pith. sign in

arxiv: 2605.02002 · v1 · submitted 2026-05-03 · 🧮 math.PR

Glauber dynamics for random field Ising models on bounded degree graphs and MLSI

Yi Han This is my paper

Pith reviewed 2026-05-08 19:33 UTC · model grok-4.3

classification 🧮 math.PR
keywords fieldrandomdynamicsglaubergraphsinequalityonlyprove
0
0 comments X p. Extension

The pith

Glauber dynamics for RFIM on bounded-degree graphs mixes in polynomial time w.h.p. under anti-concentrated random fields, with MLSI and weak Poincaré inequalities also established.

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

The random field Ising model places spins on graph vertices with random external fields at each site. Glauber dynamics is a simple Markov chain that flips one spin at a time according to local probabilities. The paper shows that when the random fields are anti-concentrated enough, this chain reaches equilibrium quickly on any graph with bounded maximum degree, with the guarantee holding for most realizations of the random fields. A modified log-Sobolev inequality is proved when fields are also bounded. For weaker fields that still give exponential decay of correlations on average, and on graphs whose volume grows at most stretched-exponentially, a weaker inequality suffices for polynomial-time sampling from a warm start. These results move beyond the integer lattice, where similar statements were known, to arbitrary bounded-degree graphs.

Core claim

When the random field distribution is sufficiently anti-concentrated, we prove that with high probability over the quenched randomness of the external field, the Glauber dynamics of this RFIM mixes in polynomial time as a consequence of a Poincaré inequality.

Load-bearing premise

The random field distribution is sufficiently anti-concentrated (and, for the MLSI part, bounded); the graph has bounded maximum degree Δ; for the weak Poincaré part, the fields satisfy weak spatial mixing in expectation and the graph has at most α-stretched exponential growth for α<1.

read the original abstract

We study the ferromagnetic random field Ising model (RFIM) on a graph $G=(V,E)$ having maximal degree $\Delta$, where the external field at each vertex is an i.i.d. random variable. When the random field distribution is sufficiently anti-concentrated, we prove that with high probability over the quenched randomness of the external field, the Glauber dynamics of this RFIM mixes in polynomial time as a consequence of a Poincar\'e inequality. This model is relevant to the Griffiths phase where the correlations decay exponentially fast in expectation over the quenched random field, but contraction does not hold point-wise due to the existence of weak fields that lead to low-temperature behavior. Previously, fast mixing of Glauber dynamics under large disorder was only proven on the integer lattice, and for RFIM on general graphs, only a sampling algorithm based on self-avoiding walks was known. Under a further technical condition that the random fields are bounded, we prove a modified log-Sobolev inequality for the Glauber dynamics. When the random field is weaker but still satisfies weak spatial mixing (exponential decay of correlations from boundary to bulk) in expectation, and the graph has at most $\alpha$-stretched exponential growth for some $\alpha<1$, then we prove a weak Poincar\'e inequality holds, which gives rise to a polynomial time sampling algorithm based on Glauber dynamics with warm start. The latter result was previously proven for the integer lattice, and we extend its scope to graphs with only a volume growth condition without assuming a local geometry.

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

0 major / 3 minor

Summary. The paper proves that for the ferromagnetic random field Ising model (RFIM) on bounded-degree graphs, when the i.i.d. random fields are sufficiently anti-concentrated, the Glauber dynamics mixes in polynomial time with high probability over the quenched fields, via a Poincaré inequality. Under the additional assumption that the fields are bounded, a modified log-Sobolev inequality (MLSI) is established. For weaker fields satisfying weak spatial mixing in expectation and graphs with at most α-stretched exponential volume growth (α<1), a weak Poincaré inequality is shown, yielding a polynomial-time warm-start sampling algorithm via Glauber dynamics. These results extend prior lattice-specific theorems to general bounded-degree graphs without assuming local geometry beyond the growth condition.

Significance. If the technical proofs hold, the results are significant: they provide the first rigorous polynomial mixing guarantees for Glauber dynamics on general bounded-degree graphs in the Griffiths phase of the RFIM, where exponential decay holds in expectation but not pointwise. The extension from ℤ^d to graphs with only a volume-growth hypothesis broadens the applicability to irregular structures arising in statistical physics and computer science. The explicit use of anti-concentration, boundedness, and weak spatial mixing assumptions aligns with known regimes where lattice proofs succeed, and the high-probability quenched statement is a natural strengthening.

