arxiv: 2604.12248 · v1 · submitted 2026-04-14 · 🧮 math.PR · math-ph· math.MP

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Localization Lengths of Power-Law Random Band Matrices

Jiaqi Fan , Fan Yang , Jun Yin

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Pith reviewed 2026-05-10 16:09 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP

keywords random band matricespower-law decaylocalization lengtheigenvector delocalizationresolvent analysisT-variablesGaussian random matrices

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The pith

Power-law random band matrices have bulk eigenvectors with localization lengths at least a power of the bandwidth W, depending on the decay exponent α.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves lower bounds on the localization length of bulk eigenvectors in large N by N random matrices whose off-diagonal variances decay as a power law with exponent α greater than minus one. For negative α the eigenvectors spread across the full matrix size N with high probability. For small positive α they spread over arbitrarily high powers of the bandwidth W, while for larger α the bounds become specific powers of W. These results confirm the delocalized regime of a longstanding physical conjecture. The proof introduces a dynamical analysis of T-variables built from pairs of resolvent entries to handle the slow variance decay.

Core claim

We study large N×N power-law random band matrices H=(H_ij) with centered complex Gaussian entries where the variances satisfy E|H_ij|^2 ∝ (|i-j|/W+1)^{-1-α} for α>-1 and W≫1. We establish the following lower bounds, with high probability, on the localization length ℓ of bulk eigenvectors: ℓ=N if -1<α<0; ℓ≥W^C for any large C if 0<α<1; ℓ≥W^{α/(α-1)} if 1<α<2; ℓ≥W^2 if α>2. These verify the physical conjecture on the delocalized side via a dynamical analysis of T-variables formed from pairs of resolvent entries.

What carries the argument

Dynamical analysis of T-variables formed from pairs of resolvent entries of H, which tracks their time evolution to produce the localization bounds without higher-order resolvent loops.

If this is right

For -1<α<0 the eigenvectors are delocalized over all N sites with high probability.
For 0<α<1 the localization length exceeds W raised to any fixed power.
For 1<α<2 the localization length is at least W to the power α/(α-1).
For α>2 the localization length is at least W squared.
The dynamical T-variable method succeeds even though the variance profile decays slowly and the model is non-mean-field.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same T-variable dynamics could be adapted to study delocalization in other long-range disordered systems.
The four regimes suggest sharp changes in delocalization behavior near α=0,1, and 2 that might be visible in finite-size scaling.
Numerical diagonalization for moderate W and selected α values could check whether the derived powers are close to optimal.
Because the proof avoids higher-order loops it may extend more readily to non-Gaussian entry distributions.

Load-bearing premise

The matrix entries are independent centered complex Gaussians whose variances are exactly proportional to (|i-j|/W +1)^{-1-α} for fixed α>-1 and large W, with the spectrum in the bulk where the resolvent concentrates.

What would settle it

A numerical computation for α=1.5 and large W that finds typical bulk eigenvector support size consistently below W^3 would falsify the claimed lower bound.

Figures

Figures reproduced from arXiv: 2604.12248 by Fan Yang, Jiaqi Fan, Jun Yin.

**Figure 5.1.** Figure 5.1: Graphs induced by the second and the seventh terms on the RHS of (5.6): x y x y On the other hand, if a term is generated from the last two terms on the RHS of (5.6) and the ∂-operator acts on another tadpole diagram, say fx2y2 (G) or fx2y2 (G), then this diagram is “pulled-in” and “glued” to the chosen diagram. In this case, the number of effective sections increases to two. An example of this situatio… view at source ↗

**Figure 5.2.** Figure 5.2: Graph induced by the seventh term on the RHS of (5.6): x1 x2 y1 y2 a1 a ′ 1 a2 b2 For the graphs in Figures 5.1 and 5.2, one can readily verify, using (5.5) together with the averaged local law (4.3), that for x 6= y the first graph in [PITH_FULL_IMAGE:figures/full_fig_p037_5_2.png] view at source ↗

**Figure 5.3.** Figure 5.3: Examples of bad graphs: a a1 b1 a2 b2 ξ ξ a a1 b1 b2 ξ ξ ′ If we sum over the ξ 2 a2b2 edges attached to a2, we obtain two new solid edges (ξ ′ a1b2 ) 2 . The vertex a1 is then incident to four solid edges. In some situations, the distance between a and b1 may be of order ≳ |x−y|, so that the waved edge (a, a1) together with the solid edges ξ 2 a1b1 must supply the decay factor Ψ2 (|x − y|). Hence, we ne… view at source ↗

**Figure 5.4.** Figure 5.4: Examples of quasi-tail sections. (ii) A quasi-body section with external vertices x, y and outer vertices a, b is a graph consisting of a free solid edge between x (resp. y) and a, referred to as the x-solid edge (resp. y-solid edge), together 18In mythology, the Ouroboros bites only its own tail; here we also allow it to bite its own body. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_5_4.png] view at source ↗

**Figure 5.5.** Figure 5.5: , where, for simplicity, the colors of the solid edges are not shown. x a b v3 v2 v1 y [PITH_FULL_IMAGE:figures/full_fig_p041_5_5.png] view at source ↗

**Figure 5.6.** Figure 5.6: shows a completion of the body section in [PITH_FULL_IMAGE:figures/full_fig_p041_5_6.png] view at source ↗

