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Mean-field random matrices thermalise at the same N^{-1/2} overlap scale in the bulk and at regular spectral edges, with a larger N^{-1/4} scale only at cusps.

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2026-08-02 08:48 UTC pith:6ETI2XGS

load-bearing objection The regular-edge 1/N variance claim is a real result if the hidden O(ρ²) cancellation in Lemma 4.9 holds, but that cancellation is exactly the part I could not verify from the text. the 2 major comments →

arxiv 2607.03546 v2 pith:6ETI2XGS submitted 2026-07-03 math-ph math.MPmath.PR

Anomalous rate of eigenstate thermalisation at singularities of the density of states

L\'aszl\'o Erd\H{o}s , Joscha Henheik , Volodymyr Riabov This is my paper
classification math-ph math.MPmath.PR MSC 60B2015B52
keywords eigenstate thermalisation hypothesisquantum unique ergodicityrandom matrix theorymatrix Dyson equationmulti-resolvent local lawscusp singularityspectral edgecorrelated random matrices
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to prove the Eigenstate Thermalisation Hypothesis (ETH) for large N×N mean-field random matrices whose entries may be correlated, identifying the microcanonical ensemble and the optimal fluctuation scale of eigenvector overlaps in every spectral regime. The central claim is that the variance of diagonal overlaps ⟨u_j,Au_j⟩ around their microcanonical average is of order 1/N in the bulk and, unexpectedly, also at regular spectral edges, while at cubic-root cusps it is of order N^{-1/2} for generic observables and only N^{-1} for a special codimension-one family. This contradicts the long-standing Feingold–Peres and Srednicki inverse-density prediction of (Nρ)^{-1} scaling. The mechanism is a delicate cancellation in the matrix Dyson equation: three nearly singular directions of the stability operators are nearly coplanar, and a corrected definition of regularity reduces the dangerous factor 1/|β| to 1/α² with α² = Δ^{2/3}+|β|.

Core claim

Under Assumptions 2.1–2.5, for any deterministic observable A and any indices j,k, with very high probability |⟨u_j,A u_k⟩ − δ_jk ⟨A ℑM(γ_j)⟩/⟨ℑM(γ_j)⟩|² is stochastically dominated by (1/N)[ ||R_E(•A)||_hs / (ẽα(γ_j,γ_k)+N^{-1/4}) + ||•A||_hs ]². In words: the natural deterministic approximation to every overlap is the self-consistent Green function ℑM(γ), and the fluctuation around it is controlled by how much of A lies along a specific discrepancy direction R_E and by the improved stability factor α. The diagonal variance is ∼1/N in the bulk and at regular edges, and ∼N^{-1/2} at exact cusps except for observables orthogonal to R, for which 1/N persists. The paper's proof shows the invers

What carries the argument

The engine is the matrix Dyson equation −M(z)^{-1}=z−D+S[M(z)] and the associated two-body stability operator B_{z,w}=1−M(z)S[·]M(w), whose smallest eigenvalue β measures the instability of the deterministic two-resolvent approximation. The proof isolates a discrepancy direction R(z,w), essentially the difference between the unstable directions of B_{z,¯z} and B_{z,z}, and shows the three critical directions (from B_{z,¯z}, B_{z,z}, B_{¯z,¯z}) are coplanar up to order ρ². This coplanarity, together with a corrected regularization of observables (solving a quadratic minimisation), replaces the naive factor 1/|β| by the improved stability factor 1/α² with α²=Δ^{2/3}+|β|, which is what converts

Load-bearing premise

The load-bearing premise is the near-coplanarity of the three critical directions of the stability operators up to an error of order ρ², together with the matching cancellation estimate (4.56); if these fail, the regular-edge bound reverts from 1/N to the larger (N|β|)^{-1} scale and the central anomaly disappears.

