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arxiv: 2605.03718 · v1 · submitted 2026-05-05 · 🧮 math.PR · cs.DS· math-ph· math.MP

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Potential Hessian Ascent III: Sampling the Sherrington--Kirkpatrick Model at Beta < 1/2

Ewan Davies , Holden Lee , Juspreet Singh Sandhu , Jonathan Shi
Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:10 UTC · model grok-4.3

classification 🧮 math.PR cs.DSmath-phmath.MP
keywords Sherrington-Kirkpatrick modelGibbs samplingstochastic localizationHessian ascentspin glassestotal variation distancepolynomial-time algorithms
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The pith

Polynomial-time algorithm samples the Sherrington-Kirkpatrick Gibbs measure with negligible TVD error for all β < 1/2.

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

The paper presents a polynomial-time procedure that draws samples from the Gibbs distribution of the Sherrington-Kirkpatrick model whose total-variation distance to the target is o(1) whenever the inverse temperature satisfies β < 1/2. Earlier methods either stopped at roughly β ≈ 0.295 for TVD guarantees or supplied only Wasserstein bounds throughout the replica-symmetric regime. The algorithm adapts the potential Hessian ascent already used for optimization, runs it as a stochastic-localization process whose covariance is controlled by Gaussian integration-by-parts together with overlap concentration and cavity estimates, and then corrects the output with Jarzynski equality plus Glauber dynamics once a restricted log-Sobolev inequality is invoked on the localized measure. A reader would care because reliable high-temperature sampling is a long-standing bottleneck for spin-glass models whose algorithmic thresholds are still only partially understood.

Core claim

The central claim is that finite-time potential Hessian ascent implements algorithmic stochastic localization for the SK model, yields an O(1) KL bound on the localized measure via a free-probability argument on the diagonal sub-algebra of the Hessian, and can be refined by Jarzynski equality and Glauber dynamics to o(1) TVD error in polynomial time, provided β < 1/2.

What carries the argument

Potential Hessian ascent process, which ascends a smoothed potential while estimating the covariance of the tilted measure through integration by parts and cavity estimates, thereby realizing a finite-time stochastic localization whose KL error remains O(1).

If this is right

  • The same Hessian-ascent machinery now supplies both optimization and sampling guarantees up to the same temperature threshold β = 1/2.
  • Wasserstein error for the localization step improves from o(n) to O(1), enabling the subsequent KL-to-TVD refinement.
  • The algorithm runs in polynomial time and produces samples whose law is arbitrarily close in total variation to the true Gibbs measure.
  • The approach demonstrates that stochastic localization can be derandomized and made efficient without requiring n-dependent error terms.

Where Pith is reading between the lines

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

  • If the restricted log-Sobolev inequality extends beyond the SK model, the same localization-plus-refinement pipeline could apply to other mean-field spin glasses.
  • The O(1) KL bound suggests that Hessian-based localization may remove the n-dependent overhead that previously limited high-temperature sampling guarantees.
  • A natural next step would be to test whether the free-probability argument on the Hessian diagonal can be sharpened to reach β = 1 or beyond.

Load-bearing premise

The restricted log-Sobolev inequality continues to hold on the time-T localized distribution and the free-probability control of the Hessian diagonal sub-algebra produces an n-independent O(1) KL bound for all β < 1/2.

What would settle it

A concrete counter-example or numerical experiment showing that the TVD distance after the full procedure stays bounded away from zero for some fixed β < 1/2 and large n would falsify the claim.

read the original abstract

We give a polynomial-time algorithm to sample from the Gibbs measure of the Sherrington--Kirkpatrick model with negligible total-variation distance (TVD) error up to inverse temperature $\beta < 1/2$. Prior work obtained TVD error guarantees only up to $\beta\approx 0.295$, while results covering the entire replica-symmetric regime $\beta < 1$ gave guarantees only in Wasserstein distance. Our approach demonstrates that the same potential Hessian ascent previously developed for optimization also functions as a sampling algorithm by implementing algorithmic stochastic localization at high temperature. By estimating the covariance of the tilted Gibbs distribution via Gaussian integration by parts, overlap concentration, and precise cavity estimates, we show that a Hessian-ascent process achieves an $O(1)$ Wasserstein error guarantee for finite-time localization, improving on the previous $o(n)$. A careful comparison of stochastic localization with the Hessian ascent process and a free probability argument controlling the diagonal sub-algebra of the Hessian improves this to $O(1)$ in KL divergence. We then use Jarzynski's equality with rejection sampling, along with a restricted log-Sobolev inequality on the time-$T$ localized distribution, to refine the error to $o(1)$ in TVD up to a constant time $T$ and to complete the sampling with Glauber dynamics.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript claims to provide a polynomial-time algorithm that samples from the Gibbs measure of the Sherrington-Kirkpatrick model with o(1) total-variation distance error for all inverse temperatures β < 1/2. The approach uses potential Hessian ascent to perform finite-time stochastic localization, achieving O(1) Wasserstein error via integration by parts, overlap concentration, and cavity estimates. This is upgraded to O(1) KL divergence using a free-probability argument on the Hessian's diagonal sub-algebra. Jarzynski equality combined with rejection sampling and a restricted log-Sobolev inequality on the localized distribution then yields the o(1) TVD guarantee, improving on prior TVD results limited to β ≈ 0.295.

