Estimate of the exit time for the Long Range Ising model on random regular graphs

Vanessa Jacquier

REVIEW 4 major objections 5 minor 16 references

Estimate of the exit time for the Long Range Ising model on random regular graphs

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper estimates how long a long-range Ising spin system on a random regular graph lingers in its metastable all-minus state before flipping to the stable all-plus state, giving explicit exponential-in-β bounds on that exit time.

desk verdict The setting is genuinely new and worth thinking about, but the proof of the main theorem has a false combinatorial lemma at its core and some algebra that does not close. read the letter →

arxiv 2509.02188 v1 pith:NNFQB7AZ submitted 2025-09-02 math.PR math-phmath.MPphysics.data-an

Vanessa Jacquier This is my paper · ORCID

classification math.PRmath-phmath.MPphysics.data-an MSC 82B2082C2082C2660K3560J1005C80

keywords long-rangeIsingmodelrandomregulargraphmetastabilityGlauberdynamicsCheegerconstantenergybarrierexittimelow-temperaturenucleation

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

The paper studies the long-range Ising model on random r-regular graphs evolving under Glauber dynamics at low temperature. Its central claim is that the all-minus state is metastable, and the first time the system reaches the stable all-plus state is exponentially large in the inverse temperature β, with a logarithmic rate lying between two explicit bounds: Γ_l = J(r/2 − o(r))n from below and Γ_u = J(r f(r,λ) + o(r))n from above. The lower bound is linear in the number of vertices n; the upper bound depends on how fast interactions decay with graph distance, reducing to the known short-range result when decay is exponential and widening when decay is a slow power law. If the estimate is right, it provides a quantitative nucleation barrier shaped by both the graph's Cheeger constant and its diameter.

What carries the argument

The load-bearing object is the distance-shell expansion of the Hamiltonian: for a plus cluster A, ∆H(σ) = −2h|A| + 2Σ_{i=1}^{d} J(i)|∂_e^{(i)}A|, where ∂_e^{(i)}A counts opposite-spin pairs at graph distance i. The proof bounds shell sizes with the growth inequality |∂_e^{(i)}A| ≤ (r−1)^{i−1}|∂_eA|, then feeds in the known w.h.p. lower bound on the Cheeger constant and upper bound on the diameter to evaluate sums of the form Σ_i J(i)(r(r−1)^{i−1} − 1). Paths from −1 to +1 are constructed by flipping spins one at a time in blocks; the combinatorial Lemma 3.5 replaces true shell counts by worst-case coefficients r(r−1)^{i−1} − 1. A general metastability theorem then converts the barrier bounds

What would settle it

Construct a small r-regular graph and a plus cluster A for which some vertex x ∈ A has no opposite spin at distance 1 but has one at distance 2. Compute the exact communication height Φ(−1,+1) by exhaustive enumeration of single-spin-flip paths: if the barrier exceeds Γ_u for such a graph, the initial-segment assumption breaks the upper bound in Theorem 2.8.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes an interval estimate for the energy barrier governing metastability: under Glauber dynamics at low temperature, the stability level of the all-minus configuration on a random r-regular graph satisfies Γ_l ≤ V_{-1} ≤ Γ_u, where Γ_l = J(r/2 − o(r))n and Γ_u = J(r f(r,λ) + o(r))n. The coefficient f(r,λ) equals r for exponential decay, r−2 for power-law decay with λ ≥ d, and ζ(λ)(r−1)^d for power-law decay with λ < d. Consequently, for every ε > 0, the first hitting time of +1 satisfies e^{β(Γ_l−ε)} < τ_{+1} < e^{β(Γ_u+ε)} with probability tending to 1 as β → ∞, and the same exponential window holds from any metastable state. This generalizes the short-ran

Load-bearing premise

The upper-bound argument assumes that, looking outward from any vertex of a plus cluster, the opposite-spin vertices occupy an initial segment of distance shells—if none at distance i, none farther away—an assumption that can fail when a cluster blocks sight of farther spins, and the main theorem statement also omits the small-field condition h < J C√r that the lower-bound proof requires.

Editorial extensions

If this is right

If the bounds hold, the all-minus state is metastable, and the exit time to the stable all-plus state is exponentially large in β with logarithmic rate inside [Γ_l, Γ_u].
For exponential decay J(i) = J r^{1−i}, the barrier coefficient matches the short-range random-regular-graph barrier up to r-dependent constants, so distant spins do not change the exponential order of the exit time.
For power-law decay with λ ≥ d, the upper coefficient is J r(r−2), still linear in n; for λ < d it becomes J r ζ(λ)(r−1)^d, bringing the graph diameter into the barrier.
Every configuration other than ±1 has stability level at most Γ_u, so no alternative state forces a longer exit time than the stated window; whether −1 is the unique metastable state is left open.

