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Rapid mixing within a phase for Ising models

Vasyl Paliy, Youngtak Sohn

arXiv:2610.23777v1math.PRcs.DSmath-phmath.MP

TL;DR

The paper asks whether a biased initialization can rapidly equilibrate within an Ising phase despite slow mixing between phases. It develops conditioned-measure inequalities and an inward-drift extension; for Z2 synchronization, spectral initialization gives approximate posterior sampling in Oλ(n log n) updates.

  • Problem

    The paper asks whether dynamics initialized with a bias toward one phase rapidly approaches the Gibbs measure conditioned on that phase, on a timescale shorter than leaving the phase.

    Source: §1 · p. 1

  • Method

    The proof uses duplicate variables to control interactions on the disagreement set under sparse spectral width, then extends core inequalities to a larger region under inward drift.

    Source: §1.2 · p. 5, §2.1 · p. 8

  • Results

    Spectral initialization and an independent global sign flip yield posterior TV error ≤n^-10 after Oλ(n log n) Glauber updates.

    Source: §1 · p. 2

  • Takeaways & Limitations

    For Z2 synchronization, a sufficiently biased start supports approximate posterior sampling without requiring worst-case mixing across phases.

    Source: §1 · pp. 1–2

    The inequalities require drift assumptions; extending them beyond the core requires inward drift toward it.

    Source: §1.1 · p. 5, §1.2 · p. 5

Abstract

from arXiv · show

We study Glauber dynamics for Ising measures conditioned to have a sufficiently large majority of $+1$ spins. Under a sparse spectral width condition on the interaction matrix, we derive a Poincar\'{e} inequality for the conditioned measures from a magnetization drift condition, and a modified logarithmic Sobolev inequality from a stronger, entropy weighted condition. As applications, we obtain approximate posterior sampling in total variation distance for $\mathbb{Z}_2$ synchronization at high enough signal-to-noise ratio using $O(n\log n)$ Glauber updates from a spectral initialization. For ferromagnetic Ising models on random $d$-regular graphs, we prove rapid phase ordering down to inverse temperatures of order $1/d$, extending the temperature range of Gheissari, Sly, and Sohn (2025). Finally, we extend these inequalities to unconditioned Ising measures under an inward drift condition outside the conditioned region. This yields $O(n\log n)$ mixing of Glauber dynamics for the Sherrington-Kirkpatrick model at every fixed inverse temperature when the external field is sufficiently large.

1 Introduction

The introduction asks whether phase transitions are the only obstacle to rapid Glauber mixing from a biased start. It develops functional inequalities for conditioned Ising measures and applies them to synchronization, the Sherrington-Kirkpatrick model, and random regular graphs.

1 Introduction

  • For fixed λ ≥ λ0, starts with mθ ≥ C/λ quasi-equilibrate within the positive-overlap phase; spectral initialization and a random sign flip yield posterior TV error ≤n^-10 after Oλ(n log n) updates.The TV guarantee holds for C0n log n ≤ t ≤ en/C0.Source: §1 · p. 2

1.1 Core functional inequalities under sparse spectral width condition

  • Under 8ωM(J) ≤ 1−ε, Theorem 1.2 establishes core functional inequalities using magnetization drift for variance control and a stronger entropy-weighted drift condition for entropy control.The entropy-weighted condition implies the magnetization drift condition.Source: §1.1 · pp. 4–5

1.2 Lifting the core functional inequalities

  • Inward drift outside the core lifts the inequalities to a larger region; the MLSI case additionally requires entropy-weighted drift and a size constraint on M.The lifting theorem combines the sparse spectral width condition with inward drift toward the core.Source: §1.2 · pp. 5–6
  • For the SK model, sufficiently large fields at every site and average field O(1) yield an MLSI constant of order 1/n and Oβ,H(n log n) Glauber mixing at every fixed β.The stated result applies with high probability over the interaction matrix.Source: §1.2 · pp. 6–7

