A CLT for the total energy of the two-dimensional critical Ising model

Jianping Jiang

REVIEW 3 major objections 3 minor 12 references

A CLT for the total energy of the two-dimensional critical Ising model

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The total energy of the critical two-dimensional Ising model on a growing rectangle converges to a Gaussian with mean zero and variance 8/π.

desk verdict The stress-test's 'false inequality' is a misreading of formula (17); with the intended fraction, gθ ≥ 0 on (0,β_c] and the paper's CLT argument holds together. read the letter →

arxiv 1908.06704 v1 pith:Z554TZJK submitted 2019-08-19 math.PR math-phmath.MP

Jianping Jiang This is my paper · ORCID

classification math.PRmath-phmath.MP MSC 60F0582B2082B27

keywords Isingmodelcriticaltemperaturecentrallimittheoremtotalenergypartitionfunctionfreeboundaryconditionperiodictwo-dimensionallattice

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a central limit theorem for the total energy (Hamiltonian) of the critical two-dimensional Ising model on a $[1,2N]\times[1,2M]$ rectangle, with periodic boundary condition horizontally and free boundary condition vertically, when both $M$ and $N$ tend to infinity simultaneously. It provides a simultaneous-limit CLT for this observable and isolates the free-boundary contribution $(4/\pi)N\ln N$ to the mean energy. After centering by $4\sqrt{2}MN-(4/\pi)N\ln N$ and scaling by $\sqrt{4MN\ln N}$, the energy has moment generating function tending to $e^{4t^2/\pi}$, hence converges weakly to a Gaussian with mean $0$ and variance $8/\pi$. The result matters because it shows precisely how boundary conditions shape the energy fluctuations at criticality and because the same mechanism is a step toward understanding the scaling limit of the energy field.

What carries the argument

The central object is the exact partition function of the periodic-free rectangle, written as a product over Fourier modes $\theta=\pi(2n-1)/(2N)$, with $\cosh\gamma_\theta=\coth(2\beta)\cosh(2\beta)-\cos\theta$ and $\gamma_\theta\ge 0$. The moment generating function is the ratio $Z_{M,N}(\beta-s)/Z_{M,N}(\beta)$, and the proof expands $\ln Z$ around the critical inverse temperature $\beta_c$. The essential mechanism is that the mode sum for $L_3$ produces the variance $8/\pi$ through a sum of $16(3-4\cos\theta+\cos^2\theta)^{-1/2}$ behaving like $4N\ln N$, while $L_4$ contributes the boundary term $-(4/\pi)N\ln N$ to the centering.

What would settle it

Compute $g_\theta=(\coth(2\beta)-\cosh(2\beta)\cos\theta)/\sinh\gamma_\theta$ on a fine grid of $\theta\in(0,\pi]$ for $\beta$ just below $\beta_c$; a negative value would disprove the unproved inequality on which Lemma 9 rests. Alternatively, simulate the critical Ising model on $2N\times2N$ periodic-free rectangles for increasing $N$ and compare the empirical distribution of $(E_{M,N}+4\sqrt{2}MN-(4/\pi)N\ln N)/\sqrt{4MN\ln N}$ with a standard Gaussian; a systematic departure in variance from $8/\pi$ would refute the theorem.

Watch

Extended reading notes

Core claim

Theorem 1 asserts that for each $t\ge 0$, $\lim_{N\to\infty}\langle e^{t\hat E_{M,N}}\rangle_{\beta_c}=e^{4t^2/\pi}$, where $\hat E_{M,N}=(E_{M,N}+4\sqrt{2}MN-(4/\pi)N\ln N)/\sqrt{4MN\ln N}$, provided $N(\ln\ln N)^2/(M\ln N)\to 0$. Consequently $\hat E_{M,N}$ converges weakly to a Gaussian with mean $0$ and variance $8/\pi$. The centering contains both the bulk term $4\sqrt{2}MN$ and the free-boundary term $(4/\pi)N\ln N$; the proof obtains the Gaussian limit by expanding the logarithm of the moment generating function into four parts $L_1,L_2,L_3,L_4$ coming from the log partition function.

Load-bearing premise

The proof assumes without proof that the function $g_\theta=(\coth(2\beta)-\cosh(2\beta)\cos\theta)/\sinh\gamma_\theta$ is nonnegative for every $\beta\in(0,\beta_c]$ and every $\theta\in(0,\pi]$, because this keeps the denominator of the second-derivative estimate at least one in absolute value.