What would settle it

Take a two-block deformed Wigner matrix (D with N/2 entries +1 and N/2 entries −1) and measure Var(⟨u_j,A u_j⟩) for A aligned with the discrepancy direction R_j at a regular edge as N grows: the paper predicts decay N^{-1}; observing N^{-2/3} would falsify it. Alternatively, numerically compute the three singular eigenvectors of B_{z,¯z}, B_{z,z}, and B_{¯z,¯z} near a regular edge and test whether their coplanarity error scales as ρ²; a larger exponent breaks the cancellation.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The Feingold–Peres inverse-density variance (Nρ)^{-1} is false for mean-field random matrices; at regular edges the variance is 1/N, the same as in the bulk.
  • The microcanonical ensemble for these systems is ⟨A ℑM(γ_j)⟩/⟨ℑM(γ_j)⟩, with the leading fluctuations set by the discrepancy direction R and the improved stability factor α.
  • In the absence of cusps the entire eigenbasis fluctuates like the columns of a Haar unitary; cusp eigenvectors, by contrast, carry strong cross-entry correlations and are not Haar-like.
  • At a physical cusp, generic diagonal overlaps fluctuate on the N^{-1/4} scale, while a codimension-one family of observables keeps the Haar-like N^{-1} scale.
  • The multi-resolvent local laws with the new notion of regular observables give a template for sharp eigenvector bounds across edges and cusps, not just in the bulk.

Where Pith is reading between the lines

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

  • Editorial inference: the coplanarity of the three critical directions looks like a hidden structural identity of the matrix Dyson equation; if it holds beyond the perturbative regimes studied here, similar cancellations should appear in other singular limits of self-consistent mean-field models.
  • Editorial inference: the variational definition of the microcanonical ensemble (minimising the deterministic two-resolvent expectation over the shift y) suggests a general principle: the 'correct' thermal average is the one that minimises the deterministic fluctuation, which could be tested in models where the MDE is replaced by a different self-consistent equation.
  • Editorial inference: the N^{-1/4} scale at cusps should be directly observable in numerical ETH tests for deformed Wigner ensembles with a two-block deformation; a measurement of the variance of ⟨u_j,A u_j⟩ across the cusp would distinguish this prediction from both the bulk 1/N and the inverse-density scaling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript proves an ETH/QUE statement for N×N mean-field random matrices with correlated entries under Assumptions 2.1–2.5. The microcanonical expectation is identified as ⟨AℑM(γ_j)⟩/⟨ℑM(γ_j)⟩, where M is the solution of the matrix Dyson equation (2.10). Theorem 2.6 gives a very-high-probability bound on squared eigenvector-overlap fluctuations in terms of an anomalous direction R_E and an improved stability factor eα(γ_j,γ_k). From this the authors derive optimal 1/N fluctuations in the bulk and at regular square-root edges, and N^{-1/2} fluctuations at cubic-root cusps for the special anomalous direction, thereby disproving the inverse-density Feingold–Peres/Srednicki scaling (Nρ)^{-1} in all small-density regimes. The proof is structured as a rigorous RMT argument: improved M-bounds (Theorem 3.6 / Proposition 4.3) feed two-resolvent local laws (Theorem 3.7), which in turn feed the ETH derivation in §3.4. The main technical novelty is the improved regularization •A defined via the variational problem (1.8) and the cancellation identities in Lemma 4.9.

Significance. If the result is correct, it is a substantial advance: it gives the first rigorous ETH with the optimal fluctuation scale for general correlated mean-field ensembles across the bulk, regular edges, and cusps, and it invalidates a widely used physics prediction in the one setting where the question can be posed rigorously. The microcanonical shift is obtained from an explicit minimization problem, the anomalous direction R_E is computed from the MDE solution rather than fitted, and Corollary 2.8 yields concrete, falsifiable N-scaling predictions; Figure 1 provides independent numerical corroboration. The paper is also methodologically valuable for the formulation and proof of two-resolvent local laws with a new regular-observable norm. The main caveat is that the central algebraic mechanism is not independently verifiable from the reviewed text.