minor comments (3)
  1. The precise quantitative form of the 'sufficiently anti-concentrated' condition on the random-field distribution (used for the high-probability Poincaré inequality) should be stated explicitly in the introduction or as a numbered assumption, rather than left implicit from the abstract.
  2. In the statement of the weak Poincaré result, clarify whether the α-stretched exponential growth condition is with respect to graph distance or another metric, and confirm that the constant α<1 is uniform over the graph family.
  3. The manuscript would benefit from a short comparison table or paragraph contrasting the new assumptions (anti-concentration, bounded fields, stretched-exponential growth) with those in the cited lattice results (e.g., the integer-lattice papers mentioned).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for recommending minor revision. The summary provided in the report accurately reflects the main contributions of the manuscript, including the extension of mixing results for Glauber dynamics on the RFIM to general bounded-degree graphs under anti-concentration and volume-growth conditions.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes Poincaré inequalities (hence polynomial mixing) for Glauber dynamics on RFIM under anti-concentration of the random field, and a weak Poincaré under weaker mixing-in-expectation plus stretched-exponential growth, on bounded-degree graphs. These are direct extensions of lattice results using explicit assumptions (bounded Δ, anti-concentration or boundedness, volume growth) that do not reduce to fitted parameters, self-definitions, or load-bearing self-citations. Prior lattice work is cited only for context and motivation; the new proofs for general graphs rest on independent analytic arguments (e.g., spatial mixing estimates and canonical-path or variance bounds) that are verifiable against external benchmarks without circular reduction. No ansatz smuggling, uniqueness theorems imported from the same authors, or renaming of known results occurs in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions from probability theory and statistical mechanics (ferromagnetic interactions, i.i.d. random fields, bounded degree) without introducing new free parameters or invented entities.

axioms (2)
  • domain assumption The graph has finite maximum degree Δ.
    Invoked throughout to control local neighborhoods and mixing.
  • domain assumption Random fields are i.i.d. and sufficiently anti-concentrated.
    Central hypothesis for the high-probability Poincaré inequality.

pith-pipeline@v0.9.0 · 5574 in / 1276 out tokens · 17944 ms · 2026-05-08T19:33:15.217190+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

28 extracted references · 5 canonical work pages · 1 internal anchor

  1. [1]

    Rounding effects of quenched randomness on first- order phase transitions

    Michael Aizenman and Jan Wehr. “Rounding effects of quenched randomness on first- order phase transitions”. In:Communications in mathematical physics130.3 (1990), pp. 489–528

    1990

  2. [2]

    Entropic independence I : Modified log- S obolev inequalities for fractionally log-concave distributions and high-temperature I sing models

    Nima Anari et al. “Entropic independence I: Modified log-Sobolev inequalities for fractionally log-concave distributions and high-temperature ising models”. In:arXiv preprint arXiv:2106.04105(2021)

    work page arXiv 2021

  3. [3]

    Log-Sobolev inequality for near critical Ising models

    Roland Bauerschmidt and Benoit Dagallier. “Log-Sobolev inequality for near critical Ising models”. In:Communications on Pure and Applied Mathematics77.4 (2024), pp. 2568–2576

    2024

  4. [4]

    Phase transition in the 3d random field Ising model

    Jean Bricmont and Antti Kupiainen. “Phase transition in the 3d random field Ising model”. In:Communications in mathematical physics116.4 (1988), pp. 539–572

    1988

  5. [5]

    # BIS-hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness region

    Jin-Yi Cai et al. “# BIS-hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness region”. In:Journal of Computer and System Sci- ences82.5 (2016), pp. 690–711

    2016

  6. [6]

    Approximate Tensorization of En- tropy at High Temperature

    Pietro Caputo, Georg Menz, and Prasad Tetali. “Approximate Tensorization of En- tropy at High Temperature”. In:Annales de la Facult´ e des Sciences de Toulouse. Math´ ematiques24.4 (2015), pp. 691–716.doi:10.5802/afst.1460

    work page doi:10.5802/afst.1460 2015

  7. [7]

    Edge-Tilting Field Dynamics: Rapid Mixing at the Uniqueness Threshold and Optimal Mixing for Swendsen-Wang Dynamics

    Xiaoyu Chen et al. “Edge-Tilting Field Dynamics: Rapid Mixing at the Uniqueness Threshold and Optimal Mixing for Swendsen-Wang Dynamics”. In:arXiv preprint arXiv:2604.10525(2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  8. [8]

    Rapid mixing at the uniqueness threshold

    Xiaoyu Chen et al. “Rapid mixing at the uniqueness threshold”. In:Proceedings of the 57th Annual ACM Symposium on Theory of Computing. 2025, pp. 879–890

    2025

  9. [9]

    Rapid mixing of Glauber dynamics via spectral independence for all degrees

    Xiaoyu Chen et al. “Rapid mixing of Glauber dynamics via spectral independence for all degrees”. In:SIAM Journal on Computing(2024), FOCS21–224

    2024

  10. [10]