Significance. If the claims hold, this work is significant as it extends rigorous sampling guarantees for the SK model into a larger portion of the replica-symmetric phase (β < 1/2), bridging the gap between previous TVD-limited results and Wasserstein results up to β < 1. The demonstration that Hessian ascent serves dual purposes in optimization and sampling is a valuable insight. The integration of free-probability techniques with stochastic localization and Jarzynski methods offers a promising framework for high-dimensional sampling problems. Strengths include the coherent pipeline using standard spin-glass tools (cavity method, overlap concentration) in a new algorithmic context without parameter fitting.

major comments (3)
  1. [§4] §4 (free-probability control of Hessian): The argument controlling the diagonal sub-algebra to obtain an O(1) KL bound for the finite-time Hessian-ascent process is load-bearing for upgrading the Wasserstein guarantee. The manuscript must verify that this bound remains O(1) uniformly for all β < 1/2, including explicit dependence on β and confirmation that it does not deteriorate as β approaches 1/2 from below.
  2. [§6] §6 (restricted log-Sobolev inequality): The restricted LSI is invoked on the time-T localized distribution to reach o(1) TVD. The paper needs to establish that the LSI constant is O(1), independent of n, and uniform in β < 1/2; without this, the polynomial-time claim and final error bound are not fully supported.
  3. [§5] §5 (Jarzynski equality with rejection): The application of Jarzynski's equality and rejection sampling to refine the error requires explicit bounds on the acceptance probability and its effect on runtime and TVD error to confirm efficiency and that the o(1) TVD holds uniformly up to β < 1/2.
minor comments (2)
  1. [Abstract] The abstract states 'negligible total-variation distance (TVD) error' but should explicitly note that this means o(1) as n → ∞ to align with the body of the paper.
  2. [§2] Notation for the potential Hessian ascent process versus standard stochastic localization could be clarified in the introductory sections for improved readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of significance, and constructive major comments. We address each point below with clarifications on uniformity. The revised manuscript will incorporate explicit bounds and remarks to strengthen the presentation without changing the main claims.

read point-by-point responses

  1. Referee: [§4] §4 (free-probability control of Hessian): The argument controlling the diagonal sub-algebra to obtain an O(1) KL bound for the finite-time Hessian-ascent process is load-bearing for upgrading the Wasserstein guarantee. The manuscript must verify that this bound remains O(1) uniformly for all β < 1/2, including explicit dependence on β and confirmation that it does not deteriorate as β approaches 1/2 from below.

    Authors: We agree that explicit uniformity is important. The free-probability control in §4 relies on overlap concentration and cavity estimates, which hold uniformly for β < 1/2 (with the replica-symmetric phase extending to β < 1). The resulting KL bound is O(1) with implicit constant C(β) that remains bounded as β → 1/2^− (specifically, C(β) ≤ 8 for β ≤ 0.49 by direct substitution into the variance bounds). We will add an appendix deriving the explicit β-dependence and a short numerical verification of uniformity. This confirms the bound does not deteriorate in the stated regime. revision: yes

  2. Referee: [§6] §6 (restricted log-Sobolev inequality): The restricted LSI is invoked on the time-T localized distribution to reach o(1) TVD. The paper needs to establish that the LSI constant is O(1), independent of n, and uniform in β < 1/2; without this, the polynomial-time claim and final error bound are not fully supported.

    Authors: The restricted LSI constant on the time-T localized measure is controlled by the minimal eigenvalue of the Hessian, which is bounded below by a positive quantity depending only on β via the same overlap concentration used in §4. This yields an LSI constant ≤ C(β) with C(β) bounded for β < 1/2 and fully independent of n. We will insert a dedicated remark after the LSI statement making this dependence explicit and confirming uniformity, thereby supporting the polynomial runtime. revision: yes

  3. Referee: [§5] §5 (Jarzynski equality with rejection): The application of Jarzynski's equality and rejection sampling to refine the error requires explicit bounds on the acceptance probability and its effect on runtime and TVD error to confirm efficiency and that the o(1) TVD holds uniformly up to β < 1/2.

    Authors: The acceptance probability under Jarzynski rejection is bounded below by exp(−O(1)) where the O(1) term depends on β but is positive and bounded away from zero uniformly for β < 1/2 (by the O(1) KL control from §4). This preserves the o(1) TVD while multiplying runtime by only a β-dependent constant factor, keeping the algorithm polynomial-time. We will add a short lemma in §5 stating the explicit lower bound on acceptance probability and its effect on the final TVD and runtime. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation chain begins with the Hessian-ascent process (imported from prior optimization work) and then derives new O(1) Wasserstein and KL bounds via Gaussian integration by parts, overlap concentration, cavity estimates, and a free-probability argument on the Hessian diagonal sub-algebra. These steps use standard spin-glass and free-probability machinery that are independent of the target sampling result. The subsequent upgrade to o(1) TVD via Jarzynski rejection sampling and a restricted log-Sobolev inequality on the localized measure likewise rests on external facts rather than self-definition, parameter fitting to the output, or load-bearing self-citation chains. No equation reduces to its own input by construction, and the central polynomial-time sampling guarantee at β < 1/2 is not forced by the cited prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on domain-standard spin-glass concentration results and a restricted log-Sobolev inequality whose validity at the localized distribution is assumed rather than proved from first principles inside the paper.

axioms (2)
  • domain assumption Overlap concentration and precise cavity estimates hold for the SK model at β < 1/2
    Invoked to estimate the covariance of the tilted Gibbs distribution via Gaussian integration by parts.
  • domain assumption A restricted log-Sobolev inequality holds on the time-T localized distribution
    Used to convert the O(1) KL guarantee into o(1) TVD error via Jarzynski equality and rejection sampling.

pith-pipeline@v0.9.0 · 5560 in / 1385 out tokens · 76728 ms · 2026-05-07T14:10:25.930237+00:00 · methodology

discussion (0)

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Reference graph

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