major comments (2)
  1. [§4.2.1, Lemma 4.9, Eq. (4.56); used in §4.2.2 around (4.81)–(4.88)] The main claimed anomaly — regular-edge variance ∼1/N, Corollary 2.8(ii) — is produced by the improved M-bound (3.31). In the proof of Proposition 4.3, the crucial coefficient is b11,t; passage from (4.88) to the required bound (4.81) uses the second-order cancellation (4.56) of Lemma 4.9. The text states that Lemma 4.9 is proved in §9.2.1, but the version made available for this review does not contain that proof, and §1.3 explicitly disclaims an independent heuristic for this coplanarity/cancellation. This is a load-bearing unverified claim: if (4.56) holds only to order O(ρ) instead of O(ρ²), then |b11,t|∼ρ1ρ2/|β| rather than ρ1ρ2/eα_t², and the regular-edge variance reverts to the non-anomalous (N|β|)^{-1}≈N^{-2/3} scale, destroying Corollary 2.8(ii). The final submission must include the full proof of Lemma 4.9 (and of Lemma 3.10 / §9.4 used at (3.50)), or state (4.53)–(4.56) as an
  2. [§3.4, Lemma 3.10 and Eq. (3.50)] The derivation of Theorem 2.6 from the two-resolvent bound (3.46) uses Lemma 3.10 to replace the (z1,z2)-regular norm |||·|||_{z1,z2} by the one-body quantities R_E and eα(E1,E2) appearing in the statement (2.23). Lemma 3.10 is asserted with the proof deferred to §9.4, which is not included in the reviewed text. Since this step is needed to obtain the exact form of the microcanonical shift and the anomalous direction in the final theorem, the final version should contain the full proof, not merely a reference to a deferred section.
minor comments (4)
  1. [§2.1, Assumption 2.4] The text correctly admits that Assumption 2.4 (bounded self-consistent Green function) does not follow from Assumptions 2.1–2.3. Since Theorem 2.6 is stated with I=R, the abstract's wording 'general mean-field random matrices' is stronger than what is actually proved: the result is conditional on an additional admissibility condition. The authors point to [6, Section 9] for sufficient conditions; the final version should make this qualification explicit in the introduction/abstract and, ideally, verify the assumption for the numerical example in Figure 1.
  2. [§7, proof of Proposition 5.5] The GFT/zag step is carried out in the real-symmetric case, with the complex-Hermitian case left to the reader. Since Theorem 2.6 is stated for both symmetry classes, the final version should either include the complex cumulant expansion or state explicitly that the argument is analogous with the standard modifications.
  3. [Appendix B (referenced in §4, after Thm 3.6)] The claim that the M-bound (3.31) is 'optimal' is supported only by a reference to Appendix B and a remark about deformed Wigner matrices. This lower-bound statement is not needed for the upper bound in Theorem 2.6, but if the paper advertises optimality, the final version should include the full proof or a precise statement of the optimality result.
  4. [Notation, Eq. (2.20) vs (3.26) vs Corollary 2.8] The symbol R is used for the one-body projector R_E, the two-body direction R(z,w), and the vector R_j in Corollary 2.8. These are mathematically distinct objects; a naming change or a table of notation would improve readability.

Circularity Check

0 steps flagged

No load-bearing circularity; the central derivation is self-contained and parameter-free.

full rationale

Theorem 2.6 is derived from two-resolvent local laws (Theorems 3.7–3.8) whose deterministic M-bounds (Theorem 3.6) are proved from the MDE solution M(z) via the characteristic flow. The stability factors beta, alpha and the direction R are constructed from M, S, D, and the shape analysis of the self-consistent density; none is calibrated to eigenvector overlap data. The microcanonical shift is the minimizer of (1.8)/(3.20), and the final bound (2.23) contains the derived quantities R_E and ealpha; Figure 1 is an independent simulation-vs-prediction check, not a fit. The paper explicitly disclaims an independent heuristic for the O(rho^2) cancellations (Lemma 4.9) and for the resulting 1/N edge rate (“we admit that we do not have an independent heuristics...”, Section 1.1; “we do not see any intuitive or conceptual reason...”, Section 1.3). A missing heuristic or unverified algebra is a correctness risk, not circularity. The text also cites prior work, including the same authors’ results (e.g., [36] for eigenvalue rigidity and zigzag ingredients), but these are parameter-free external results under stated assumptions (Assumptions 2.1–2.5) and do not assume the target ETH fluctuation rates; hence they do not make the derivation circular. Assumption 2.4 is admitted not to follow from Assumptions 2.1–2.3, but this is a domain restriction, not a dependence on the conclusion. No step reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 3 invented entities

The central claim rests on imported machinery (MDE uniqueness [43], shape analysis [5,6], zigzag framework [25,16,36]) and on one paper-specific algebraic miracle (coplanarity of the three critical directions). No fitted parameters enter: β, α and R are computed from the MDE solution, and the Figure 1 curve is a genuine prediction, not a fit. The paper is self-citation-heavy (this group built most of the framework), but the cited results are independently established, so circularity is low.

