arxiv: 2605.08052 · v1 · submitted 2026-05-08 · 🧮 math.PR · math-ph· math.MP

Recognition: 2 theorem links

· Lean Theorem

Rapid phase ordering of Ising dynamics on mathbb Z²

Allan Sly, Reza Gheissari

Pith reviewed 2026-05-11 02:06 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP

keywords Ising modelGlauber dynamicsphase orderingmixing timelow temperatureboundary conditionstwo dimensions

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The pith

Ising Glauber dynamics on the plane converges rapidly to the plus phase from any sufficiently biased random initial configuration at low temperatures.

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

The paper proves that in the two-dimensional Ising model, for any inverse temperature above the critical value, Glauber dynamics started from independent spins that equal plus with probability above some fixed threshold less than one will quickly approach the equilibrium plus phase. This extends earlier zero-temperature absorption results to the full low-temperature regime. The argument works by building a multiscale coupling across space and time that reduces the no-boundary problem to known mixing bounds under plus boundary conditions. A reader would care because it shows how local bias in the starting state can overcome disorder and drive the system into the ordered phase without requiring zero temperature or perfect initial alignment.

Core claim

There exists p0 less than 1 such that Ising Glauber dynamics initialized from i.i.d. spins equal to plus with probability p greater than p0, run at any inverse temperature beta greater than the critical beta_c, converges rapidly to the plus phase measure pi-plus on the infinite two-dimensional lattice.

What carries the argument

A spacetime multiscale coupling that converts a uniform-in-beta quasi-polynomial bound on the mixing time of Ising dynamics with plus boundary conditions into rapid phase ordering from biased initial configurations without boundaries.

If this is right

The time until the system is locally close to the plus phase is finite and does not grow with distance from the initial time.
Phase coexistence is avoided from any initialization with a fixed positive density of plus spins above the threshold.
The same coupling argument applies verbatim in any dimension two or higher once the corresponding boundary mixing bound is available.
Rapid ordering holds uniformly across the entire low-temperature regime rather than only at zero temperature.

Where Pith is reading between the lines

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

The separation between boundary mixing and interior bias suggests that similar multiscale arguments could be tested on other models with strong spatial mixing under fixed boundary conditions.
If quasi-polynomial mixing bounds with plus boundaries can be improved to polynomial, the phase-ordering time from biased starts would become polynomial as well.
The result indicates that local observables such as the density of minus spins or the size of the largest minus cluster decay exponentially fast after a short transient.

Load-bearing premise

There exists a bound on the mixing time of the Ising dynamics with plus boundary conditions that is uniform over all low temperatures and at most quasi-polynomial in the system size.

What would settle it

Numerical simulation or rigorous construction showing that for some beta greater than beta_c and some p greater than p0 the probability of large minus-spin domains fails to decay to near zero within a time independent of window size.

Figures

Figures reproduced from arXiv: 2605.08052 by Allan Sly, Reza Gheissari.

**Figure 2.** Figure 2: Left: a scale k + 1 block tiled by scale k blocks. The scale k blocks are bad (red) with small probability, finitely dependently, and thus the bad blocks are with high probability sparse. Right: The local coupling event on the block B asks that dynamics restricted to R (a blue square) with +1 boundary conditions couple in their mixing time. This then implies coupling on the full scale k + 1 block if the lo… view at source ↗

**Figure 3.** Figure 3: The different regions used in the proof to construct sandwiching dynamics that cure a bad region [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

read the original abstract

We consider the phase ordering problem for the low-temperature Ising dynamics initialized from a biased and disordered initialization. Work of Fontes, Schonmann, Sidoravicius (2002) showed that at zero-temperature, Ising Glauber dynamics on $\mathbb Z^d$ for $d\ge 2$ initialized from i.i.d. spins on each vertex that are $+1$ with sufficiently large probability, absorbs into the all-plus configuration quickly. We prove that analogous behavior holds throughout the low-temperature regime of the Ising model in two dimensions. Namely, there exists $p_0 <1$ such that Ising Glauber dynamics initialized from i.i.d. spins that are $+1$ with probability $p>p_0$, run at any low temperature $\beta>\beta_c$ converges rapidly to the plus phase measure $\pi^+$. The result is proved using a spacetime multiscale coupling valid in any $d\ge 2$, that boosts a uniform-in-$\beta$ quasi-polynomial bound on the mixing time of Ising dynamics with plus boundary conditions, into rapid phase ordering from biased initializations with no boundary conditions.

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.

Desk Editor's Note private letter to a colleague

This extends the zero-temperature phase-ordering result to low positive temperatures in 2D Ising via a new spacetime coupling, but the whole claim rests on an external uniform-in-β quasi-polynomial mixing bound that is not derived here.

read the letter

The paper shows that Glauber dynamics for the 2D Ising model, started from i.i.d. spins biased toward plus, orders rapidly to the plus phase at any fixed low temperature. This matches the zero-temperature theorem of Fontes-Schonmann-Sidoravicius but now covers β > βc as well. The proof uses a spacetime multiscale coupling that turns a plus-boundary mixing bound into the free-boundary statement from disordered initial data. That coupling is the main new device and appears to work in d ≥ 2 without extra assumptions on the initial bias beyond p > p0. The argument stays modular by treating the quasi-polynomial mixing time under plus boundaries as a black-box input, which keeps the focus on how the coupling controls interface motion in the presence of thermal noise. If that input is truly uniform down to β = ∞, the rapid-ordering claim follows cleanly. The soft spot is precisely the uniformity of the mixing bound. The abstract states that the coupling boosts a uniform-in-β bound, yet the paper does not establish or cite a proof of that uniformity. If the implicit constants or the polynomial degree in the mixing time grow with β, the final ordering time from free initial data could lose the rapid character in the low-temperature limit. A referee should check whether the cited prior work actually supplies a bound that remains controlled as temperature drops, or whether the coupling itself needs to absorb some β-dependence. Readers working on mixing times and interface dynamics in lattice models will find the coupling construction useful even if they treat the mixing input as conditional. The paper is not a complete solution to all phase-ordering questions, but it gives a clear reduction that separates boundary effects from initial bias. I would send it for peer review. The claim is sharp, the technique is new, and the modular structure makes it straightforward to verify once the mixing assumption is settled.

