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arxiv: 2511.06196 · v3 · submitted 2025-11-09 · 🧮 math.PR · math-ph· math.MP

Central limit theorem for high temperature spin models via martingale embedding

Xiao Fang , Yang Xie , Yi-Kun Zhao This is my paper

Pith reviewed 2026-05-18 00:24 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP MSC 60F0582B2060E15
keywords central limit theoremmartingale embeddingPoincaré inequalityIsing modelSherrington-Kirkpatrick modelWasserstein distancehigh temperaturespin systems
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The pith

Martingale embeddings yield non-asymptotic Wasserstein bounds for central limit theorems on projections of spin vectors obeying a Poincaré inequality.

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

The paper proves a central limit theorem for one-dimensional projections of high-dimensional random vectors taking values in {-1,1}^n that satisfy a Poincaré inequality. Using martingale embeddings, the authors derive an explicit non-asymptotic error bound in 2-Wasserstein distance controlled by two-point and three-point functions. The result is applied to the Ising model with finite-range interactions, the ferromagnetic Ising model under the Dobrushin condition, and the Sherrington-Kirkpatrick model at high temperature, all with heterogeneous external fields allowed. A reader would care because the bounds give quantitative rates of convergence to normality for dependent discrete systems without invoking strong mixing or uniform bounds.

Core claim

For any unit vector v, the law of the linear functional <X, v> is within an explicit 2-Wasserstein distance of the standard Gaussian, where the distance is bounded by a combination of the L1 norms of the two-point and three-point correlation functions of the coordinates of X.

What carries the argument

Martingale embedding of the projection process, which constructs a continuous-time martingale whose increments reproduce the conditional variances and permits transfer of Gaussian approximation bounds via coupling or Stein-type arguments.

If this is right

  • The central limit theorem holds for Ising models with finite-range interactions at sufficiently high temperature.
  • The result applies to ferromagnetic Ising models satisfying the Dobrushin condition.
  • The Sherrington-Kirkpatrick model obeys the central limit theorem at high enough temperature.
  • All three applications remain valid under arbitrary heterogeneous external fields.

Where Pith is reading between the lines

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

  • The correlation-function form of the error may allow direct numerical verification of the bound on moderate-sized lattices.
  • The same embedding construction could be examined for other discrete distributions that obey Poincaré inequalities, such as certain Markov random fields on graphs.
  • Quantitative CLT rates of this type supply a route to error control in mean-field approximations for statistical mechanics models.

Load-bearing premise

The high-dimensional vector in {-1,1}^n satisfies a Poincaré inequality that bounds the variance of any function by its average squared gradient.

What would settle it

A concrete high-temperature Ising configuration on a finite graph where the measured 2-Wasserstein distance from the projected law to the matching Gaussian exceeds the explicit upper bound formed from its two-point and three-point functions.

read the original abstract

We use martingale embeddings to prove a central limit theorem (CLT) for one-dimensional projections of high-dimensional random vectors in $\{-1,1\}^n$ satisfying a Poincar\'e inequality. We obtain a non-asymptotic error bound involving two-point and three-point functions for the CLT in 2-Wasserstein distance. We present three illustrative applications: Ising model with finite-range interactions, ferromagnetic Ising model under the Dobrushin condition, and the Sherrington-Kirkpatrick spin glass model at sufficiently high temperature. In all the examples, we allow heterogeneous external fields.

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

1 major / 2 minor

Summary. The paper proves a central limit theorem for one-dimensional projections of random vectors in {-1,1}^n that satisfy a Poincaré inequality. It uses martingale embeddings to obtain a non-asymptotic error bound in 2-Wasserstein distance expressed explicitly in terms of two-point and three-point correlation functions. Three applications are presented: the Ising model with finite-range interactions, the ferromagnetic Ising model under the Dobrushin condition, and the Sherrington-Kirkpatrick model at high temperature, all allowing heterogeneous external fields.

Significance. If the central derivation holds, the result supplies a concrete quantitative CLT with explicit error terms for projections of dependent spin systems. The explicit dependence of the bound on two- and three-point functions, together with the verification of the Poincaré assumption in three standard high-temperature regimes, is a strength that makes the bound directly usable. The martingale-embedding approach adds a new technical route to such non-asymptotic statements in probability theory.

major comments (1)
  1. §3, Theorem 3.2: the non-asymptotic W2 bound is stated to depend on the Poincaré constant only through the correlation functions, but the proof sketch does not display the precise factor by which the constant multiplies the three-point term; this factor must be tracked explicitly to confirm that the bound vanishes in the high-temperature regime for the SK model.
minor comments (2)
  1. §2.1, Definition 2.1: the precise normalization of the two-point function (whether centered or not) should be restated when it first appears, to avoid ambiguity when the external fields are heterogeneous.
  2. Figure 1: the caption does not indicate whether the plotted error is the theoretical bound or a numerical approximation; this should be clarified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation for minor revision. We respond to the major comment below.

read point-by-point responses

  1. Referee: [—] §3, Theorem 3.2: the non-asymptotic W2 bound is stated to depend on the Poincaré constant only through the correlation functions, but the proof sketch does not display the precise factor by which the constant multiplies the three-point term; this factor must be tracked explicitly to confirm that the bound vanishes in the high-temperature regime for the SK model.

    Authors: We thank the referee for this observation. In the proof of Theorem 3.2 the Poincaré constant enters the three-point term through the L² bound on the martingale increments obtained from the embedding construction and the associated integration-by-parts identity for functions on the hypercube. The precise multiplicative factor in front of the summed three-point correlations is exactly the Poincaré constant C_P (with no additional numerical prefactor beyond the universal constants already present in the two-point term). We will revise the proof sketch in Section 3 to display this dependence explicitly. With the factor stated, the bound vanishes in the high-temperature regime for the SK model because C_P remains O(1) while the three-point functions are controlled by the high-temperature assumption. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external martingale embedding and Poincaré inequality

full rationale

The paper establishes a non-asymptotic W2 CLT bound for projections of {-1,1}^n vectors satisfying a Poincaré inequality, using martingale embedding to control dependence and expressing errors explicitly via two- and three-point correlation functions. The Poincaré assumption is verified separately in the three high-temperature regimes (finite-range Ising, Dobrushin, high-temp SK), but the central bound itself is not obtained by fitting parameters to the target CLT or by reducing to self-citations. No load-bearing step reduces by construction to the paper's own inputs; the embedding technique and inequality are independent external tools, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the random vector satisfies a Poincare inequality, which is a standard but non-trivial domain assumption for the models considered; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The high-dimensional random vector in {-1,1}^n satisfies a Poincare inequality.
    Invoked to control dependence and enable the martingale embedding argument for the CLT bounds.

pith-pipeline@v0.9.0 · 5392 in / 1257 out tokens · 36674 ms · 2026-05-18T00:24:30.193909+00:00 · methodology

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Forward citations

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    stat.ML 2026-05 unverdicted novelty 5.0

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5 extracted references · 5 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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