random variable X on the circle, write

The Hilbert–Schmidt estimate below has the same dimension-independent scale as the dissipative-resolvent bound in [2, Lemma 3.1 and Appendix C]. We prove the disk estimates directly using a Hilbert-space-valued Calderón–Zygmund decomposition [6, Theorem 1].

We record a useful extension of the same proof. If a rational analytic d\times d matrix function F has nonnegative Hermitian real part in the disk, its representing positive measures have total mass \operatorname{Re}F(0). Their Hilbert–Schmidt variation is at most \tr(\operatorname{Re}F(0))\le d\norm{F(0)}. Restoring the constant i\operatorname{Im}F(0), the preceding argument gives

We need only the resulting polynomial dependence on dimension and the norm at the center.

3.4 Determinant phases and artificial absorptions

To average inverse traces over a real diagonal shift, we integrate the phase of the determinant. The estimate must also allow artificial absorptions that vary with that shift.

Two variants will be used without changing this argument. If the real derivative is a Hermitian matrix K(s)\ge cP_J for a fixed c>0, its contribution is \tr(K\operatorname{Im}G)\ge c\tr_J\operatorname{Im}G. Thus Equation (3.17) holds with constants also depending on c. Additional real diagonal derivatives can instead be retained explicitly in Equation (3.15) and estimated using available absorption bounds. This is how the calibrated root derivative will be handled.

3.5 The artificial probe profile

Separate from the cell ports, we allow at most one artificial probe at each site. A prescribed mask chooses among absence, strength \tau, and strength \tau\vartheta(v_i), where v_i is always the true site potential, including when the real reference value differs from it. Masks do not depend on the potentials. They may be fixed after conditioning on spatial offsets or other independent prescriptions. Zero strength means absence.

Here \vartheta:[-h,h]\to[0,1] is fixed, of class C^{1,1}, polynomial on finitely many pieces, positive on (-h,h), and identically one on a plateau containing [-h/2,h/2] and the compact interior potential settings of the cell construction. Such a profile can be chosen after those compact settings are fixed and before the later scale and width parameters are selected. To construct it, choose a<h large enough that the required compact set lies in [-a,a], and use 3t^2-2t^3 on each endpoint transition, with t=(v+h)/(h-a) on [-h,-a] and t=(h-v)/(h-a) on [a,h]; put \vartheta=1 on [-a,a]. The endpoint values are zero, the derivatives match at the joins, and

Constants may depend on the fixed profile. In particular, \vartheta^{-q} is integrable on [-h,h] for every q<1/2.

For estimates at a single cell of size L, artificial strengths will obey 0\le\tau\le e^{-c_*L} with a sufficiently large fixed c_*>0. The screening estimates will also allow 0\le\tau\le1. All spatial masks remain arbitrary subject to the independence convention above.

4 The resonant cell law

Throughout this section, h>0 and E\in(-4-h,4+h) are fixed. All choices may depend on these two parameters. A polynomial bound in L means C(1+L)^C, with constants independent of the exterior system, its dimension, and the artificial-probe mask. The construction below specifies an auxiliary law on one cell. Its incorporation into the true product law is deferred to Proposition 7.1.

4.1 Clipping the corners

Fix an integer margin b\ge3. In a tile of odd side b_l=3^l, use coordinates centered at its central site and put

Only sufficiently large l are used. Let

Thus p is the adjacency column from the root 0 to J. Let M:\R^B\longrightarrow\R^J be the adjacency matrix from B to J; when used on V, it has an additional zero row at the root. Define

We identify a projection with its range when specifying a matrix domain. The cell and its corner coupling are shown in Figure 1.

4.2 Calibrating the root with a decaying profile

We choose the root reference value so that the Dirichlet cell has a mode at energy E. For arbitrary real values v|_J, write

Whenever A is invertible, define

The Dirichlet matrix on V, with root potential c(v), then satisfies

Thus invertibility of the punctured matrix A permits resonance of the full Dirichlet cell. We must choose the interior law so that A^{-1} is controlled, c(v) stays inside the physical potential interval, and this root-normalized mode is small at the boundary. The transmission estimates will later control how it couples to an arbitrary passive exterior.

We first construct a comparison profile \psi, normalized by \psi_0=1, and potentials v^0 satisfying its local equation. The profile is defined also on B, where it is small but nonzero, so it need not be a Dirichlet eigenvector. A deformation of this profile will give a polynomial inverse bound for the punctured matrix at v^0. These two properties will control the exact calibrated mode after the interior values are randomly perturbed.

Choose a rational multiple \theta of \pi and a real phase \phi such that

The first choice is possible because the energy is in (-4-h,4+h) and rational multiples of \pi are dense. The sequence of cosine values is periodic, so the second condition excludes only finitely many phases modulo 2\pi. Put

Define the real sequence f on all integers by

Recursion in either direction gives |f_k|\le|k|. For small fixed \nu,c_u>0, and |u|\le c_u/L, set

Every neighbor in the last sum belongs to V\cup B and \psi_0=1. Uniformly over the allowed parameters,

Indeed |u f_{x_1+x_2}| is bounded by a fixed multiple of c_u and the periodic carrier is bounded away from zero. For neighbors x,y the additional ratio factors differ from 1 by O(\nu+c_u): their radial exponents differ by at most \nu, and |u(f_{y_1+y_2}-f_{x_1+x_2})|\le Cc_u. At \nu=u=0, the cosine addition formula gives v_x^0=E-4\cos\theta. We therefore fix \nu,c_u small enough, once and for all, that

From now on u=u_L is fixed and suppressed in the notation. Set

In particular, |\psi_x| is comparable to \epsilon on B and on Z^+, with constants independent of L. We have chosen the port strength \sigma as the square of this designed boundary-profile scale. The exact calibrated mode will satisfy the polynomially lossy comparison in Equation (4.23).

4.3 The branch sampler

We first describe the boundary conditionings used by the sampler. For each corner triple use the fixed basis in Equation (4.4) and put

These matrices satisfy

Thus a common shift changes the real diagonal in the missing direction n, while a shift along \mathsf D_j changes its coupling to r_j without changing that diagonal. The three diagonal directions I,\mathsf D_1,\mathsf D_2 are linearly independent: multiplying a vanishing linear combination by n gives a vanishing linear combination of the orthonormal vectors n,r_1,r_2. Thus centered coordinates (s,t_1,t_2) specify physical potentials by

The map has a fixed nonzero Jacobian. In fact its three columns, as vectors of diagonal entries, are orthogonal and have length \sqrt3. Use this same coordinate convention at all levels, including at exactly coincident triples.

A prescribed corner box is the image under Equation (4.17) of a product of three centered intervals. Fix an exponent \kappa. The boxes allowed below have all three side lengths at least c(1+L)^{-\kappa} and at most C, lie at distance at least c(1+L)^{-\kappa} from the boundary of (-h,h)^3, and keep each physical coordinate in a single polynomial piece of the fixed artificial profile \vartheta. Constants may depend on \kappa; in the hierarchy this exponent is fixed by K_0. Uniform measure in these box coordinates is also uniform measure in the physical parallelepiped. Because \vartheta is positive in the interior and comparable to squared distance from the support endpoints, every nonzero artificial profile a_i\in\{1,\vartheta\} on a permitted box obeys

Increasing the exponent absorbs any fixed comparison constants.

Choose D_1>C_2+2. A shell-depth parameter D_2 will be fixed after the polynomial thresholds in the boundary and zero-jitter transmission estimates have been fixed. Finally choose D_3 sufficiently large depending on D_1,D_2 and these thresholds. On J sample

where the \zeta_x are independent centered uniforms of half-width

They are independent of r and all boundary sampling.

For these interior draws, use the matrix A and the calibrated value c(v) from Equation (4.5). At the root, put an open cell port a of strength \sigma, and use c(v) as its real reference potential. To define the hidden true potential, fix a smooth cutoff \chi:\R\to[0,1] with \chi=1 on [-d,d] and \supp\chi\subset[-2d,2d], and sample

This is exactly the cutoff Cauchy law used in Lemma 3.4. The open reference c(v) does not depend on the hidden draw \delta.

The boundary law is as follows. On each corner triple sample uniformly in its prescribed box, independently between the four triples. On every single side site x sample

using independent \xi_x and one common s_0. These draws are independent of the triples and the interior draws. Sites of the tile outside V\cup B retain their independent uniform [-h,h] laws, independently of all draws just described. Apart from the common shifts r,s_0 and the dependence of the root center c(v) on v|_J, all sampling is independent.

We record explicitly the order of choices used later. The carrier, d,\nu,c_u,C_1,C_2, the compact cutoff and plateau, and the corner-box exponent are fixed first. Next fix D_1. The polynomial threshold in Proposition 5.1 and the regular inverse thresholds in the zero-jitter part of Proposition 6.1 can then be fixed without D_2,D_3. At depth L-|x|_\infty=\lfloor D_2\log L\rfloor, Equation (4.10) gives

Choose D_2 large enough that this dominates the already fixed polynomial errors and reflected-field bounds in that argument. Finally choose D_3 to satisfy both the residual requirements above and the inverse and path-entry stability requirements of Proposition 6.1. Enlarging D_3 changes neither the exponents in Equation (4.23) nor the earlier zero-jitter bounds. All parameters are now fixed; only after that do we increase the lower cell-size threshold. This order prevents the depth choice from depending on a polynomial exponent that itself grows with the depth.