Reading between the lines

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

Editorial extension: For λ < d, Γ_u is not merely linear in n: combining the paper's formula with the diameter bound gives (r−1)^d = Θ(n log n), so the upper barrier can grow like n^2 log n, making the interval between the bounds extremely wide for slowly decaying interactions.
Editorial extension: The proof logic suggests a crossover criterion: short-range-like behavior persists when the shell sum Σ_i J(i) r(r−1)^{i−1} stays bounded by fixed r-dependent constants; when small λ makes this sum grow with diameter, genuinely long-range effects dominate the barrier.
Editorial extension: One could probe the initial-segment assumption numerically: on small cubic graphs with planted blocking clusters, simulate or exactly compute the first-passage exponent; if it exceeds Γ_u, the upper bound needs a correction for non-convex plus clusters.

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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

Colleague,

This paper is about the long-range Ising model on random r-regular graphs, a combination I don't think anyone has treated before. The author frames it as an extension of Dommers's short-range result and uses standard path-wise metastability estimates plus known bounds on the Cheeger constant and diameter. That is the right toolkit, and the paper is honest that it only produces an interval for the energy barrier.

The problem is that the interval bounds are not actually established. The load-bearing step is Lemma 3.5, which says that for a decreasing interaction, if the counts c_i = number of opposite spins at distance i from a given site have the property that a zero at i forces zeros at all larger distances, then Σ c_i J(i) ≥ Σ J(i). The proof of Lemma 3.3 (and symmetrically 3.4) asserts that for a connected set A, the counts c_i = |∂^i_x(A^c)| have exactly that initial-segment property. They don't. In an r-regular graph, take A = {x} ∪ N(x). Then there are no opposite spins at distance 1 from x (c_1 = 0), but there can be opposite spins at distance 2 (c_2 > 0). That configuration is perfectly compatible with connectedness. More generally, a connected set can have 'holes' in its distance profile to the complement. Without the initial-segment property, the inequality can fail; I checked a simple case with r=10, f(i)=10^{1-i}, c=(0,10,0,...), and Σ c_i f(i) < Σ f(i). So (5.1) and (5.5) do not follow, and the upper bound Γ_u is unsupported.

The algebra in Proposition 2.7 is also shaky. The step from (4.1) to the claimed bound involves an inequality of the form n(h+Σ)(S+h-A)/S ≤ n(2h+Σ-A) that doesn't hold in the small-h regime relevant here. That would need a rewritten argument even if Lemma 3.5 were fixed.

There are smaller issues too. Lemma 3.1 assumes h < J C √r, but Theorem 2.8 states the result without that condition. And in the same lemma, the inequality 2|A|(-h+J i(G)) ≥ n(...) uses |A| ≥ n/2, while the lemma only has |A| ≤ n/2; this is repairable by noting the path crosses the half-filling manifold, but as written it is wrong.

What the paper does well: the new setting is real, the use of Cheeger and diameter estimates is sensible, and the author does not hide the looseness of the bounds. The reference list is appropriate. The flaws are technical, not conceptual.

Who is this for? People working on metastability on random graphs, especially those who want to see whether long-range interactions can be handled in that setting. I would not cite the bounds as they stand, but the question is worth pursuing.

My recommendation: send it to referees. The errors are substantive but possibly repairable, and a careful referee could tell whether the approach can be patched. Expect heavy revision.

Referee Report

4 major / 5 minor

Summary. The paper studies Glauber dynamics for a ferromagnetic long-range Ising model on random r-regular graphs at low temperature. It claims that the homogeneous -1 state is metastable, with stability level bounded between Γ_l = J(r/2 - o(r))n and Γ_u = J(r f(r,λ) + o(r))n, and that the first hitting time of +1 satisfies P_{-1}( e^{β(Γ_l-ε)} < τ_{+1} < e^{β(Γ_u+ε)} ) → 1 as β→∞. The proof uses the path-wise approach, the Cheeger constant, diameter estimates, and combinatorial lemmas controlling the energy change when flipping spins. The paper would generalize earlier work by Dommers on short-range interactions, with explicit dependence on the interaction range and graph degree.