Editorial extensions

If this is right

The same CLT holds whenever $M\ge N/(\ln N)^\alpha$ for $\alpha\in[0,1)$, so the result covers squares and moderately rectangular boxes; in particular $M=N$ is allowed.
The variance grows as $(32/\pi)MN\ln N$, a factor $\ln N$ larger than the $MN$ bulk scale, showing that the free vertical boundaries dominate the energy fluctuations.
The proof identifies the exact leading free-boundary contribution $(4/\pi)N\ln N$ to the mean energy, consistent with conformal-covariant behavior of the energy density near the boundary.
The Gaussian limiting fluctuation, together with the $|z_1-z_2|^{-2}$ decay of energy correlations, suggests that the critical energy field does not have a finite white-noise scaling limit in the usual probabilistic sense.
The theorem gives the first simultaneous-limit statement of this type for the critical Ising energy, removing the need to send $M\to\infty$ before $N\to\infty$ as in earlier work.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the denominator bound used in Lemma 9 can be extended to both signs of $t$, the same Taylor-expansion scheme would likely yield the CLT for all real $t$ and, as the paper conjectures, for other boundary conditions such as free or all-plus.
A natural testable extension is to subtract the boundary term and study the distributional scaling limit of the energy field after renormalization; the paper only conjectures a white-noise limit, so a precise theorem would require new mode-sum estimates.
The ratio-of-partition-functions method may transfer to other exactly solvable lattice models whose partition function factors over Fourier modes, converting a CLT for additive observables into a Taylor-expansion estimate for mode sums.
The unproved sign condition on $g_\theta$ is checkable by direct numerical evaluation; if it fails, the proof has a gap, although the theorem itself might still be true by a sharper two-sided estimate.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Jianping Jiang proves a simultaneous-limit CLT for the total energy of the critical 2D Ising model on a 2N×2M rectangle, with the free-boundary N ln N term made explicit. The math looks right to me, and the stress-test's main objection does not survive reading the actual formula. Equation (17) is plainly meant to be (coth(2β) − cosh(2β) cosθ)/sinh γθ, not coth − (cosh cos)/sinh. With the correct reading, at β_c one gets gθ = √2(1−cosθ)/√((1−cosθ)(3−cosθ)) ≥ 0, and since the numerator decreases in β, gθ ≥ 0 on (0, β_c]. So the 'gθ < 0 for small θ' example is an artifact of misparsing the fraction. The same correct reading makes (54) and (74) consistent.

Now the substance. The new content: De Coninck had the iterated limit M→∞, N→∞; here the limits are simultaneous under M ≥ N/(ln N)^α (α<1), and the expected energy has the boundary term (4/π)N ln N subtracted. That term is derived from the partition function, not inserted. The method is a clean use of the McCoy–Wu product formula: Taylor expand the three parts L1, L3, L4, control sums with Lemma 7's exponential-in-M estimates and harmonic bounds. I did not find fitted parameters or circular reasoning.

Soft spots, in proportion: The passage from (10) to weak convergence is one sentence, citing Problem 30.4 of Billingsley. That is terse; the argument requires continuity of the limiting MGF at 0 to rule out mass escaping to −∞. I believe the cited problem covers it, but a referee will want the one-line tightness explanation. Lemma 4's proof is compressed and the algebra behind (34)–(35) is easy to stumble over, but it is correct. Nothing else looks broken.

The only real presentation issue: the paper only proves the MGF for t≥0, and Remark 5 acknowledges that t<0 needs more work. For a CLT, one-sided convergence plus continuity at 0 is enough, but the authors should say so explicitly.

Who is it for: people working on critical Ising fluctuations, energy field scaling limits, and numerical studies of finite-size corrections. It deserves a serious referee and probably acceptance after minor revisions.

Referee Report

3 major / 3 minor

Summary. The paper proves a central limit theorem for the total energy of the two-dimensional critical Ising model on a (2N) x (2M) rectangle, with periodic boundary condition in the horizontal direction and free boundary condition in the vertical direction. Theorem 1 states that, under the growth condition N(ln ln N)^2/(M ln N) -> 0, the normalized energy (E_{M,N} + 4√2 MN - (4/π)N ln N)/√(4MN ln N) has moment generating function converging to e^{4t^2/π} for each t ≥ 0, and hence converges weakly to a Gaussian with variance 8/π. The proof follows the classical partition-function-ratio method: the moment generating function is written as a ratio of partition functions at shifted inverse temperatures, and the exact McCoy-Wu formula (15) is expanded in Taylor series around β_c. The main work is in four estimates (26)-(29) for the components L_1,...,L_4 of the log partition function.