5 A weighted projected impedance estimate

We retain the boundary geometry and the conditional boundary law of Lemma 4.1 and Proposition 4.3. In this section the interior of the cell does not enter the calculation. Its only relevant feature is that it couples to the boundary space through the projection P. We write

where P_0 projects onto the single side sites and Q_a projects onto the two-dimensional plane of the ath corner triple. The four complementary directions in the triples are denoted by n_a. In the prescribed ordering of each triple, n_a=(-1,-1,1)/\sqrt3. Thus \rank Q=8, whereas the dimension of the whole boundary space is O(L).

All norms below use the fixed real orthonormal bases of these spaces. Constants may depend on the fixed disorder, profile, and corner-box parameters. They will not depend on the exterior matrices, their norms, the source vector, or the artificial-probe mask. Until an exponent H is selected below, every expression denoted by \operatorname{poly}(L) means a bound C(1+L)^C whose constants are independent of H. A final polynomial bound is allowed to depend on that selected, fixed exponent.

The weight in Equation (5.1) matters: a small unweighted exceptional set could contain nearly all the response of a weakly coupled exterior mode. We first control the inverse on the corner planes Q while leaving unit absorption on the single side coordinates P_0. The corner estimates must retain a fixed fraction of the response weight. We then average the common real side shift to bound the whole projected inverse P\mathcal T^{-1}P with the auxiliary P_0 absorption removed.

5.1 Elementary reductions and polynomial estimates

We first record the comparison that permits bounded changes of the absorbed coordinates.

The inverses used in this section exist almost everywhere in the stated boundary law. To see this without any nonsingularity assumption on T, recall that a kernel vector of a passive matrix is annihilated by both its Hermitian real and imaginary parts. Absorption on P restricts such a vector to the four missing directions. On that space the Hermitian real part has the form of a fixed real symmetric matrix plus the independent common corner shifts on its diagonal. Its determinant is a nonzero polynomial, with the product of those shifts as its leading monomial. The same reasoning applies after one missing direction is deleted or after bounded changes in plane blocks. With absorption only on P_0, a possible kernel lies on the full corner triples. The four common shifts act there as scalar shifts on their respective three-dimensional blocks; the determinant again has a nonzero leading monomial. Finally, \mathcal T itself has physical real diagonal variables with a joint density, so its Hermitian real part is nonsingular almost everywhere. These observations also justify almost-everywhere slicing below.

The response weights are integrable. For example, writing G=(\mathcal T-iP)^{-1}, the Ward identity implies

The joint law of the physical boundary diagonals has a bounded density on a compact set; the bound here may depend on L. Slice that set along a common real shift of all the physical diagonals. The phase estimate Lemma 3.6, including its artificial-diagonal derivative term, bounds the integral of \tr\operatorname{Im}G along each slice. Integration over the transverse coordinates proves finiteness. One may add -i\zeta\one first and pass to the limit by Fatou. In particular, after some boundary variables are frozen, all integrals below are finite for almost every such choice.

We shall use two elementary facts about polynomials. Their degree bounds are fixed throughout this section.

The following one-dimensional observation will prevent an extremely narrow response peak from defeating a short-exception estimate.

5.2 Eight corner tests

Remove the common shift from the single side variables and omit their artificial absorptions. Denote the resulting boundary matrix by \mathcal T_0, and set

The independent single side uniforms are still present. By Lemma 5.2, w_0 and \rho are pointwise comparable by fixed factors. We now hold those independent side uniforms fixed.

Choose an exponent H, to be fixed after the polynomial losses below, and put

For each vector b' in the eight prescribed real orthonormal corner-plane bases, define

Outside their union,

Indeed, if \norm{QG_\eta Q}<(2\eta)^{-1}, inverse difference and a Neumann series give the bound in Equation (5.7). Hence failure of that bound forces \norm{QG_\eta Q}\ge(2\eta)^{-1}. Ward on Q then gives

If all eight tests fail to occur, the left side is less than 8/40<1/4, a contradiction.

Fix one test and condition on all boundary variables outside its corner triple. Use centered box coordinates (s,t_1,t_2) there, where s is the common shift and t_j is the shift whose diagonal matrix maps the missing direction n to the plane basis vector r_j. In particular, the t_j shift has zero n,n entry. Write t=(t_1,t_2). Every side length of the box is at least \ell\ge(1+L)^{-C} and at most a fixed constant. All probabilities and averages on this box are normalized uniform ones.

There is first an unweighted bound

For fixed t, apply Lemma 3.6 while varying s. Its real derivative is the identity on the tested triple. The artificial derivatives satisfy the profile condition and are controlled by their actual absorption, so the integral of the triple partial trace of \operatorname{Im}G_\eta is polynomially bounded. Since b' is a unit vector in that triple, its quadratic form is bounded by this trace. Divide by the interval length, then average over t, to obtain Equation (5.8). Neither the phase range nor this polynomial loss depends on H or on the exterior norm.

We must strengthen Equation (5.8) to a bound for w_0-mass.

5.3 A fixed remainder at one corner

The exceptional test still uses the actual inverse G_\eta. To measure its weight, we replace w_0 by a comparable response whose remainder after deleting the tested missing direction n is fixed. In \mathcal T_0-iP, freeze the real variations of the tested triple within its plane-plane block at the box center. Replace the artificial absorption in that plane-plane block by \tau times the plane identity. Retain every fixed term from T and retain all actual mixed terms and the missing-direction diagonal. Call this modified matrix \widetilde D.

The modification is bounded and supported on the tested plane. It also preserves absorption at least P plus the original artificial absorption of the tested triple. In fact the change of artificial absorption is

because 0\le a_i\le1. Consequently

The constant is fixed by Lemma 5.2.

Delete the tested missing direction n, and denote the inverse of the remaining block of \widetilde D by G_{\mathrm a}. This inverse is independent of (s,t). The off-diagonal column from n and its Schur denominator have the form

The fixed vector p_{10} may have components outside the tested triple; no bound on it is assumed. All variable parts of p_1 lie in the tested plane. The constants in Equation (5.11) include the fixed diagonal at n; there is no real t-dependent contribution to that diagonal.

Write k_{\widehat n} for k with its n coordinate deleted, and set

The block inverse formula yields

The norm of d_1 measures the response on the absorbed coordinates induced by unit amplitude in the missing direction. The scalar Schur denominator s-e controls how sharply W can concentrate. It need not agree with the corresponding denominator in the actual test G_\eta. Each profile lies on one fixed polynomial piece throughout the corner box. Hence p_1,d_1,\beta,e,N_1 are polynomials in (s,t) of bounded degree, independently of all exterior data. Their coefficients need not be bounded.

5.4 Weighted exceptions when the coupling is appreciable

Put

All maxima are taken over the closed box, which does not affect its conditional law. The bound \operatorname{Im}e\ge\norm{d_1}^2 ties the width of the response to this coupling. When D\ge\eta^{\alpha_0}, we can isolate the few transverse parameters where the coupling becomes small and use the unweighted test bound elsewhere. The case where D is small throughout the box will instead require comparing the length of each exceptional slice with its response width.

We first treat D\ge\eta^{\alpha_0}.

Suppose initially that \abs{s-e}\ge1 throughout the box. The maximum and minimum of this denominator are then comparable by a fixed factor. Indeed, Equations (5.15) and (5.16) show that d_1 varies by at most a fixed constant. If D is large, then \operatorname{Im}e\ge cD^2 throughout, whereas the variation of s-e is at most C(1+D). If D is bounded, the same variation bound and the assumed lower bound one suffice. By Equations (5.3) and (5.8),

It remains to consider the case where \abs{s-e}<1 somewhere. At that point \norm{d_1}<1; the derivative bounds then imply that D is bounded throughout the box. Since s is bounded, the same argument shows that e and s-e are bounded as well. Set

The bounded denominator and Equation (5.3) give

Furthermore,

Here is the required quantitative argument. By the product estimate,

Differentiate N_1=a(s-e)+d_1\beta in s and use Equations (5.4), (5.14) and (5.15). As D is bounded, this gives

Substitute the preceding estimate for B_1. Since D\ge\eta^{\alpha_0}, the coefficient C\tau(1+D^{-1}) is less than one half for sufficiently large L after H is fixed. Absorption proves Equation (5.21). Its polynomial exponent is independent of H.

Choose \Delta=\eta^{1/8} and let

Then

To prove this, Equation (5.10) and Ward show that d_1(s,t) is within C\tau of an affine function of t, uniformly in s. Rescale the two t intervals to unit intervals. If \mathcal A is nonempty, the affine function has a value of norm at most \Delta+C\tau, and its maximum norm is at least D-C\tau. Project it by a real unit linear functional attaining that latter norm. Because \Delta/D\to0 and \tau\ll\Delta, one of the two scalar slopes has magnitude at least cD. Slicing in that coordinate bounds the fraction where the scalar magnitude is at most \Delta+C\tau by C\Delta/D. This includes \mathcal A.