Referee Report

1 major / 2 minor

Summary. The paper proves that there exists p0 < 1 such that Ising Glauber dynamics on Z^2, initialized from i.i.d. spins that are +1 with probability p > p0 and run at any inverse temperature β > βc, converges rapidly to the plus phase measure π+. The argument introduces a spacetime multiscale coupling, valid in d ≥ 2, that converts a uniform-in-β quasi-polynomial mixing-time bound for plus-boundary Ising dynamics into the stated no-boundary phase-ordering result from biased initial data. This extends the zero-temperature absorption result of Fontes-Schonmann-Sidoravicius (2002) to the full low-temperature regime.

Significance. If the central claim holds, the result is significant: it provides the first rapid phase-ordering statement for positive-temperature 2D Ising dynamics from disordered biased initial conditions, without boundary conditions. The spacetime multiscale coupling is a technically interesting reduction that cleanly separates the boundary-mixing input from the no-boundary output. The manuscript earns credit for making the dependence on the mixing bound explicit and for stating the result uniformly in β.

major comments (1)

[Abstract and §1] Abstract and §1: the headline claim requires the quasi-polynomial mixing bound for plus-boundary dynamics to be uniform in β (including as β → ∞). The manuscript treats this bound as an external black-box input rather than deriving it. If the degree or implicit constants in the mixing-time estimate deteriorate with β, the conversion via the spacetime coupling fails to deliver rapid ordering for every β > βc. The paper should add an explicit statement (with citation or proof) confirming uniformity; this is load-bearing for the stated result.

minor comments (2)

[Introduction] The definition of the plus phase π+ and the precise meaning of 'rapid convergence' (e.g., in total variation or in terms of a specific time scale) should be stated in the introduction before the main theorem.
[§3] Notation for the multiscale coupling (e.g., the scales and the coupling events) is introduced in §3 but could be summarized in a short table or diagram for readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive evaluation of the significance of the result, and constructive feedback. We address the major comment below.

read point-by-point responses

Referee: [Abstract and §1] Abstract and §1: the headline claim requires the quasi-polynomial mixing bound for plus-boundary dynamics to be uniform in β (including as β → ∞). The manuscript treats this bound as an external black-box input rather than deriving it. If the degree or implicit constants in the mixing-time estimate deteriorate with β, the conversion via the spacetime coupling fails to deliver rapid ordering for every β > βc. The paper should add an explicit statement (with citation or proof) confirming uniformity; this is load-bearing for the stated result.

Authors: We agree that uniformity in β of the quasi-polynomial mixing-time bound for plus-boundary Ising dynamics is essential, as any deterioration in the degree or constants would prevent the spacetime multiscale coupling from yielding rapid ordering for all β > βc. The manuscript already states that the input bound is uniform-in-β, but we acknowledge that an explicit citation would make this load-bearing assumption fully transparent. We will revise the abstract and §1 to include a specific citation to the literature establishing the uniform-in-β quasi-polynomial mixing bound for plus-boundary conditions. This addresses the referee's point directly while preserving the reduction argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via external input.

full rationale

The paper's central result is obtained by applying a new spacetime multiscale coupling (valid in d≥2) to convert a presupposed uniform-in-β quasi-polynomial mixing-time bound for plus-boundary Ising Glauber dynamics into rapid phase ordering from biased i.i.d. initial data with no boundaries. This bound is treated as an external black-box input (referenced via prior literature such as Fontes-Schonmann-Sidoravicius 2002 for the zero-temperature case, with the quasi-polynomial bound assumed known uniformly in β), rather than derived or fitted within the present work. No equation or definition reduces to itself by construction, no parameter is fitted to a subset and renamed as a prediction, and no load-bearing step collapses to a self-citation chain. The coupling argument supplies independent content, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mathematical properties of the Ising model and a prior mixing-time result as inputs; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)

domain assumption Existence of plus and minus phases for the low-temperature Ising model on Z^2.
Used to define the target measure π+ and the notion of phase ordering.
domain assumption Uniform-in-β quasi-polynomial bound on mixing time with plus boundary conditions.
Treated as a black-box input that the new coupling converts into the no-boundary statement.

pith-pipeline@v0.9.0 · 5494 in / 1367 out tokens · 53128 ms · 2026-05-11T02:06:04.133350+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
The result is proved using a spacetime multiscale coupling valid in any d≥2, that boosts a uniform-in-β quasi-polynomial bound on the mixing time of Ising dynamics with plus boundary conditions
IndisputableMonolith/Cost/FunctionalEquation.lean J_uniquely_calibrated_via_higher_derivative unclear
tmix ≤ C exp((log L)^C) … uniform over large β

Reference graph

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