Significance. If the theorem is correct, it provides the first simultaneous-limit CLT for the critical Ising energy with a precise free-boundary contribution (4/π)N ln N, going beyond the earlier iterated limit of De Coninck. The derivation is coherent in outline and uses no fitted parameters: the constants 4/π and 8/π arise from the asymptotics of the exact partition function, and the paper connects the result to the expected non-existence of a usual probabilistic scaling limit for the energy field. These are genuine strengths. However, the proof as written contains a false inequality and an algebraic inconsistency in the two most delicate estimates, so the central claim is not currently established.

major comments (3)

[Lemma 9, Eq. (72)] The assertion that g_θ ≥ 0 for every β ∈ (0, β_c] and every θ ∈ (0, π] is false. From (17) and Lemma 2, at β = β_c one has g_θ = √2 − √2 cos θ / √((1−cos θ)(3−cos θ)), which tends to −∞ as θ ↓ 0. Moreover, the same negativity occurs for β immediately below β_c and sufficiently small θ: at θ = 0, g_0 = coth(2β) − cosh(2β)/√(A(A−2)) with A = coth(2β) cosh(2β) > 2, and this tends to −∞ as β ↑ β_c. Since (72) is used to obtain |1 + e^{−4Mγ} + (1 − e^{−4Mγ})g_θ| ≥ 1, the bound (73) and the final estimate (78) are not established. Lemma 9, and therefore the proof of (29), rests on an invalid inequality and needs a replacement argument.
[Eq. (54), Lemma 8] Formula (54) for f'_θ at β_c is inconsistent with (17), (50), and (52). At β_c, (52) gives g'_θ = −2(1 + cos θ) csch(γ_θ), so (50) yields f'_θ|β_c = −2(1 − e^{−4Mγ_θ})(1 + cos θ) csch(γ_θ) / [1 + e^{−4Mγ_θ} + (1 − e^{−4Mγ_θ}) g_θ], with g_θ = √2 − √2 cos θ csch(γ_θ). The denominator written in (54) instead contains √2 (1 − cos θ)^{1/2} (3 − cos θ)^{-1/2}, which is not equal to g_θ (for small θ the former tends to 0 while g_θ tends to −∞). Therefore the decomposition in (64)-(69) and the proof of Lemma 8 do not establish the claimed limit (63). This is a load-bearing algebraic error, not a mere typo, because Lemma 8 supplies the free-boundary shift in (29).
[Theorem 1, proof after (10)] The deduction of weak convergence from (10) is not justified as written. The paper cites Problem 30.4 of [2], but the standard result requires convergence of moment generating functions on an interval around 0 (or for all real t). Here (10) is proved only for t ≥ 0, and convergence of positive-t moment generating functions does not control left tails and does not by itself imply tightness or weak convergence. The author should either extend the moment generating function computation to t in a neighborhood of 0 or provide a separate tightness/characteristic-function argument.

minor comments (3)

[Proof of (28), around Eq. (47)] In the sentence defining the continuous function, the denominator is printed as 3 − 4 cos θ − cos² θ; it should be 3 − 4 cos θ + cos² θ, as in (47) and Lemma 2.
[Eq. (78)] The last term in (78), C21 M² csch²(γ_θ)/(M N ln N), is more clearly written as C21 M csch²(γ_θ)/(N ln N); the displayed form is confusing though mathematically equivalent.
[Lemma 4, inequality (33)] The proof of (33) says it follows from the monotonicity of csch and the mean value theorem; for clarity, the reader needs the explicit bound on |β − β_c| in terms of 1/√(4M N ln N), which is implicit and could be stated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CLT derivation is self-contained apart from the standard external McCoy–Wu partition-function formula.

full rationale

The proof of Theorem 1 is a direct analytic derivation: Lemma 3 exactly expresses the moment-generating function as Z(βc - t/√(4MN ln N))/Z(βc), and Lemma 1 supplies the only external ingredient, the standard McCoy–Wu partition function. The subsequent proof expands ln Z into L1-L4 and obtains (26)-(29) by Taylor expansion and harmonic-sum estimates. No parameter is fitted to the claimed CLT; the mean shift 4√2MN and the variance 8/π arise from the asymptotics rather than being inserted. There are no author self-citations in the derivation, no imported uniqueness theorem, and no renaming of a known result; the cited iterated-limit result of De Coninck is used only as background. The only flagged weakness, the unproved bound g_θ ≥ 0 used in Lemma 9 at equation (72), is a correctness gap in the remainder estimate, not a circularity: it does not identify the CLT with an input of the derivation. Therefore no circular step exists.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The proof rests on the exact partition function formula (15) taken from [11], standard Taylor and harmonic-number estimates, and an unproved positivity assertion for g_θ in Lemma 9. No free parameters are fitted and no new entities are introduced.