For each t\in\mathcal A, Equations (5.13) and (5.18) gives

On the complement of \mathcal A, absorption gives \abs{s-e}\ge\norm{d_1}^2\ge\Delta^2, so W\le\Delta^{-4}M_1^2. Using Equation (5.8) there and Equation (5.22) on \mathcal A, we obtain

We used the boundedness of D and Equation (5.20) in the last step. If M_1=0, all the weighted integrals vanish and the inequality holds directly. In particular, this argument never divides by a possibly zero expected weight. The powers in Equation (5.23) satisfy

5.5 Weighted exceptions when the coupling is small

We now assume D<\eta^{\alpha_0} and write b'=r_j for the tested basis vector. Vary t_j between two points separated by at least half its interval length. From Equations (5.10) and (5.12),

This can be made as small as a prescribed fixed constant by choosing H sufficiently large, then L sufficiently large. The inference uses variation over the whole t_j interval; it does not assume that the exterior has a lower bound on its coupling.

Next delete n in the actual matrix G_\eta^{-1}, without freezing its plane block. Let G_{\eta,\mathrm{rem}} denote the remaining inverse, put

and write s-e_\eta for its Schur denominator at n. Its off-diagonal column is exactly the same p_1 as before. The remaining matrices for G_{\mathrm a} and G_{\eta,\mathrm{rem}} differ by a bounded matrix supported on Q. Moreover, \norm{QG_{\eta,\mathrm{rem}}Q}\le\eta^{-1} by Ward. Inverse difference and symmetry therefore give

For clarity, the inverse identity used in both estimates is G_{\eta,\mathrm{rem}} =G_{\mathrm a}+G_{\mathrm a}\Delta G_{\eta,\mathrm{rem}}, where \Delta=Q\Delta Q has bounded norm. The first left factor is thus controlled by PG_{\mathrm a}b', regardless of the exterior norm.

Take the bound in Equation (5.24) small enough that the first term in the Schur inverse formula cannot by itself cause \mathcal E_{b'}. On that exception,

On the other hand, absorption on the actual Schur extension yields

Together with Equation (5.27), this implies \norm{Qd_{1,\eta}}\le C\abs{b'^\top d_{1,\eta}}. Insert Equation (5.26) and absorb its small last coefficient. We conclude that, on the exception,

The exceptional set is short on every almost-everywhere s slice. To establish this rigorously, let D_t=\max_s\norm{d_1(s,t)}. Differentiating the actual Schur formula shows that, on the exception,

The first term comes from the real derivative of the tested plane block. Its artificial derivative is absorbed in the same bound; the mixed and missing-direction derivatives give the terms containing \tau. Thus \operatorname{Re}(s-e_\eta) has derivative at least one half on the exception.

There are only boundedly many components of that exceptional slice away from poles. Indeed, with all other variables fixed, the matrix varies polynomially only on the three coordinates of the tested triple. In a basis containing those coordinates, expansion of the determinant uses variable factors from at most three rows. Its degree, and the degrees of the relevant cofactors, are bounded by a constant depending only on the profile degree, not on the dimension or norm of the exterior. The same assertion holds after deleting n. Clearing the squared absolute value of a denominator converts Equation (5.6) into a real polynomial inequality of bounded degree. Such a set has a bounded number of intervals and isolated points, except for poles, which also have bounded number on an almost-everywhere slice.

On each interval component, Equation (5.28) confines \operatorname{Re}(s-e_\eta) to an interval of length C\eta D_t^2, while its derivative is at least one half. Summing over the bounded number of components gives

If D_t=0, Equation (5.28) and the Schur formula rule out the exception off its null set of poles.

For D_t>0, the denominator of the modified weight has a matching minimum scale:

Here is the point requiring the artificial-absorption floor. By Equation (5.15), the oscillation of d_1 on the slice is O(\tau). If its norm is everywhere at least D_t/2, absorption proves the first bound immediately. Otherwise D_t\le C\tau. In this latter case an artificial term must be present, since without one d_1 is independent of s. Now Equation (5.17) gives \operatorname{Im}e\ge\tau/\operatorname{poly}(L)\ge cD_t^2 for sufficiently large L. The derivative bound follows from Equation (5.14) and D<\eta^{\alpha_0}.

The exceptional length O(\eta D_t^2) must be compared with the resolution \min(\ell,D_t^2) supplied by the modified denominator. The real parts of the two Schur denominators need not vanish at the same point. Apply Lemma 5.4 to N_1 and s-e, using Equation (5.30). It gives

Combining with Equation (5.29),

The last inequality uses D_t\le D<1 and the inverse-polynomial lower bound on \ell. Averaging over t proves

in the small-coupling case. This estimate remains uniform as the coupling vanishes; no positive lower bound for D was used here.

5.6 Restoring the side shift

We now choose the parameters. All polynomial losses in Equations (5.19), (5.23) and (5.31), and the box width exponent, have been fixed independently of H. Choose H large enough that their products with

tend to zero, and that Equation (5.24) is uniformly small in its case. With this H fixed, take L sufficiently large that \tau\ll\Delta, \tau/\eta^{\alpha_0}\ll1, and all preceding absorptions are valid. The assumption \tau\le e^{-c_*L} ensures every one of these requirements for any fixed c_*>0.

For each of the eight tests, condition on the other corner variables, apply the appropriate estimate, and compare W with the same weight w_0 by Equation (5.9). Integration and summation give

for sufficiently large L. This is still conditional on the independent side uniforms. The threshold can be chosen uniformly in those frozen values and in all exterior data. Zero conditional weights cause no difficulty, since every estimate was an inequality between integrals.

Fix data in \mathcal G, before restoring the common side shift. On the space P consider the rational matrix function

It is analytic there and has nonnegative imaginary part. To check analyticity, a kernel at one point of the lower half-plane would be annihilated by P_0 and by \mathcal T_0, hence would also be a kernel at z=-i, where the inverse exists. On \mathcal G, Equation (5.7) bounds the Q,Q block of F(-i) by \eta^{-1}. Ward gives a bound one on the P_0,P_0 block and

Transpose symmetry gives the same bound on the other mixed block. Thus

For completeness, map the unit disk to the lower half-plane by z=-i(1+w)/(1-w). Then -iF(z(w)) has positive semidefinite Hermitian real part, and its value at zero obeys Equation (5.33). The positive-real-part weak estimate Equation (3.14) gives a boundary weak-L^1 bound at most C(\rank P)\eta^{-1}. One can obtain this bound directly from the positive matrix measure in its Herglotz representation: its total trace mass is at most (\rank P)\norm{F(-i)}, while the constant imaginary part has the same polynomial bound. The boundary change of variable sends Haar measure to \dd x/(\pi(1+x^2)).

The Cauchy density has a positive lower bound on the fixed interval [-h/4,h/4]. Therefore there is a polynomial R_L=C(1+L)^C, now allowed to depend on H, such that for at least half the common side shifts x in their prescribed uniform law,

This conclusion uses only the norm at -i, not the norm of T.

Restore the artificial absorptions at the single side sites. Their change has norm at most \tau and is supported on P_0. For a shift satisfying Equation (5.34), the inverse difference identity on P and a Neumann series give

as soon as \tau R_L\le1/2. This holds for sufficiently large L. Throughout the common-shift average, with or without these side absorptions, Lemma 5.2 compares the weight to w_0 by fixed factors. Hence at least a fixed fraction of the \rho-weight survives the good shifts for every corner setting in \mathcal G. Integrate this statement, use Equation (5.32), and finally average over the initially frozen independent side uniforms. Since \E\rho and \E w_0 are comparable, this proves Equation (5.2). The pointwise comparison assertion in Proposition 5.1 is already contained in Lemma 5.2. \square

5.7 The Schur form used by the interior

The following formulation makes the interface with the transmission calculation explicit.

6 Transmission through a cell

Throughout this section, the geometry and branch parameters are those of Lemmas 4.1 and 4.2 and Proposition 4.3. Constants may depend on the fixed energy, disorder, profile, corner-box exponents, and parameters already chosen, but not on exterior matrices or source vectors. A polynomial bound means C(1+L)^C with this convention; different occurrences may have different exponents. All assertions concern sufficiently large L after the parameters have been fixed.

6.1 Exterior data and the two branch estimates

Suppose that the root is the only cell port in V\cup B, and that the source port lies outside this set. Artificial probes with a prescribed mask may be present. Eliminate the exterior physical sites. Its effect on B is a fixed complex-symmetric passive matrix T, and the source gives a fixed vector k, including its input multiplier. The remaining boundary matrix is

The exterior is held fixed throughout the cell integration, as are the four corner boxes. In particular, no bound on \norm{T} or \norm{k} is assumed.