Significance. The problem is relevant, and a correct result would be a meaningful extension of the short-range metastability theory on random regular graphs to long-range interactions. The paper is clearly written and properly situates itself in the literature. However, the proof as written contains several load-bearing gaps: the initial-segment property used to bound per-flip energy changes is false, the Cheeger-based estimates discard long-range boundary contributions, and a key algebraic inequality in Section 4 is not valid. The central claims are therefore not established by the arguments presented. If the gaps can be repaired, the result would be significant, but substantial work is needed.

major comments (4)

[§3.2, Lemma 3.7] The line before (3.13) states that for P ∈ I one has ΔH(σ) ≤ n(−h + J i′(G_n)). But from (3.1), ΔH(σ) = −2h|P| + 2 Σ_{i=1}^d J(i)|∂_e^{(i)}P|, and the terms with i ≥ 2 are nonnegative and typically positive. Thus one obtains only the lower bound ΔH(σ) ≥ n(−h + J i′(G_n)) (using |∂_eP|/|P| = i′(G_n)). The subsequent estimate Φ(σ,-1) ≤ H(-1) + Γ_u therefore does not follow. This is load-bearing for Proposition 2.6 and the upper bound in Theorem 2.8.
[§3.1, Lemma 3.1] The proof contains the chain ΔH(σ) ≥ 2|A|(−h + J i(G_n)) ≥ n(−h + J(r/2 − √(log 2)√r)). The second inequality uses 2|A| ≥ n, while the standing assumption in the lemma is |A| ≤ n/2; for |A| < n/2 the inequality is reversed. The argument should be restricted to the first configuration on a path with |A| = n/2. As written, the lower bound V_{-1} ≥ Γ_l is not proved for configurations with smaller |A|, so the proof of the lower bound is incomplete.
[§5, Lemmas 3.3–3.5] The bound (5.1)/(5.5) relies on Lemma 3.5 through the assertion that for a connected set A the sequence c_i = |∂_e x^{(i)}(A^c)| satisfies 'if c_i = 0 then c_{i+1} = 0'. This is false: for A = {x} ∪ N(x) (the closed neighbourhood of x), c_1 = 0 while c_2 > 0; for a ball of radius 2 around x, c_2 = 0 while c_3 > 0. Hence the inequality Σ_i J(i)c_i ≥ Σ_i J(i) used to derive (5.1) and (5.5) is not valid in general (e.g. c = (0,d) with decreasing J already violates it). Without (5.1) and (5.5), the per-flip energy cost in Lemmas 3.3/3.4 is not controlled, and the upper bound Γ_u in Lemma 3.7/Proposition 2.6 is not established.
[§4, Eq. (4.1)] In Proposition 2.7 the displayed algebraic step n(h+Σ)(D+h−A)/D ≤ n(2h+Σ−A), with Σ = Σ_i J(i)(r(r−1)^{i−1}−1), D = Σ_i r(r−1)^{i−1}J(i), A = J(r/2−√(log 2)√r), is not valid. The difference between the left and right sides is n(A−h)(D−h−Σ)/D, which is positive in natural parameter regimes (e.g. small h, fixed r, exponential interaction). Consequently the conclusion V_σ ≤ Γ_u for all σ ≠ +1 is not supported by the displayed algebra; this affects the extension of Theorem 2.8 to all metastable states.

minor comments (5)

[§1, Organization] The text says 'Sections 3 and 3 are devoted to their proofs'; this should be 'Sections 3 and 4'.
[§4, first paragraph] The sentence 'we will prove that, for all σ ≠ +1, V_{+1} ≤ Γ_u' should read V_σ ≤ Γ_u.
[§3.1, Lemma 3.1] The small-field condition 0 < h < J C√r is stated in the lemma but not included in Condition 2.4 or in Theorem 2.8; the theorem should either include this condition or the proof should indicate where it is used.
[§2.2, Eq. (2.10)] The symbol n is used both for the number of vertices and for the distance in J(n). Later J(i) is used for a distance argument; this overloaded notation should be harmonized.
[§5, Lemma 3.2] The proof says 'connected set of edges' where it should say 'connected set of vertices'; the non-connected case is dismissed without detail.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof uses external graph estimates and standard metastability theorems; self-citation is not load-bearing.