1.3 Ferromagnetic Ising model on random regular graphs

  • For ferromagnetic Ising models on random d-regular graphs, rapid phase ordering extends down to β ≍ 1/d, with initial bias of order β in the stated temperature range.This removes the log d factor in the prior temperature threshold.Source: §1.3 · pp. 7–8
  • The order-β initial bias improves on the bound of [GSS25] when β ≤ c d^-1/2; whether these bias scales are optimal remains open.The improvement follows from a refined magnetization drift estimate verifying the inward drift condition.Source: §1.3 · p. 8

2 Proof overview

The proof overview combines conditioned-measure functional inequalities with magnetization drift to obtain rapid phase ordering and posterior sampling for synchronization. Boundary terms make localization applicable to conditioned measures, while drift conditions allow those terms to be absorbed.

2.1 Proof of the core functional inequalities

  • Conditioned product measures need not have uniform variance tensorization across external fields, obstructing a direct localization argument on the conditioned space.The proof instead localizes on the ambient space and restricts the evolving measures to the core.Source: §2.1 · p. 9
  • A quadratic boundary term supplies the uniform variance bound for conditioned product measures needed at the endpoint of localization.Taking the infimum over the extension constant gives Var_bν(f) ≤ nE∂_ν(f).Source: §2.1 · p. 10
  • Boundary-form variance and entropy inequalities extend to conditioned measures through stochastic localization and covariance bounds uniform over linear tilts.The entropy argument uses approximate tensorization of conditional entropy to establish MLSI.Source: §2.1 · pp. 11–12
  • Sparse spectral width yields conditioned-measure variance and entropy bounds, provided the corresponding boundary terms can be absorbed using drift assumptions.The variance bound uses (H1); the entropy bound requires the stronger (H1-E).Source: §2.1 · p. 12

2.2 Phase ordering and posterior sampling for synchronization

  • The unrestricted chain agrees with the reflected chain until exit, and positive magnetization drift makes such exits unlikely over the mixing interval.The proof couples both chains using the same Glauber updates.Source: §2.2 · pp. 13, 15
  • For fixed λ ≥ λ0, the reflected dynamics has modified log-Sobolev constant at least 1/(Cλn), supporting rapid mixing within the conditioned region.The proof uses the entropy-weighted drift condition and a coupling to control exits from that region.Source: §2.2 · p. 15
  • For every Cλn log n ≤ t ≤ e^γλn, the total-variation distance ∥P_x(X_t∈·) − bπ_S∥_TV is at most n^-12 + e^-cλn.Source: §2.2 · p. 15
  • The sign of the leading eigenvector provides a sufficiently biased initialization with probability 1-o(1) at fixed λ > 1.Its overlap with the planted vector is used to show |mθ(sign(v))| ≥ 2m0.Source: §2.2 · p. 15

3 Spectral independence of the core-conditioned measure

The section bounds covariance for the core-conditioned measure by analyzing two independent samples through their agreement and disagreement sets. Duplicate-variable cancellation reduces the disagreement spins to a constrained Ising measure, where spectral-width control yields a covariance bound.

3 Spectral independence of the core-conditioned measure

  • The disagreement-set representation links covariance control to the probability that each coordinate belongs to the disagreement set of two independent samples.The argument conditions on the agreement set and uses independence of the two samples.Source: §3 · pp. 16–17
  • Duplicate variables cancel the external field and interactions between agreement and disagreement sets, leaving an Ising interaction on the disagreement set.Conditioned on the agreement set and its spins, the disagreement spins have interaction matrix 2J_B,B and zero external field.Source: §3 · p. 16
  • Under the spectral-width condition, the constrained disagreement-spin measure has covariance operator norm at most 2/ε.This bound is used to control covariance for the core-conditioned measure.Source: §3 · p. 17

4 Stochastic localization with a boundary form

The section adapts stochastic localization to conditioned measures by tracking the restricted process and adding boundary forms. These forms remain supermartingales under a conditioned probability law, enabling variance and entropy bounds with boundary terms.