Let P also denote its range, equipped with a real orthonormal basis. Eliminating the four missing boundary directions gives

These formulas are used at their almost-everywhere inverses. The missing-direction block is generically invertible by Corollary 5.6. Schur complementation preserves symmetry and passivity, and gives the identity

Thus \rho is exactly the weight in Proposition 5.1, and

Hereafter M is restricted to P, and is extended by a zero row at the root when it is regarded as a map into V.

Let G be the inverse on V\oplus P with the root open, reference potential c(v), and root strength \sigma=\epsilon^2. Its internal artificial absorptions are constant during the branch integration: the true values lie on the plateau specified in Proposition 4.3. Set

Since k includes the source input multiplier, the root-to-source scattering amplitude satisfies \abs{S_{\mathrm{source},a}}^2=4\abs t^2. The sign convention for the source does not affect this identity. The inverse exists almost everywhere even though c(v) depends on other potentials. Indeed, absorption forces a kernel vector to vanish at the root. Restoring its true real diagonal therefore preserves that vector. After eliminating the fixed boundary data, the real part is a fixed matrix plus the true real diagonals on V; their absolutely continuous joint law makes its determinant nonzero almost everywhere. This also justifies the almost-everywhere slices used below.

6.2 The cutoff error and the small-response region

We first derive Equation (6.5) assuming Equation (6.4). Fix the boundary data and all jitters, and vary the common real shift r on Z^+. Write P_+ for the coordinate projection onto Z^+, extended by zero on the root and on P. Since c(v)=E+p^\top A^{-1}p and A'=P_+ on J, differentiation gives

The profile estimate in Equation (4.23), together with \abs{\psi_i}\le C\epsilon on Z^+, implies \abs{c'}/\sigma\le C(1+L)^C. Root absorption and Lemma 3.1 imply

Let I_r=[-L^{-D_1},L^{-D_1}]. The derivative of a continuous determinant phase of G^{-1} is

The phase has range at most \pi\dim(V\oplus P) after strict regularization. Integrating, using Equation (6.7), and then removing the regularization as in Lemma 3.6 gives

In particular, Ward and symmetry bound the integral of \norm{P_+u}^2 by the same polynomial. We need an additional factor when g is small.

Put D=-\operatorname{Im}G^{-1}. Complex symmetry makes D real symmetric and gives

For each coordinate, Cauchy–Schwarz in the form D bounds the square of the corresponding entry of G^*Du by (\operatorname{Im}G)_{ii}\operatorname{Im}g. Consequently, with entrywise imaginary part on vectors,

Take q=\sqrt\sigma=\epsilon, and set \Omega_q=\{r\in I_r:\abs{g(r)}\le q\}, excluding null singular sets. Equations (6.8) and (6.9) give

There are only polynomially many component intervals of \Omega_q. To see this, A is affine in r, c is a rational function obtained by inverting A, and G is a further matrix inverse in cell dimension O(L^2). Thus numerator and denominator degrees of g are bounded by a fixed polynomial in L. The inequality \abs g^2\le q^2 becomes a real polynomial inequality after clearing the squared denominator. Its intervals and the poles are therefore polynomially many; the values of the exterior coefficients do not affect this degree bound. Integrate the real part of Equation (6.6) on these intervals, first truncating away from any endpoints where necessary. Every endpoint value of \operatorname{Re}g has modulus at most q. Hence

Using \norm{P_+u}^2=\operatorname{Re}(u^\top P_+u) +2\norm{P_+\operatorname{Im}u}^2, we obtain

Normalized uniform integration in r only introduces the polynomial factor (2L^{-D_1})^{-1}.

The Z rows of the equation G^{-1}u=\sqrt\sigma e_0 involve only u_P and P_+u. Their coefficients are uniformly bounded, and M is bounded below on P. The P rows then give, respectively,

It follows from symmetry and Equation (6.1) that

The cutoff bounds in Lemma 3.4 now yield

The first term treats \abs g>q; the second follows from Equations (6.10) and (6.11) on \Omega_q. Since q=e^{-\nu L}, substitution of Equation (6.4) absorbs the factor e^{C(\log L)^2} and proves Equation (6.5) for some fixed c>0.

6.3 Regular boundary and shift data

We turn to Equation (6.4). By Proposition 5.1, it is enough to work on boundary data satisfying

since this event carries a fixed positive fraction of \E_B\rho. Fix such data, assume \rho>0, and initially set every jitter to zero. Define the boundary Schur matrix

We claim that at least half of the uniform r-interval satisfies

All exponents in this assertion can be fixed before D_2,D_3.

Indeed,

For the last inequality, put w=A^{-1}Mx. The Z rows of Aw=Mx have bounded coefficients, use only P_+w, and give \norm{Mx}\le C\norm{P_+w}. The lower singular-value bound for M gives \norm{P_+w}\ge c\norm x, which proves the assertion. For \xi>0 let K_\xi=(\mathcal B-i\xi I_P)^{-1}. If \norm{\mathcal B^{-1}}>\xi^{-1}, inverse difference implies \norm{K_\xi}\ge(2\xi)^{-1}. Ward then implies

The same conclusion follows at a singular matrix from a null vector. The determinant phase derivative of \mathcal B-i\xi I_P is at least c\tr\operatorname{Im}K_\xi. Thus phase integration bounds the Lebesgue measure of the failed event in Equation (6.14) by C\xi\dim P. Dividing by 2L^{-D_1} and choosing \xi to be a sufficiently small inverse polynomial proves the claim. The only dependence on the exterior here is through coefficients of a passive matrix; its norm never enters.

For regular zero-jitter data, omit both the root and artificial absorption in J, and write

The subscript 0 denotes the absence of internal artificial absorption; the boundary matrix \widehat T is unchanged. These definitions will also be used after the jitters have been varied. The field z_0 is the response in J to the exterior source. Define its normalized coupling to the root by

If we adjoin the calibrated open root, still omitting the artificial absorptions on V, block inversion gives the analogue of t for this auxiliary system:

The prescribed internal absorptions will be restored below. We first obtain a lower bound for this numerator using the shell jitters, then control the denominator. A diagonal change at a site j changes F_0 through the product (U_0)_j(z_0)_j. We will find a source component near the boundary and a large component of U_0 at logarithmic depth, and use the jitters to propagate the source component to that deeper site.

Block inversion through A and \mathcal B gives polynomial bounds on R_0 and \Gamma. Moreover,

has polynomial norm by Equation (6.13).

The source normalization requires some care because \widehat k and \widehat T need not be bounded. Put C_{\rm ext}=\widehat T-iI_P, \Delta=\mathcal B-C_{\rm ext} =iI_P-M^\top A^{-1}M, and b_0=\mathcal B^{-1}\widehat k. The identities

use only polynomially bounded \Delta,\mathcal B^{-1} and the norm bound Equation (6.2). Consequently,

Since Az_0=-Mb_0, the sparse Z rows also imply \norm{P_+z_0}\ge c\norm{b_0}. There is therefore a site j_0\in Z^+ with

The other field has the representation

The profile estimate and \abs{\psi_i}\le C\epsilon on Z show \norm{M^\top A^{-1}p}\le C(1+L)^C\epsilon. Thus the reflected term in Equation (6.21) has polynomial norm, with exponent fixed independently of D_2. At depth d=\lfloor D_2\log L\rfloor, the profile has

Choose D_2 large enough that this exceeds both the reflected term and the profile error in units of \epsilon, with fixed slack. Every site i at that depth then satisfies

6.4 A path of resolvent entries

Join j_0 to a site at depth d by a physical nearest-neighbor path of length O(1+\log L) in J, moving inward by decreasing absolute coordinates. The corner clipping guarantees that the path can avoid Z after its first vertex. For each successive pair y,j, with j\notin Z, Equation (6.17) gives

We next replace this physical edge by a uniformly bounded number of large entries of \Gamma. This replacement is needed because \Gamma need not be spatially local.

Write B_\Gamma=\Gamma^{-1}, n_J=\abs J, and K_\Gamma=\max(1,\norm{B_\Gamma}). Let

Form the undirected graph on S_y\cup S_j with an edge between a\in S_y and b\in S_j whenever \abs{\Gamma_{ab}}\ge\delta_\Gamma. The endpoints y,j belong to this vertex set because (B_\Gamma)_{yj}=1. If they were not connected, let F be the diagonal indicator of the component of y, extended by zero outside the graph. The inverse commutator identity would give

Indeed every nonzero summand with F_{aa}\ne F_{bb} must cross components, so its \Gamma entry is below threshold. This contradiction proves connectivity.

Every graph edge meets S_j, which consists of j and at most its four physical neighbors: j\notin Z makes the nonlocal term in Equation (6.17) vanish in its column. Thus S_j is a vertex cover of size at most five. A simple connecting path has at most ten edges, since a simple path with no two successive vertices outside a set of size five has at most eleven vertices. The other support S_y lies in the physical star of y together with Z. All these vertices lie in the shell L-\abs i_\infty\le D_2\log L+5; the fixed buffer accommodates the physical stars.