assumptions (3)

standard math Partition function formula (15) from [11] is exact for Λ_{M,N} with periodic-free boundary conditions.
External result from the McCoy-Wu textbook; Remark 4 checks the β=0 case against the formula in [4].
domain assumption g_θ := coth(2β) − cosh(2β) cos(θ)/sinh(γ_θ) ≥ 0 for all β ∈ (0, β_c] and θ ∈ (0, π].
Invoked at equation (72) in Lemma 9 to lower-bound the denominator of f''_θ by 1 in absolute value; stated without proof.
standard math Moment generating function convergence on t ≥ 0 plus Problem 30.4 of [2] implies weak convergence.
Used to pass from (10) to the Gaussian limit; the argument is cited but not demonstrated in the paper.

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Cite this review

Pith. "Pith review of A CLT for the total energy of the two-dimensional critical Ising model." pith.science (2026). https://pith.science/paper/Z554TZJK

@misc{pith2026190806704,
  author       = {Pith},
  title        = {Pith review of: A CLT for the total energy of the two-dimensional critical Ising model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z554TZJK}},
  note         = {Machine review of arXiv:1908.06704}
}

abstract

Consider the Ising model on $([1,2N]\times[1,2M])\cap\mathbb{Z}^2$ at critical temperature with periodic boundary condition in the horizontal direction and free boundary condition in the vertical direction. Let $E_{M,N}$ be its total energy (or Hamiltonian). Suppose $M$ is a function of $N$ satisfying $M\geq N/(\ln N)^{\alpha}$ for some $\alpha\in[0,1)$. In particular, one may take $M=N$. We prove that \begin{equation*} \frac{E_{M,N}+4\sqrt{2}M N-(4/\pi)N\ln N}{\sqrt{(32/\pi)MN\ln N}} \end{equation*} converges weakly to a standard Gaussian distribution as $N\rightarrow\infty$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[2]

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F. Camia , C. Garban and C.M. Newman (2015). Planar Ising magnetization ﬁeld I. Uniqueness of the critical scaling limits. Ann. Probab. 43 528-571

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J. De Coninck (1987). On limit theorems for the bivariate (magnetization, energy ) variable at the critical point. Commun. Math. Phys. 109 191-205

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De Coninck and C.M

J. De Coninck and C.M. Newman (1990). The magnetization-energy scaling limit in high di- mension. J. Stat. Phys. 59 1451-1467

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[7]

Di Francesco , H

P. Di Francesco , H. Saleur and J.B. Zuber (1987). Critical Ising correlation functions in the plane and on the torus, Nuclear Phys. B 290 527-581

work page 1987
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C. Hongler and S. Smirnov (2013). The energy density in the planar Ising model. Acta Math. 211 191-225. 13

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [2]

Billingsley (1995)

P. Billingsley (1995). Probability and Measure . 3rd ed., John Wiley & Sons, Inc

work page 1995

[2] [1]

Abraham (1978)

D.B. Abraham (1978). Block spins in the edge of an Ising ferromagnetic half-plane . J. Stat. Phys. 19 553-556

work page 1978

[3] [3]

Camia , C

F. Camia , C. Garban and C.M. Newman (2015). Planar Ising magnetization ﬁeld I. Uniqueness of the critical scaling limits. Ann. Probab. 43 528-571

work page 2015

[4] [4]

De Coninck (1984)

J. De Coninck (1984). Scaling limit of the energy variable for the two-dimensional I sing ferromag- net. Commun. Math. Phys. 95 53-59

work page 1984

[5] [5]

De Coninck (1987)

J. De Coninck (1987). On limit theorems for the bivariate (magnetization, energy ) variable at the critical point. Commun. Math. Phys. 109 191-205

work page 1987

[6] [6]

De Coninck and C.M

J. De Coninck and C.M. Newman (1990). The magnetization-energy scaling limit in high di- mension. J. Stat. Phys. 59 1451-1467

work page 1990

[7] [7]

Di Francesco , H

P. Di Francesco , H. Saleur and J.B. Zuber (1987). Critical Ising correlation functions in the plane and on the torus, Nuclear Phys. B 290 527-581

work page 1987

[8] [8]

Hecht (1967)

R. Hecht (1967). Correlation functions for the two-dimensional Ising mode l. Phys. Rev. 158 557- 561

work page 1967