4 Stochastic localization with a boundary form

  • Conditioning the localization process on the core produces a localization process for the conditioned prior under the conditioned probability law.The restricted process is formed by conditioning each evolving measure on the core.Source: §4 · p. 18
  • The Dirichlet and conditional entropy forms with boundary terms are supermartingales under the conditioned law.This extends a property previously used in localization arguments without boundary terms.Source: §4 · p. 18
  • Covariance bounds uniform over linear tilts yield approximate conservation of variance and entropy in expectation along localization.The proof applies this principle to the conditioned process and its boundary forms.Source: §4 · pp. 19–20

5 Absorbing the boundary form

The section controls quadratic and entropic boundary terms by comparing boundary exit drift with upward magnetization drift. Under sparse spectral width and the relevant drift assumptions, these terms can be absorbed into the Dirichlet and entropy forms.

5 Absorbing the boundary form

  • A downward-to-upward drift ratio at most 4/5 across the boundary layers permits comparison of boundary forms with interior Dirichlet energy.The proof applies the comparison lemma with α = 4/5.Source: §5 · pp. 23, 25
  • The cited argument states that absorbing the implicit constants into C proves the two boundary estimates, while Lemma 5.1 is stated under ωM(J) < 1/2.Lemma 5.3 is stated under ωM(J) < 1/2 and (H1-E).Source: §5 · pp. 20, 24, 25
  • The proof compares the conditioned measure with a product measure and uses bounded perturbation to transfer the product estimates to general interactions.The interaction matrix is separated into its off-diagonal part before the comparison.Source: §5 · pp. 26–27

6 Lifting the core functional inequalities

A Foster–Lyapunov drift condition controls excursions beyond the conditioned cap, allowing the core Poincaré and modified logarithmic Sobolev inequalities to be lifted to a larger state space.

6 Lifting the core functional inequalities

  • A Foster–Lyapunov drift bounds excursion duration outside the target set, with λ setting the decay rate, and can combine with a Poincaré inequality on that set.The stated tail bound decays as (1−λ)^t, scaled by V(x).Source: §6 · p. 27

6.1 Poincar´e lifting

  • The lifted Poincaré inequality follows by splitting variance into cap and exterior contributions, controlling the cap with Theorem 1.2 and the exterior with a drift estimate.The proof combines these bounds to establish Theorem 1.3(a).Source: §6.1 · pp. 28–29

6.2 Entropy lifting

  • The MLSI lift uses entropy-chain decomposition and Rothaus’ inequality, with an exponential Lyapunov function needed for the exterior term’s additional factor of n.The cap contribution uses Theorem 1.2(b), while the exterior quadratic form is controlled by a Lyapunov drift argument.Source: §6.2 · pp. 29, 31
  • Under Theorem 1.3(b)’s assumptions, the argument establishes the claimed modified logarithmic Sobolev inequality on the expanded region.The proof concludes by bounding entropy by the entropy Dirichlet form.Source: §6.2 · pp. 29, 31

7 Application estimates and proofs

The application proofs verify the paper’s conditions in synchronization, random-regular-graph, and high-field settings. They also record the authors’ use of AI assistance and their responsibility for checking the work.

7.1 Z2 synchronization

  • In the synchronization application, the estimates verify (H1-E) and establish a uniform bound over the stated configurations.The (H1-E) verification uses R ≥ n/(2λ^3) for sufficiently large n.Source: §7.1 · p. 33

7.2 Ferromagnetic Ising model on a random regular graph

  • For random d-regular graphs, the proof establishes the theorem’s bound (1.10) for states with m(σ) ≥ 2β over times Cn log n ≤ t ≤ e^(n/C).The argument uses the resulting MLSI bound and couples the conditioned and unconditioned chains until the chain exits the conditioned region.Source: §7.2 · p. 36

7.3 High-field functional inequalities

  • Under the stated field conditions, the unconditioned measure has spectral gap at least order 1/n and an MLSI constant at least order 1/n.The gap follows by taking the outer cutoff S = −n; the entropy result uses the additional assumption stated for the entropy estimate.Source: §7.3 · p. 37

Statement of AI use

  • The authors report that AI-assisted calculations and suggestions were checked by the authors, who take responsibility for the paper’s contents.They also state that they wrote the initial manuscript and revised it with assistance.Source: Statement of AI use · p. 38
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