Concatenate the paths for the physical edges and erase loops. We obtain distinct sites

all carrying shell jitters, with j_s at depth d. Fixed tie rules make all choices measurable functions of the regular zero-jitter data alone.

6.5 Propagation by the independent jitters

Choose D_3 sufficiently large after D_2. For every allowed simultaneous jitter perturbation, its diagonal operator norm is at most L^{-D_3}. Inverse difference applied to R_0 therefore preserves its polynomial norm, the bounds for \Gamma^{-1}, and the path-edge lower bounds, with constant-factor slack. In particular, if superscripts denote perturbed and zero-jitter values, then

where \Delta_J is the jitter diagonal. Equations (6.19) and (6.20) show that the norm and starting-component bounds also retain constant-factor slack. The terminal bound retains, say, \abs{(U_0)_{j_s}}\ge1 throughout these perturbations. For this last assertion use Equation (6.21) and the uniform profile estimate again: the large total size of U_0 is not controlled by an additive small-perturbation argument.

Condition on all nonpath jitters. Expose the path jitters successively, setting unexposed ones to zero only in the intermediate calculations. Let w=L^{-D_3} be their common half-width. At step a<s, changing the diagonal at j_a by \zeta\in[-w,w] gives the exact rank-one formula

Values on the right refer to the state just before this update. The denominator has modulus between 1/2 and 3/2 after increasing D_3 if necessary. For \zeta\in[-w,-w/2] and \zeta'\in[w/2,w], the two values on the right differ by

If one of the two intervals contains a value below one quarter of this separation in modulus, every value in the other interval is above that threshold. If it contains no such value, that interval itself is successful. Hence the conditional success probability is at least 1/4, and success gives

Constants and the exponent here include the fixed D_3.

Finally expose the jitter at j_s. Applied to the scalar F_0, the update is

Since \abs{(U_0)_{j_s}}\ge1, the same two-interval argument gives a final success of probability at least 1/4 and size at least (1+L)^{-C}\abs{(z_0)_{j_s}}. Only the current component is used at each step; later jitters need not preserve previously propagated components. There are O(1+\log L) updates, all at distinct sites and hence using independent uniforms. Combining Equations (6.20) and (6.25), we conclude that

This holds conditionally on every regular zero-jitter boundary and r datum, and uniformly after conditioning on nonpath jitters.

6.6 Restoring absorption and the root

Reinsert the prescribed artificial absorptions in J. If their diagonal is \tau A_{\rm art}, with 0\le A_{\rm art}\le I_J, the rootless inverse becomes

Write \Gamma_\tau=(R_\tau)_{JJ} and z_\tau=(R_\tau)_{JP}\widehat k. Inverse difference and the polynomial bounds already established give

Thus the error in p^\top z_0/\epsilon is at most \epsilon^{-1}\tau C(1+L)^C\sqrt\rho.

Now restore the open root. Its scalar Schur denominator is

For the auxiliary system with internal artificial absorptions omitted, the calibration and block formula for \Gamma give

The boundary profile estimate gives \norm{M^\top A^{-1}p}\le C(1+L)^C\epsilon, and hence the second term is polynomially bounded. Reinstating absorption changes the normalized denominator by at most \sigma^{-1}\tau C(1+L)^C, including the absorption at the root. Because c_*>2\nu, this is small for large L. We have therefore proved, uniformly on all the data being used,

Passivity and the open root also give -\operatorname{Im}d\ge\sigma, so this denominator is nonzero.

Block inversion now identifies the amplitude, with an irrelevant sign, as

The numerator error in Equation (6.27) is exponentially smaller than the successful lower bound in Equation (6.26). The polynomial denominator and the success probability can therefore be incorporated into e^{-C(\log L)^2}. For each good boundary datum, integration over the regular r event, of probability at least 1/2, and then all jitters gives

When \rho=0, invertibility in Equation (6.1) gives \widehat k=0, and the inequality is immediate. Finally integrate over the weighted good boundary event from Proposition 5.1. This proves Equation (6.4) and completes the proof of Proposition 6.1.

For clarity, the choices above have no circular dependence. First fix the profile and its plateau, the box-width exponents, and D_1 as in Proposition 4.3. The polynomial in Proposition 5.1, the regular zero-jitter inverse thresholds, and the exponents for the profile residual and reflected field can then be fixed without D_2,D_3. Choose D_2 to obtain the profile height in Equation (6.22). Choose D_3 afterwards to guarantee the profile estimates and all the perturbation margins used above. The constants in Equation (6.26) may depend on this last choice. Any fixed c_*>2\nu then makes the absorption errors small for sufficiently large L; no polynomial threshold is chosen by using the norm of an exterior matrix.

6.7 An envelope for arbitrary interior choices

For comparison between different interior laws we need a bound that does not require the branch on V. Keep the same geometry, boundary data, exterior source, and prescribed artificial mask. Allow any set D\subset V of internal cell ports, at most one per site, with strengths s_d\in[e^{-C_0L},1] and real reference potentials bounded in modulus by a fixed C_0. All other internal real diagonals are their true values, and there are no cell ports on B. The set D, strengths, and references may depend arbitrarily on the true interior potentials and on auxiliary choices.

When this envelope is compared with the branch lower bound, the corner-box conditionings must coincide. The single-side laws on B may differ: their settings and artificial absorptions change \rho only by the pointwise bounded factors in Proposition 5.1. This is the precise boundary comparison needed when an adaptive collection of descendant ports is replaced by an open parent root.

7 A hierarchy of probability splits and energy comparisons

Throughout this section the disorder strength h>0 and the interior energy E are fixed. All parameters in the cell construction are fixed as well. Constants may depend on these choices, including the corner-box exponent chosen below, but not on the outer torus, the upper thinning level, or the prescribed artificial-probe mask. We write

Only levels above a sufficiently large fixed l_0 will be used. We increase l_0 whenever necessary to apply all the preceding cell estimates.

7.1 An iid implementation of the rare cell laws

On the torus of side b_N, choose one uniform lattice offset, independently of the true iid potentials. For each l_0\le l\le N, partition the torus into squares of side b_l whose centers are congruent to that offset modulo b_l. These partitions are nested. In each tile use the translated geometry and branch law of Lemma 4.1 and Proposition 4.3, with L=L_l. The word ancestor will always refer to strict containment in this nested family unless the contrary is stated.

We first record the geometric feature that makes conditioning compatible across levels. The corner triples belonging to different tiles and levels are disjoint except when they coincide exactly. At a coincident triple the ordering of its three sites, and the two real plane-basis vectors used for its box coordinates, are the same. In fact, the missing corner and its two companions have fixed displacements from the corresponding two ends of a tile. Passing to an extreme child preserves those positions:

Consequently, if an ancestor triple meets a descendant tile, it is one of that descendant’s own triples. Also, the set V\cup B of a descendant is contained in that of its ancestor, and every descendant center belongs to the ancestor’s V. The latter inclusion will ensure that all descendant reference-potential changes are interior changes when we use the parent-cell envelope.

Put K_0=10^6. For each distinct corner triple, give its three true potentials a single box label as follows. In each product of polynomial pieces of \vartheta lying in (-h,h)^3, use the fixed linear box coordinates from Proposition 4.3. Partition the resulting region into maximal dyadic cubes whose concentric doubles lie in the region. Their inverse images are the corner boxes. All labels are defined from the true potentials, but we will expose only specified labels. At exact corner coincidences we use the same label, rather than successively refining it. Null boundaries may be assigned arbitrarily.

Conditional on a box label, the three true variables are uniform on that box: the original density is constant, and the coordinate change is linear with constant nonzero Jacobian. Labels of disjoint triples are independent. If a box has width less than \xi, its points lie within C\xi of the boundary of one of the finitely many piece regions. Indeed, away from those boundary strips a dyadic cube of width comparable to \xi containing the point has its concentric double inside the region, so a maximal cube containing it cannot be smaller. Thus

A level-l tile has valid own labels when the widths of all four corner boxes are at least b_l^{-K_0}. The doubled-box requirement also provides the polynomial clearance from support endpoints and piece boundaries required in the cell estimates.

Assign to every tile U of level l an independent selector \eta_U\in\{0,1\} of success probability q_l. These selectors are independent of all true potentials. A tile is eligible if it is selected and has no selected strict ancestor. Eligible tiles are pairwise disjoint, even when a selected tile later fails its branch test. An eligible tile is allowed if its own labels and those of every strict ancestor are valid at their respective levels.

For an allowed tile U, let \mathcal L_U be the likelihood of its cell branch law relative to the true product law with its four corner boxes fixed. It changes the law on V and the noncorner single-side coordinates of B, leaves the conditional laws on the four corner triples unchanged, and leaves all remaining sites of the tile unchanged. Its root factor is the conditional density centered at the reference c(v) constructed from the true values on J. Apart from the allowance gate, the likelihood uses only data in U and its own corner boxes. In particular it does not use ancestor labels in any other way. The density bound of Proposition 4.3 gives, uniformly over valid boxes,

After increasing l_0, we have p_l\mathcal L_U\le1.