**Figure 5.7.** Figure 5.7: Molecular graphs of a headless snake (left) and a snake with a weight-head (right). Definition 5.7 (Graph operations). Consider a graph G = Sq·Fq+1 · · · Fp, where each Fi ∈ {fxiyi (G), fxiyi (G)}, i ∈ [[q + 1, p]], is called a free tadpole diagram. The graph Sq = S1 · · · Sq is a headless snake with q effective sections, where the external vertices of Si, i ∈ [[q]], are also denoted by xi , yi. (Recall … view at source ↗

read the original abstract

We study large $N\times N$ power-law random band matrices $H=(H_{ij})$ with centered complex Gaussian entries, where the variances satisfy a power-law decay $\mathbb{E}|H_{ij}|^2\propto (|i-j|/W+1)^{-1-\alpha}$, for some exponent $\alpha>-1$ and bandwidth $W\gg 1$. We establish the following lower bounds, with high probability, on the localization length $\ell$ of bulk eigenvectors in the different regimes of $\alpha$: (1) $\ell=N$ if $-1<\alpha<0$; (2) $\ell \ge W^{C}$ for any large constant $C>0$ if $0 < \alpha <1$; (3) $\ell \ge W^{\alpha/(\alpha-1)}$ if $1 < \alpha <2$; (4) $\ell \ge W^{2}$ if $ \alpha > 2$. These results verify the physical conjecture of arXiv:cond-mat/9604163 on the delocalized side. The main difficulty in the proof lies in handling the interplay between the non-mean-field nature of the model and the slow decay of the variance profile. To address this issue, a key technical ingredient is a new dynamical analysis of $T$-variables formed from pairs of resolvent entries of $H$. In contrast to the fundamental works on regular random band matrices with fast-decaying variances in arXiv:2501.01718 and arXiv:2506.06441, this approach does not rely on higher-order resolvent loops.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives the first rigorous lower bounds on localization lengths for power-law band matrices in the delocalized regimes, using a new dynamical T-variable method on resolvent pairs.

read the letter

The main point is that they prove lower bounds on the localization length ℓ of bulk eigenvectors for these N by N matrices with power-law variance decay (|i-j|/W +1)^{-1-α}. The bounds are ℓ = N for α between -1 and 0, at least W to a large power for 0 < α < 1, W to the power α/(α-1) for 1 < α < 2, and at least W squared for α > 2. This lines up with the delocalized side of the 1996 conjecture for long-range disordered systems.

Referee Report

0 major / 3 minor

Summary. The manuscript studies N×N power-law random band matrices with centered complex Gaussian entries whose variances decay as E[|H_ij|^2] ∝ (|i-j|/W + 1)^{-1-α} for α > -1 and W ≫ 1. It claims to prove high-probability lower bounds on the localization length ℓ of bulk eigenvectors in four regimes: ℓ = N for -1 < α < 0; ℓ ≥ W^C for any large C when 0 < α < 1; ℓ ≥ W^{α/(α-1)} when 1 < α < 2; and ℓ ≥ W^2 when α > 2. The argument relies on a dynamical analysis of T-variables formed from pairs of resolvent entries, addressing the non-mean-field regime and slow variance decay without higher-order resolvent loops, thereby verifying the conjecture of arXiv:cond-mat/9604163 on the delocalized side.

Significance. If the claimed bounds hold, the work supplies rigorous verification of a physical conjecture on eigenvector delocalization for power-law band matrices across multiple regimes, including explicit W-dependence. The introduction of a dynamical T-variable analysis that avoids higher-order loops is a technical contribution with potential applicability beyond this model. The paper delivers a full rigorous proof (no fitted parameters or data-driven claims), which strengthens its value in the random-matrix literature.

minor comments (3)

[Abstract and §1] The abstract states the bounds hold 'with high probability' but does not specify the precise form (e.g., 1 - N^{-c} or 1 - exp(-W^c)); adding this in the introduction or Theorem statements would clarify the result's strength.
[Main theorems] The localization length ℓ is used throughout but its precise definition (e.g., via eigenvector mass or Green-function decay) should be recalled explicitly in the statement of the main theorems for self-contained reading.
[§3 or §4] A short heuristic paragraph explaining why the dynamical T-variable evolution sidesteps higher-order loops (in contrast to the cited works arXiv:2501.01718 and arXiv:2506.06441) would improve accessibility without lengthening the paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report contains no specific major comments, so we have no individual points to address. We will incorporate any minor editorial suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper is a rigorous mathematical proof deriving lower bounds on eigenvector localization lengths directly from the power-law random band matrix model definition, using resolvent analysis, Gaussian concentration, and a new dynamical treatment of T-variables. No parameter fitting occurs, no predictions reduce to inputs by construction, and load-bearing steps do not rely on self-citations or ansatzes imported from prior author work. The cited conjecture (arXiv:cond-mat/9604163) is external and the result verifies rather than assumes it; contrasts with other papers are non-load-bearing. The derivation chain remains independent of its target bounds.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the definition of the power-law variance profile and standard tools of random matrix theory; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption Centered complex Gaussian entries are independent with the given variance profile.
The model is defined with these entries; all concentration and moment calculations rely on Gaussian properties.
standard math The resolvent of H exists and its entries satisfy standard bounds and concentration in the bulk.
The dynamical analysis of T-variables is built on resolvent identities that hold for any matrix with the given spectrum assumptions.

pith-pipeline@v0.9.0 · 5591 in / 1699 out tokens · 62576 ms · 2026-05-10T16:09:33.675762+00:00 · methodology

discussion (0)

Reference graph

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