MathAudit

Paper metadata

Original submission
July 19, 2023
Latest revision
October 19, 2026 · version 2
Claimed MSC
60J10, 37A25
arXiv categories
math.PR
Paper license
CC BY 4.0 ↗
Comments
17 pages, 3 figures. Minor update for typos, correct compile

Abstract

We analyze the absolute spectral gap of Markov chains on graphs obtained from a cycle of nn vertices and perturbed only at approximately n1/ρn^{1/ρ} random locations with an appropriate, possibly sparse, interconnection structure. Together with a strong asymmetry along the cycle, the gap of the resulting chain can be bounded inversely proportionally by the longest arc length (up to logarithmic factors) with high probability, providing a significant mixing speedup compared to the reversible version.

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated October 23, 2026
01Statements2 reported findingsNo statement issues found

The two principal absolute-spectral-gap bounds follow under the stated polynomial size separation and symmetric interconnection-gap hypotheses. The paper explicitly distinguishes this spectral conclusion from a finite-time mixing bound for nonnormal chains.

Theorems 5–6No issue found

The exponent threshold and conditioning losses are consistent

PDF pp. 3, 12–14 · arXiv:2307.09949v2

Lemma 11 gives δk=O(log⁡−(γ−1−α)k)\delta_k=O(\log^{-(\gamma-1-\alpha)}k). Choosing 3<η<(γ−1−α)/23<\eta<(\gamma-1-\alpha)/2 makes the large-angle discrepancy incompatible with Lemma 19 exactly when γ>7+α\gamma>7+\alpha. The large-angle failure probability is exponentially small in k/log⁡θkk/\log^\theta k plus O(k1−M)O(k^{1-M}). Taking M>1+ρuM>1+\rho_u absorbs the conditioning factor nn. The small-angle conditioned argument uses the monotonicity coupling rather than multiplying a mere O(1/k)O(1/k) error by nn.

Proposition 7; Lemmas 11, 15, 17No issue found

The compressed eigenproblem and the conditioned model match

PDF pp. 4, 7 and 9–11 · arXiv:2307.09949v2

The endpoint variables give CAx=D(μ)xCAx=D(\mu)x without needing to divide by μ\mu, so even the zero-eigenvalue case is covered. Symmetry of AA supplies the singular-value contraction on the perpendicular component. Rooting a removed edge removes the rotational ambiguity between uniform compositions and uniformly cut cycles. The weighted coupling has row sum mm and column sum m+1−km+1-k, hence uniform marginals and coordinatewise domination. These are the properties required by the two model comparisons.

02Proofs3 reported findingsNo proof issues found

The moment, eigenvector, coupling, and block-concentration arguments support both main theorems. Some explanatory symbols and a rate sentence need local correction; the proof later uses the correct rate and complement event.

Sections 3–4No issue found in the reviewed proof

The small-angle obstruction is uniform over all candidate eigenvalues

PDF pp. 5–12 · arXiv:2307.09949v2

On the bounded-arc event, the geometric-series error is at most half its linear term. Cauchy–Schwarz and the perpendicular-component estimate then force the squared empirical mean to be at most 12δk12\delta_k times the empirical second moment. Geometric moments exclude this in the independent model. For the cycle model the mean is fixed, while Lemma 17 and the Berry–Esseen bound control the conditioned second moment. This avoids an insufficient direct conditioning of Chebyshev's inequality.

Lemmas 19–20; Proposition 21No issue found in the reviewed proof

The large-angle argument uses a matching prior probability lemma

PDF pp. 12–14 · arXiv:2307.09949v2

The angle lemma in the cited earlier research paper is stated as Proposition 10 in its final public preprint, arXiv:1509.01431v4, with the same geometric arc distribution, angle cutoff, block length and cosine-positive-part deficit. Its proof uses the same polynomial relation between LL and kk. Independent blocks amplify probability at least 1/31/3 to the required exponential tail. The earlier paper's Lemma 15 also gives the needed asymptotic lower bound 1/n1/n for the conditioning event. These inputs, together with Lemma 19, yield the claimed contradiction.

Gerencsér–Hendrickx, final public preprint, Proposition 10 and Lemma 15 ↗
Lemma 11 discussion; Proposition 12(ii)Suspected typo

A rate sentence and complement-event symbols should be corrected

PDF pp. 7 and 12 · arXiv:2307.09949v2

For α>0\alpha>0, the denominator defining δk\delta_k is not bounded away from zero: its order is log⁡−αk\log^{-\alpha}k, so δk\delta_k is not generally comparable to ϵk\epsilon_k. The correct rate is explicitly used on p. 14. Also the vanishing probability on p. 12 is P(S(M)c)\mathbb P(S(M)^c), not P(S(M))\mathbb P(S(M)); the surrounding argument consistently concerns failure of the bounded-arc event. These corrections do not change the estimates or the principal conclusions.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2307.09949v2
Authors listed
Balázs Gerencsér, Julien M. Hendrickx
Audit date
October 23, 2026
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