Give each tile a further independent uniform variable t_U on [0,1]. Its acceptance indicator is

The selectors, acceptance uniforms, and offset are auxiliary randomness; the true potentials are sampled first from their unchanged iid law.

For a stage n, l_0\le n\le N, place active cell ports at precisely the accepted roots of levels at least n. Use their branch strengths and reference potentials. Every site without an active cell port has its true real potential. Because eligible tiles are disjoint, there is at most one cell port at a site. Denote the scattering matrix without artificial probes by S_{n,N}.

We may also prescribe at most one artificial probe per site, absent or of strength \tau or \tau\vartheta(v_i). Its mask may depend on the offset and on external randomness independent of the potentials and branch decisions. The profile is always evaluated at the true value, including at an active cell root. When comparing stages n<m, we assume

This is sufficient for the cell estimate at every removed level and at its immediate parent. In particular all constants below are uniform over these masks. For notational convenience, expectations denoted by \E_{\rm off} may include any external independent randomness used in the geometry or deterministic data prescriptions.

7.2 Thinning passes and admissible data

Fix n<m\le N. A thinning pass acts on a designated union of level-m tiles, chosen from the offset and any independent geometric randomness. In that region remove all accepted root ports of levels n,\ldots,m-1, restoring their true real values. Process these levels in increasing order. At a given level interpolate by retaining each eligible accepted root independently with probability \theta, with \theta decreasing from one to zero. Use fresh independent thinning marks; never recompute branch choices. One may subsequently perform a second pass on the complementary union of level-m tiles. Each such pass is covered separately by the estimates below, with its own starting system. A prefix may end between levels or at an intermediate value of \theta.

Let f be deterministic finite real scalar or Euclidean-vector site data, possibly depending on the offset. Its restriction supplies data on every port set. Let g be data fixed at the beginning of the pass such that, for a deterministic c_g<\infty,

We permit g to depend on selectors, branch statuses, and the start-of-pass response, and on external independent randomness. The precise admissibility requirement is that, at each root closure to which the energy comparison is applied, g is measurable in the conditioning that preserves that root’s hidden law in Lemma 7.2. In particular it must not inspect still hidden true values needed for future closures, and it must not depend on the thinning marks of the pass. This formulation allows response-dependent adjustments while retaining the conditional closure identity.

For a level-l tile U, let D(U) be its immediate parent, and put

Define the nonnegative charge

The sum includes all tiles of a level, even when the pass acts on only part of the torus; this only enlarges the bound.

We prove the theorem by first establishing the estimate that controls the interpolation near \theta=0. In a current level-l system, let \mathcal A_l be the thinnable roots in the pass region that are still present. For a\in\mathcal A_l, write U(a) for its level-l tile. Set

Port data in these formulas are evaluated at their physical sites. Trivially,

7.3 The two useful scales of the additive charge

8 Screening and affine energies at a fixed stage

Throughout this section the starting level n is fixed and sufficiently large. Constants bearing a subscript n may depend arbitrarily on n, as well as on the fixed h,E. They do not depend on the torus size once that size is sufficiently large. We use the stage-n construction of Proposition 7.1. We also allow the prescribed artificial probes: at each prescribed site there is at most one such probe, of strength \tau or \tau\vartheta(v_x), where 0\leq\tau\leq1. A mask specifies which of these choices is made at each site. Masks, observation grids, and test centers may depend on the hierarchy offset and on external independent randomness, but not on the potentials or on the branch decisions. When conditioning on the offset below, we also fix any such external randomness. Uniform assertions about these prescriptions mean estimates for every individual prescription, not a single event on which all prescriptions work simultaneously.

Write \Lambda_r(x) for the max-norm ball of radius r, with shortest coordinate distances on the torus. A tile of level l>n is hazardous if its selector is present, regardless of acceptance, or if one of its own corner labels fails the level-l validity test. The construction gives, conditional on the offset,

Consequently, for a square of radius A, the probability that a hazardous tile of side greater than D meets that square is at most

All the counting estimates below are used within a small fixed fraction of the torus side, where squares may be unwrapped in the usual way.

8.1 Prospective observations and a generic observation bound

Choose a large power of three Q_1, and tile the torus by observation cores of side Q_1. Set \beta=1/12 and require Q_1^\beta\geq b_n. For a core centered at z, test candidate level-n roots in \Lambda_{3Q_1}(z) using the prospective rule from Proposition 7.1: retain all local acceptance rules, but omit ancestor selectors and validity tests at sides exceeding Q_1^\beta. The resulting test for one candidate uses randomness within distance C Q_1^\beta of that candidate, conditional on the offset.

Each test succeeds with probability at least c q_n p_n>0. To see this, first expose the required selector statuses and the needed corner-box identities. Conditional on the candidate being eligible and all required labels being valid, its acceptance likelihood integrates to one, so its selector and acceptance contribute q_np_n. The probability of the required strict-ancestor selector absences and the own and ancestor validity conditions is bounded below by a positive absolute constant: their failures have a summable union bound along the ancestor chain, made small by the choice of the starting level. This also explains why the local likelihood is not additionally conditioned on finer corner data.

Cover \Lambda_{2Q_1}(z) by mesh squares of side comparable to Q_1^{1/4}. In each such square choose candidates separated by a sufficiently large multiple of Q_1^\beta. There are at least c Q_1^{2(1/4-\beta)}=cQ_1^{1/3} candidates with disjoint randomness neighborhoods, for sufficiently large Q_1 at fixed n. Their success indicators are independent conditional on the offset. The probability that some mesh square contains no successful root is therefore at most

In particular, with arbitrarily high probability, the prospective observation roots form a C Q_1^{1/4}-net of \Lambda_{2Q_1}(z), also for nearest-neighbor graph distance after changing the absolute constant.

The next argument applies to every possible observation set with this net property. Its equations always use the true potentials.

8.2 Projected resolvents and shield comparisons

The decay estimate below uses an exponential-conjugation argument of Combes–Thomas type [8]. We first establish the projected inverse bound on the observed coordinates; no bound on the full inverse is required.

Call an observation core good if the prospective roots form the specified net, the deterministic observation bound holds, and no hazardous tile of side greater than Q_1^\beta meets \Lambda_{4Q_1} about its center. The prospective roots of a good core are actual stage-n ports: all omitted ancestor requirements hold. Moreover, every real reference change in its tested rows occurs at a cell port of side at most Q_1^\beta. Such a port has strength bounded below by a positive number depending on Q_1.

We may count only a subset of these good cores. In a partly thinned or Dirichlet-restricted system, we count only cores for which the physical neighborhood \Lambda_{3Q_1+1}(z) remains present, and for which all cell ports, their strengths, and the real reference values in the tested rows \Lambda_{3Q_1}(z) agree with the full stage-n system. We call these tests intact. The neighbor layer preserves every hopping term in the true equations; artificial probes may still have any of the allowed masks. Let J_{\mathrm g} be the coordinate embedding of the sites in the counted cores, and set

where Y includes every current port, whether or not it lies in a counted core.

8.3 Crossing costs and the wall variant

By Equation (8.3) and Lemma 8.1, the probability of a local test failure can be made arbitrarily small, first by taking Q_1 large and then C_{\mathrm{obs}} large. The additional hazard cost for one core is at most

which tends to zero as Q_1 grows. Consequently, in any neighborhood of radius a fixed multiple of Q_1, the probability that some core meeting that neighborhood is bad is arbitrarily small. All these probabilities are conditional on the offset, uniformly in that offset.

8.4 Jump tails and fixed-stage limits

At full stage n, with the possible extra probes, define

Here \dist_2 is Euclidean distance formed from the shortest coordinate displacements. Let d_{n,N}^{\leq D}(x) and d_{n,N}^{>D}(x) denote the same sum restricted respectively to jump lengths at most D and greater than D.

8.5 Compact profiles and convergence of affine energies

To approximate affine coordinate data on a torus without choosing a discontinuous coordinate branch, put

Its chord distances satisfy

and their ratio to \dist_2(x,y) tends to one uniformly for bounded nonzero jumps as N\to\infty. Indeed, for shortest coordinate displacements \delta_i, the squared chord distance is \sum_i(b_N\sin(\pi\delta_i/b_N)/\pi)^2. Truncate the jumps, use this local limit, and then use Equation (8.13) for the tails. Translation after offset averaging and Equation (8.14) give

There are no artificial probes in this identity.

The order of parameters in this section will be used explicitly below. One fixes n before choosing observation scales, quantiles, jump cutoffs, or local approximation radii. Their possibly very large values impose only fixed thresholds on subsequent spatial scales. With fixed additional parameters, one can then take m>n so large that scales of order \exp(b_m^2) exceed every such threshold. A probe strength \tau=C_0/R at those scales tends to zero and eventually satisfies \tau\leq\exp(-c_*b_m), as required by Theorem 7.3. Finally N is increased after these choices. Neither this procedure nor Equation (8.23) supplies or uses a rate of convergence uniform in n.