axioms (5)
  • domain assumption Shape analysis: the self-consistent density ρ (under Assumptions 2.1–2.4) has only regular square-root edges and cubic-root cusps
    Imported from [5,6]; the whole regime classification in Corollary 2.8 (bulk/edge/cusp/almost-cusp/small-gap) presupposes that no other singularity types occur.
  • standard math Existence, uniqueness and stability of the matrix Dyson equation solution M(z) for the data pair (D,S) [43]
    Foundation of the deterministic approximation; the stability operators B_{z,w} and their spectral properties are derived from it.
  • domain assumption Assumptions 2.1–2.5: bounded expectation, self-energy norm bounds, fullness, bounded self-consistent Green function, higher-order cumulant bounds with sparse correlation graph (|N(α)| ≤ N^{1/2−μ})
    Delimit the class of correlated mean-field models. Assumption 2.4 is admitted (§2.1) not to follow from 2.1–2.3; it makes the admissible set I non-empty and is needed for the I=R version of Theorem 2.6.
  • standard math Zigzag machinery: characteristic-flow dynamics (4.2)–(4.5), Ornstein–Uhlenbeck zig/zag flows (5.1)–(5.2), and multivariate cumulant expansion (Prop. 7.3) behave as established in [25,16,36,34]
    Proof framework; the paper adapts, rather than re-derives, these tools.
  • ad hoc to paper Coplanarity of the three critical directions and the cancellation identities (4.53)–(4.56) hold at the stated O(ρ²)/O(ρ_1 ρ_2) accuracy for general S
    The paper's own centerpiece ('we do not see any intuitive or conceptual reason why these three directions remain coplanar in general, but they do', §1.3). Load-bearing: the anomalous 1/N edge rate is the output of these cancellations; the authors admit there is no shortcut heuristic.
invented entities (3)
  • Anomalous fluctuation mode direction R(z_1,z_2) and its real-axis projection R_E independent evidence
    purpose: Identifies the codimension-one subspace of observables whose overlap fluctuations do not follow the generic 1/N scale; drives the N^{-1/4}-variance cusp fluctuations.
    Defined from the MDE solution via (3.24)–(3.26); predicts a falsifiable N^{-1/2} variance at cusps, verified by the authors' own GUE simulation (Figure 1).
  • Improved regularization •A and microcanonical correction y_min independent evidence
    purpose: Correct microcanonical ensemble at small density; the shift from the naive ˚A to •A converts the (N|β|)^{-1} bound into (Nα²)^{-1}.
    Derived as the minimizer of (1.8) from first principles, not postulated ad hoc; it is the object whose real-axis limit is ⟨A ℑM(γ_j)⟩/⟨ℑM(γ_j)⟩.
  • Improved stability factor α² = Δ^{2/3} + |β| independent evidence
    purpose: Replaces the inverse-stability factor 1/|β| in the two-resolvent M-bounds; the single scalar encoding the edge-vs-cusp anomaly.
    Explicit function of derived quantities (gap Δ and stability eigenvalue β); α^{-2} is the theoretical curve matched in Figure 1, i.e., an out-of-sample prediction.

pith-pipeline@v1.3.0-alltime-deepseek · 79110 in / 18757 out tokens · 185378 ms · 2026-08-02T08:48:43.679479+00:00 · methodology

0 comments
read the original abstract

We prove the Eigenstate Thermalisation Hypothesis (ETH), also known as Quantum Unique Ergodicity (QUE), for large $N\times N$ mean-field random matrices with general correlation structure. We identify the microcanonical ensemble and establish the optimal fluctuation scale of eigenvector overlaps around it. Our results invalidate the inverse-density scaling predicted by Feingold and Peres [Phys. Rev. A 34, 591 (1986)] (and incorporated into Srednicki's ansatz [Phys. Rev. E 50, 888-901 (1994)]) in the physics literature of quantum chaos, based upon popular semiclassical theory, and uncover the genuine mechanism which relies on multi-resolvent local laws. Although fluctuations are expected to increase as the density of states vanishes, and indeed scale as $N^{-1/2}$ in the special cusp regime, rather than $N^{-1}$ in the bulk, we find, unexpectedly, that the same $N^{-1}$ rate persists at regular spectral edges. Hence, generically, in the absence of cusps, the entire eigenbasis fluctuates on the same scale as a Haar unitary. This anomaly stems from delicate cancellations in the solution of the underlying matrix Dyson equation, which form the core of our analysis.

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Reference graph

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