full rationale

The derivation chain is self-contained with respect to external benchmarks. The bounds on V_-1 are obtained from the Bollobás–Fernandez de la Vega diameter bound (Theorem 2.1), the Bollobás lower Cheeger bound and the Alon upper Cheeger bound (Theorems 2.2–2.3), the general tunneling theorem of Manzo–Nardi–Olivieri–Scoppola (used in the proof of Theorem 2.8), and explicit algebraic estimates on sums of J(i) (Lemma 3.6). None of these inputs is the target exit-time estimate. No parameter is fitted to the exit time; the notation f(r,λ) in (2.28) simply names the coefficient produced by Lemma 3.6. The only self-citation, [12] in the introduction, is related work on a biaxial perimeter and is not used in the proofs of Propositions 2.6–2.7 or Theorem 2.8. There is no imported uniqueness theorem, no ansatz smuggled in by citation, and no renaming of a known empirical pattern. The skeptical concern about Lemma 3.5's initial-segment hypothesis, if valid, is a mathematical gap in the proof of the upper bound, not a circularity: the lemma is stated and proved independently, and its failure would destroy the bound rather than make it equivalent to an assumption. The omitted magnetic field condition in Lemma 3.1 is likewise a statement-level correctness issue, not a circular step. Therefore no circular step is identified and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters or new physical entities. Its central claim rests on external results about random regular graphs, on a cited metastability theorem, and on an unproven structural assumption about distance sets that is used to simplify energy-increase estimates.

assumptions (5)

domain assumption Random r-regular graphs have diameter bounded as in (2.2) w.h.p. (Theorem 2.1 of [4]).
Used in Lemma 3.3/3.4 to assert Σ_i |∂_e^{(i)}_x(A^c)| ≥ n/2 > d, and in Lemma 3.6 to control sums up to the diameter.
domain assumption The Cheeger constant of random r-regular graphs satisfies i(Gn) ∈ [r/2 - √(log 2)√r, r/2 - C√r] w.h.p. (Theorems 2.2, 2.3 from [3], [1]).
Used in Lemma 3.1 for the lower bound on V_{-1} and in Lemma 3.7 for the upper bound on the communication height.
domain assumption The path-wise metastability theorem [15, Theorem 4.1] applies to -1 and to any metastable state η.
Invoked in the proof of Theorem 2.8 without verifying that -1 is a maximizer of the stability level or that other hypotheses of the theorem hold.
ad hoc to paper For any vertex x and set A (or A^c), the sequence c_i = |∂_e^{(i)}_x(A^c)| satisfies: if c_i=0 then c_{i+1}=0.
Needed for Lemma 3.5 to apply in Lemmas 3.3 and 3.4. This is not proven for arbitrary subsets and is false in general (a separating cluster can create gaps in the distance sequence).
ad hoc to paper The magnetic field is small: 0 < h < J C √r with C < √(3/2) - √(log 2).
Introduced in Lemma 3.1 for the lower bound Γ_l ≤ V_{-1}, but omitted from Condition 2.4 and the statement of Theorem 2.8.

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Cite this review

Pith. "Pith review of Estimate of the exit time for the Long Range Ising model on random regular graphs." pith.science (2026). https://pith.science/paper/NNFQB7AZ

@misc{pith2026250902188,
  author       = {Pith},
  title        = {Pith review of: Estimate of the exit time for the Long Range Ising model on random regular graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NNFQB7AZ}},
  note         = {Machine review of arXiv:2509.02188}
}

read the original abstract

We investigate the metastable behavior of the long-range Ising model on random regular graphs under Glauber dynamics at low-temperature. We estimate the energy barrier and exit time from the metastable state using a nontrivial path-wise approach that explicitly accounts for the spatial decay of the interactions and the structural properties of the graph, such as the Cheeger constant and known estimates of the diameter. Our results generalize those of Dommers \cite{dommers2017metastability} for the short-range case, providing a unified framework for understanding metastability in systems with long-range interactions.

Figures

Figures reproduced from arXiv: 2509.02188 by the authors.

**Figure 1.** A schematic representation of the communication height between two configurations σ and η. Given a configuration σ, we consider the set Iσ of all configurations η with H(η) < H(σ). Note that Iσ = ∅ if σ is a global minimum of the Hamiltonian function. The stability level of a configuration σ ̸∈ X s is defined as Vσ := Φ(σ, Iσ) − H(σ). (2.21) 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

N. Alon. On the edge-expansion of graphs. Combinatorics, Probability and Computing, 6(2):145–152, 1997

work page 1997
[2]

Bissacot, E

R. Bissacot, E. O. Endo, A. C. van Enter, B. Kimura, and W. M. Ruszel. Contour methods for long- range Ising models: weakening nearest-neighbor interactions and adding decaying fields. InAnnales Henri Poincaré, volume 19, pages 2557–2574. Springer, 2018

work page 2018
[3]