9 Vanishing of the limiting energy in two dimensions

Throughout this section, the disorder strength and the interior energy are fixed. All auxiliary constructions are those of the preceding sections. In particular, Proposition 8.7 provides finite numbers d_n with d_n\to d\geq0. No convergence rate uniform in n is assumed.

Suppose for contradiction that d>0. We first thin the gaps between retained inner squares and grid walls. A localized change of the affine data disappears after the complementary thinning pass. Since the expected initial and final affine energy densities approach the same limit, choosing the sign of this change forces its local first variation to be small in expectation.

We then place weak probes in a square annulus between one inner square and its wall. A logarithmic voltage profile makes total transmission to the probes small, so the cell-to-cell scattering block is nearly unitary. Its surviving internal jump energy gives an analytic trace a nonzero central value. Closing the probes identifies the boundary modulus with the preceding first variation evaluated after a support-preserving shift of the true potentials. A bounded cost for changing the law transfers smallness to boundary probabilities, and dimension-independent moment bounds turn this into a small boundary integral, contradicting submean.

9.1 Geometry and order of scales

Fix a number M_0>20, to be enlarged below. After choosing n, choose m>n and a power of three L_* such that

The torus side b_N will be much larger than L_*. Since L_* is a power of three, the torus admits an aligned partition into grid tiles D of side L_*. The alignment is specified by the hierarchy offset. Write c_D for a tile center and use max-norm radii about this center.

Let I_D be the union of level-m tiles whose centers have radius at most R, and let O_D be the union of level-m tiles whose centers have radius at least L_*/3. Thus the inner square has radius R+O(b_m), and the outer walls begin at radius L_*/3+O(b_m). The remaining level-m tiles form the gap. Thin levels n,\ldots,m-1 in all gaps, leaving the inner squares and walls at stage n. The result is called the intermediate system. A complementary pass on the retained regions finishes the transition to uniform stage m. Both pass regions are unions of whole level-m tiles, as required by Theorem 7.3.

Define the selector event

This event involves selectors only. Conditional on the offset,

Indeed q_l=e^{-b_l^6} and L_*<3e^{b_m^2}, so the right side tends to zero faster than any fixed negative power of R. On \mathcal P_D, the intermediate cell ports in D lie only in I_D\cup O_D; after the complementary pass there are no cell ports in D.

Independently of the potentials, selectors, and acceptance variables, choose a uniformly random grid tile, conditional on the offset. For this test tile write D,c,I,O,\mathcal P instead of D,c_D,I_D,O_D,\mathcal P_D. All expectations below include this choice. It is a permitted independent prescription for the screening and comparison estimates. The retained regions and the probe annulus used below are shown in Figure 2.

9.2 An adaptive adjustment of the affine data

For the moment use no artificial probes. Let S' be the intermediate scattering matrix and let f=f_N be the four-component torus embedding used to define the affine energy in Proposition 8.7. Its chord distances are globally comparable to shortest Euclidean torus distances, and its nearest-neighbor increments are bounded. Define four-component data on the intermediate cell ports by

As elsewhere, real scalar data inside a matrix expression denote the corresponding diagonal matrices. The trace H^0 is real, since every diagonal entry of S'F_\nu S'^* is real. It is the local first variation of the affine energy: for fixed S',

Here the perturbation is data on the intermediate ports. The proof below extends it by a cutoff to site data with bounded increments and chooses its sign separately in each grid tile. On the selector event, the affected ports disappear in the complementary pass.

9.3 An annulus of artificial probes

Add probes at all sites of

and nowhere else, using strength \tau\vartheta(v_x), where

The constant C_0 will be fixed before n,m,N are chosen. For sufficiently large m, Equation (9.1) implies \tau\leq e^{-c_*b_m}, so this mask is allowed throughout the first comparison pass. Use the same underlying hierarchy choices as before. The resulting intermediate scattering matrix has block form

The first block consists of all cell ports, and the second consists of the new probes. We also use C for their index set. In this section I denotes the cell ports in the retained inner square and J denotes all other cell ports; these symbols are local to this section. On \mathcal P, every port in J is outside the probe annulus toward the exterior. Set

The data F,F_I still refer to Equation (9.4), now restricted to the cell ports of Equation (9.10).

We first give the deterministic use of a good annular barrier. A site is called good for this purpose when \vartheta(v_x)\geq d_0, where d_0>0 is fixed sufficiently small. Count only edges with both endpoints good and lying in a buffered square band strictly inside C, for example with both endpoint radii in [1.2M_0R,1.8M_0R]. Give such an edge cost one, and all other edges cost zero. Denote by \ell(x) the resulting cost distance from the region |x-c|_\infty\geq 2M_0R. Let \mathcal B be the event

where \gamma>0 is a sufficiently small fixed constant.

The next event combines small total transmission with positive internal jump energy and a bound on transmission weighted by F. The weighted bound is needed because F grows on distant cell ports: small K alone does not control those coordinate factors in the trace.

9.4 An analytic trace with nonzero central value

Close all inserted probes with the common phase z. For real s, the corresponding physical shifts and the phase are

By Lemma 3.2, the scattering matrix on cell ports is

Define H(z) on the unit circle by replacing S' with S_z in Equation (9.4). Its matrix expression contains the adjoint S_z^*. To remove that obstruction to analyticity, replace the adjoint by the adjugate and multiply the constant term by the determinant:

This function is analytic in the disk. Indeed B is a contraction, and the scattering entries in Equation (9.21) are rational analytic functions there. Possible boundary poles are removable: away from finitely many points their boundary values are unitary, which excludes a genuine pole of a rational entry on the circle. Thus \mathfrak h extends continuously to the boundary. There,

because \operatorname{adj}S_z=(\det S_z)S_z^*.

9.5 Changing the potential law under a bounded twist

The artificial profile vanishes at the support endpoints and belongs to C^{1,1}([-h,h]). For sufficiently small |s|, the map

is therefore a diffeomorphism of [-h,h] onto itself, with derivative at least 1/2. If r_s is the density of its pushforward of the uniform law relative to that same law, then

Consequently

Here \mu is the uniform law on [-h,h].

Fix W<\infty and assume |s|/\tau\leq W. Conditional on the offset and the test tile, change every true variable in C by T_s and leave all other true variables, selectors, and auxiliary uniforms unchanged. There are at most C(M_0R)^2 changed sites. By independence and Equation (9.29), the resulting joint law has a density \mathscr D_s relative to the original joint law with

This also holds after averaging over the offset and the test tile. The bound is uniform in m,N once they are sufficiently large for the maps to be diffeomorphisms. Its constant may be large; only its independence of the later scales matters.

We next check why this change of true potentials reproduces the closed-probe system despite the potential-dependent hierarchy. This check concerns recomputed statuses, not an assumption that statuses are fixed by definition.

On \mathcal P, selected tiles of level at least m do not meet the test tile. Every tile of level below m touching C lies wholly in the thinned gap, since b_m\ll R and the annulus is separated from both retained regions by a distance comparable to R. Thus a selected tile capable of carrying an intermediate cell port is disjoint from C. Its own true variables, own corner labels, local likelihood, and reference potential are unchanged by the shift. Eligibility depends only on selectors and is unchanged as well.

There is one remaining possibility: a changed coordinate may alter the validity of an ancestor label used by a retained tile. Any such ancestor contains both the retained root and a coordinate in C. Their separation is at least a fixed multiple of R, so the ancestor side is at least cR. The probability that any such large tile has an invalid own label is bounded by

This counts only tiles meeting C, which also covers every relevant changed triple. The original bound is uniform in the offset and torus size. After the shift the probability is at most D_W\sqrt{u_R} by Equation (9.30). Exclude both events. All potentially changed ancestor validity tests are then valid before and after the shift. Ancestor box identities themselves need not be the same: the acceptance rules use those identities only through the validity gates, whereas each retained tile’s own box and likelihood are unchanged.

On the remaining part of \mathcal P, all intermediate cell ports and references are therefore unchanged when the hierarchy is recomputed with the same selectors and auxiliary uniforms. Changes of acceptance inside the gap have no effect, because those ports have already been thinned, and \mathcal P excludes the larger ones. Before closure the real potentials on C were the true ones; the only added terms there were the artificial absorbers. Hence Lemma 3.2 and Equation (9.20) imply that H(z) is exactly the no-extra-probe intermediate trace H^0 evaluated at the shifted data. Almost-everywhere inverse statements suffice throughout.

Let a_{n,m,N}=\E[\one_{\mathcal P}|H^0|] in the original law. Since \mathcal P uses only unchanged selectors, Markov’s inequality and Equation (9.30) give, for every fixed \delta>0,

where one may take e_R(W)=u_R+D_W\sqrt{u_R}. This tends to zero as m grows at fixed W,M_0,C_0. The estimate is uniform over bounded twists; it does not assert simultaneous invariance for all twists on a single realization. Only the probability bound for each twist will be integrated.

9.6 The contradiction

10 A fixed-energy fractional-moment estimate

We return to the true product law and remove the hierarchy from the operator whose decay is to be estimated. Throughout this section, h>0 and E\in(-4-h,4+h) are fixed. All choices may depend on these two parameters. Distance means the shortest-coordinate max-norm distance on the torus \T_N^2=(\Z/b_N\Z)^2. The notation for the probe sets below is local to this section.

We prove the theorem by first retaining arbitrary deterministic masks of artificial probes. At each site a mask prescribes absence, strength \tau, or strength \tau\vartheta(v_i), with at most one probe per site. The profile \vartheta is the one fixed earlier; in particular it equals one on [-h/2,h/2], is C^{1,1}, and vanishes quadratically only at the endpoints of the potential interval. Write \eta_i for a prescribed positive probe strength and let Y have columns \sqrt{\eta_i}e_i. In this section

Thus S is symmetric and unitary, and Y^\top GY is a contraction by Lemma 3.1. All exceptional algebraic singularities below can be discarded under the continuous product law.

The parameter order is

The barrier threshold d_0>0 will be small; C_0 will be large. After they have been fixed, n and then m>n are made sufficiently large. Finally N is sufficiently large in terms of all these choices; all ensuing estimates hold for every larger N. We always require b_N>100R and \tau\le1.

10.1 Filling and twisting an annulus

Let X,Y' be arbitrary deterministic subsets of the probe sites in the radius-R squares centered at x,y, respectively, where \dist(x,y)\ge10R. Let

Fill every previously unoccupied site of C with a probe of strength \tau\vartheta(v_i), keeping existing probes and their strengths. The filled mask is again deterministic. Superscripts {\rm old} and {\rm fill} distinguish the resulting scattering matrices. Identify the probes in C with their sites and put

We seek a bound by an arbitrarily small multiple of \E V^s. Thus Equation (10.3) leaves two tasks: attenuate D across the filled annulus, and combine a small mean return U with a resampling bound for U^sV^s. An unweighted estimate for barrier failure would not suffice when \E V^s is small. We therefore control that failure also under the V^s-weighted law.

10.2 Two barriers under a transmission tilt

Call a site good when \vartheta(v_i)\ge d_0. Give cost one to an edge precisely when both endpoints are good and both radii about x belong to [2.2R,2.8R]; all other edges have cost zero. Let d_{\rm in} be the associated path-cost distance from the radius-2R square. Independently of this definition, count good-good edges with both radii in [3.2R,3.8R], and let d_{\rm out} be the resulting distance from \{i:\dist(i,x)\ge4R\}. Integer roundings of the four band boundaries make no difference to the constants. For a sufficiently small absolute \gamma>0, define

These two strictly buffered bands will allow separate resampling of the inside and outside of radius 3R.

10.3 Cutoffs and the small return estimate

We record explicitly the consequence of Lemma 9.3 that will be used several times. Suppose \Psi=GY_J is a matrix of fields sourced at a probe set J, and a scalar cutoff \chi changes only across edges whose two endpoints belong to C and carry probes of strengths at least \tau d. If \abs{\chi_i-\chi_j}\le L, then the hopping commutator can be written as a probe source:

Only endpoints of changing edges need be included on the right. Indeed, in the coefficient at an endpoint one divides by its square root strength, and replaces each neighboring field by its absorbed field divided by that neighbor’s square root strength. Bounded lattice degree gives the stated operator bound. Ward then yields

The output here includes all probes. These estimates also compare two systems agreeing on the cutoff support and its changing edges: cut off a field of the first system and correct its commutator residual in the second. No estimate on the exterior resolvent is involved.

10.4 Direct transmission and independent resampling

We next prove two pointwise estimates:

For the first, use fields sourced at Y', symmetry of S, and a cutoff equal to one inside radius 2R, zero outside radius 4R, with gradient at most K/R. Its source term vanishes, while both endpoints of every changing edge have strength at least \tau d_{\min}. Equations (10.8) and (10.9) give D\le K(R\tau d_{\min})^{-1}V; here C_0\ge1. In particular, if \E V^s=0, then D=0 almost surely, and no tilted law is required.

For the improved estimate choose an integer

with K_1 sufficiently large. For j\ge1 use the cutoff

All original sources have d_{\rm out}=0. Nonzero gradients have both endpoint distances at least (j-1)\ell, because cost distances are integer valued and change by at most one across an edge. Gradients only occur on counted good-good edges in the outer band. For the field \Psi=G^{\rm fill}Y_{Y'}, put

The cutoff equations and Ward give

Start with B_0=V/2. On \mathcal B, the set X has d_{\rm out}\ge\gamma R, so it remains beyond every one of the \lfloor\gamma R/\ell\rfloor cutoffs. For sufficiently large fixed C_0, followed by sufficiently large R, this number is at least c(d_0)C_0. The iteration proves the second estimate in Equation (10.11). Notice that A_j includes every absorbed output, including X; the proof does not require X\subset C.

To separate the product V^sU^s, enlarge the probability space with two independent fresh families of true potentials. Let A=\{i:\dist(i,x)<3R\}. Form one comparison configuration by retaining the original potentials on A^c and replacing those on A by the first fresh family. Form a second by retaining the original potentials on A and replacing those on A^c by the second fresh family. Keep the filled deterministic mask in both configurations, and recompute its strengths from the corresponding potentials. Denote the first configuration’s transmission to Y' by V^{\rm ext}, and the second’s transmission from X by U^{\rm int}.

On \mathcal B, the cutoff (1-d_{\rm out}/(\gamma R))_+ is supported in the shared exterior A^c, equals one on Y', and has all changing edges in its buffered outer band. Applying it to the fresh-interior fields and correcting in the original system proves

Likewise (1-d_{\rm in}/(\gamma R))_+ is supported in the shared interior A and gives

Take C_0 large enough that both factors are at most two. The two comparison configurations are independent unconditionally: they use disjoint parts of the original product sample and independent fresh families. Each has the original true product distribution. Dropping \one_{\mathcal B} before factoring therefore yields

This factorization makes no assertion of independence conditional on the original barrier event.

The exceptional event is controlled under the tilted law from Lemma 10.3. Unitarity gives U\le K R, and Equations (10.11) and (10.6) give, respectively,

Both quantities tend to zero uniformly over all the prescriptions. Combining these estimates with Lemma 10.2, we obtain

The constants K_s,c>0 are independent of large C_0. First choose C_0 to make its exponential term small, then use Lemma 10.4 with n,m,N in order to make the return term small, and enlarge m,N as necessary for \epsilon_R. Thus the entire multiplier can be made arbitrarily small. These are uniform expectation bounds for each choice of mask and centers; no simultaneous good event over all choices is needed.

10.5 Contraction over all masks

The passage from finite-scale smallness to exponential fractional-moment decay parallels the finite-volume criteria of [4]. We prove the contraction here for the probe transmission moments and deterministic masks.

10.6 Removing the two endpoint probes

Choose a mask consisting only of two constant-strength probes \tau at distinct sites x,y. Let G^0=(H_N-E)^{-1}, and let g be its two-by-two compression to \{x,y\}. Resolvent subtraction gives the damped compression

For a symmetric two-by-two matrix, direct multiplication by the adjugate shows

Since S_{xy}=2i\tau(g^{\rm damp})_{xy}, the exact identity is

For completeness, the moments needed to remove the determinant are uniform in the exterior and the volume. Conditional on all other potentials, Schur complementation gives

with A a fixed real symmetric matrix. Put t=(v_x+v_y)/2, u=(v_x-v_y)/2. The integration square becomes \abs u\le h, \abs t\le h-\abs u, with Jacobian two. For each fixed u, let \lambda_1(u),\lambda_2(u) be the real eigenvalues of A+\diag(u,-u). If 0<q<1,

Integration in u, with the normalized joint uniform density, therefore proves

The bound is uniform over all real symmetric A, so no exterior norm enters. The deleted system is generically invertible; its exceptional set has zero product measure. A one-site Schur complement and the same scalar integral also give \E\abs{G^0_{xx}}^q\le K_{h,q}.

Since \abs{\det(1-i\tau g)}\le(1+\tau\norm g)^2, Equation (10.16) with q=2s<1 gives a uniform bound on \E\abs{\det(1-i\tau g)}^s. Apply Cauchy–Schwarz to Equation (10.15):

The first factor of expectations decays exponentially by Equation (10.14), applied to singleton subsets. The diagonal is covered by the one-site bound. The now fixed R,\tau and all constants are absorbed into A(h,E,s) and a(h,E,s)>0, proving Theorem 10.1. This conclusion remains a fixed-real-energy finite-volume estimate; the spectral passage uses it separately at each energy.

11 Infinite-volume spectral type

We now remove the fixed-energy restriction from the spectral conclusion. The probabilistic input is the true finite-torus estimate Theorem 10.1. We use the bounded self-adjoint spectral theorem and its Borel functional calculus in the form of [16, Theorems 3.1 and 3.7]. We include the spectral-averaging step to make the quantifiers explicit; it is the rank-one mechanism of [15, Theorem 2’, Proposition 7 and Theorem 8(i)].