Bollobás

B. Bollobás. The isoperimetric number of random regular graphs. European Journal of combinatorics, 9(3):241–244, 1988

work page 1988
[4]

Bollobás and W

B. Bollobás and W. Fernandez de la Vega. The diameter of random regular graphs.Combinatorica, 2:125–134, 1982

work page 1982
[5]

Bovier, S

A. Bovier, S. Marello, and E. Pulvirenti. Metastability for the dilute Curie–Weiss model with Glauber dynamics. Electronic Journal of Probability, 26:1–38, 2021

work page 2021
[6]

Cassandro, P

M. Cassandro, P . A. Ferrari, I. Merola, and E. Presutti. Geometry of contours and Peierls estimates in d= 1 Ising models with long-range interactions. Journal of Mathematical Physics, 46(5), 2005

work page 2005
[7]

E. N. Cirillo and F. R. Nardi. Relaxation height in energy landscapes: an application to multiple metastable states. Journal of Statistical Physics, 150:1080–1114, 2013

work page 2013
[8]

Den Hollander and O

F. Den Hollander and O. Jovanovski. Glauber dynamics on the Erd˝ os-Rényi random graph.In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius, pages 519–589, 2021

work page 2021

Show all 16 references

[9]

S. Dommers. Metastability of the ising model on random regular graphs at zero temperature. Probability Theory and Related Fields, 167:305–324, 2017

2017
[10]

Dommers, F

S. Dommers, F. Den Hollander, O. Jovanovski, and F. R. Nardi. Metastability for Glauber dynamics on random graphs. 2017

2017
[11]

F. J. Dyson. Existence and nature of phase transitions in one-dimensional Ising ferromagnets. In SIAM-AMS Proceedings, volume 5, pages 1–12, 1972

1972
[12]

Jacquier, W

V . Jacquier, W. M. Ruszel, and C. Spitoni. Isoperimetric inequality for nonlocal bi-axial discrete perimeter. arXiv preprint arXiv:2412.13005, 2024

2024
[13]

Lazarides and A

A. Lazarides and A. Pizzi. Apparent bistability from weak long-range interactions. arXiv preprint arXiv:2506.10068, 2025

2025 arXiv
[14]

Littin and P

J. Littin and P . Picco. Quasi-additive estimates on the Hamiltonian for the one-dimensional long- range Ising model. Journal of Mathematical Physics, 58(7), 2017

2017
[15]

Manzo, F

F. Manzo, F. Nardi, E. Olivieri, and E. Scoppola. On the essential features of metastability: tunnelling time and critical configurations. Journal of Statistical Physics, 115(1-2):591–642, 2004

2004
[16]

A. C. van Enter, B. Kimura, W. M. Ruszel, and C. Spitoni. Nucleation for one-dimensional long-range Ising models. Journal of Statistical Physics, 174:1327–1345, 2019. 20

2019

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

N. Alon. On the edge-expansion of graphs. Combinatorics, Probability and Computing, 6(2):145–152, 1997

work page 1997

[2] [2]

Bissacot, E

R. Bissacot, E. O. Endo, A. C. van Enter, B. Kimura, and W. M. Ruszel. Contour methods for long- range Ising models: weakening nearest-neighbor interactions and adding decaying fields. InAnnales Henri Poincaré, volume 19, pages 2557–2574. Springer, 2018

work page 2018

[3] [3]

Bollobás

B. Bollobás. The isoperimetric number of random regular graphs. European Journal of combinatorics, 9(3):241–244, 1988

work page 1988

[4] [4]

Bollobás and W

B. Bollobás and W. Fernandez de la Vega. The diameter of random regular graphs.Combinatorica, 2:125–134, 1982

work page 1982

[5] [5]

Bovier, S

A. Bovier, S. Marello, and E. Pulvirenti. Metastability for the dilute Curie–Weiss model with Glauber dynamics. Electronic Journal of Probability, 26:1–38, 2021

work page 2021

[6] [6]

Cassandro, P

M. Cassandro, P . A. Ferrari, I. Merola, and E. Presutti. Geometry of contours and Peierls estimates in d= 1 Ising models with long-range interactions. Journal of Mathematical Physics, 46(5), 2005

work page 2005

[7] [7]

E. N. Cirillo and F. R. Nardi. Relaxation height in energy landscapes: an application to multiple metastable states. Journal of Statistical Physics, 150:1080–1114, 2013

work page 2013

[8] [8]

Den Hollander and O

F. Den Hollander and O. Jovanovski. Glauber dynamics on the Erd˝ os-Rényi random graph.In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius, pages 519–589, 2021

work page 2021