    Rapid mixing on random regular graphs beyond uniqueness

    Xiaoyu Chen et al. “Rapid mixing on random regular graphs beyond uniqueness”. In: arXiv preprint arXiv:2504.03406(2025). REFERENCES 31

    work page arXiv 2025

  11. [11]

    Localization schemes: A framework for proving mix- ing bounds for Markov chains

    Yuansi Chen and Ronen Eldan. “Localization schemes: A framework for proving mix- ing bounds for Markov chains”. In:2022 IEEE 63rd Annual symposium on foundations of computer science (FOCS). IEEE. 2022, pp. 110–122

    2022

  12. [12]

    Optimal mixing of Glauber dynamics: Entropy factorization via high-dimensional expansion

    Zongchen Chen, Kuikui Liu, and Eric Vigoda. “Optimal mixing of Glauber dynamics: Entropy factorization via high-dimensional expansion”. In:Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. 2021, pp. 1537–1550

    2021

  13. [13]

    Long range order for three-dimensional ran- dom field Ising model throughout the entire low temperature regime

    Jian Ding, Yu Liu, and Aoteng Xia. “Long range order for three-dimensional ran- dom field Ising model throughout the entire low temperature regime”. In:Inventiones mathematicae238.1 (2024), pp. 247–281

    2024

  14. [14]

    Exponential decay of correlations in the two-dimensional random field Ising model

    Jian Ding and Jiaming Xia. “Exponential decay of correlations in the two-dimensional random field Ising model”. In:Inventiones mathematicae224.3 (2021), pp. 999–1045

    2021

  15. [15]

    The total progeny in a branching process and a related random walk

    Meyer Dwass. “The total progeny in a branching process and a related random walk”. In:Journal of Applied Probability6 (1969), pp. 682–686

    1969

  16. [16]

    Fast relaxation of the random field Ising dynamics

    Ahmed El Alaoui et al. “Fast relaxation of the random field Ising dynamics”. In:The Annals of Probability54.1 (2026), pp. 99–136

    2026

  17. [17]

    A spectral condition for spectral gap: fast mixing in high-temperature Ising models

    Ronen Eldan, Frederic Koehler, and Ofer Zeitouni. “A spectral condition for spectral gap: fast mixing in high-temperature Ising models”. In:Probability theory and related fields182.3 (2022), pp. 1035–1051

    2022

  18. [18]

    Time-dependent statistics of the Ising model

    Roy J Glauber. “Time-dependent statistics of the Ising model”. In:Journal of math- ematical physics4.2 (1963), pp. 294–307

    1963

  19. [19]

    Geoffrey Grimmett.Percolation. 2nd ed. Springer, 1999

    1999

  20. [20]

    Approximation algorithms for the random field Ising model

    Tyler Helmuth et al. “Approximation algorithms for the random field Ising model”. In:SIAM Journal on Discrete Mathematics37.3 (2023), pp. 1610–1629

    2023

  21. [21]

    The ground state of the three-dimensional random-field Ising model

    John Z Imbrie. “The ground state of the three-dimensional random-field Ising model”. In:Communications in mathematical physics98.2 (1985), pp. 145–176

    1985

  22. [22]

    Random-field instability of the ordered state of continuous symmetry

    Yoseph Imry and Shang-keng Ma. “Random-field instability of the ordered state of continuous symmetry”. In:Physical Review Letters35.21 (1975), p. 1399

    1975

  23. [23]

    Levin, Yuval Peres, and Elizabeth L

    David A. Levin, Yuval Peres, and Elizabeth L. Wilmer.Markov Chains and Mixing Times. 2nd ed. Providence, RI: American Mathematical Society, 2017

    2017

  24. [24]

    Martinelli

    Fabio Martinelli. “Lectures on Glauber Dynamics for Discrete Spin Models”. In:Lec- tures on Probability Theory and Statistics. Vol. 1717. Lecture Notes in Mathematics. Berlin: Springer, 1999, pp. 93–191.doi:10.1007/978-3-540-48115-7_2

    work page doi:10.1007/978-3-540-48115-7_2 1999

  25. [25]

    Approach to Equilibrium of Glauber Dynamics in the One Phase Region. I. The Attractive Case

    Fabio Martinelli and Enzo Olivieri. “Approach to Equilibrium of Glauber Dynamics in the One Phase Region. I. The Attractive Case”. In:Communications in Mathematical Physics161.3 (1994), pp. 447–486

    1994

  26. [26]

    Jim Pitman.Combinatorial Stochastic Processes. Vol. 1875. Lecture Notes in Mathe- matics. Springer, 2006

    2006

  27. [27]

    The logarithmic Sobolev inequality for discrete spin systems on a lattice

    Daniel W Stroock and Boguslaw Zegarlinski. “The logarithmic Sobolev inequality for discrete spin systems on a lattice”. In:Communications in Mathematical Physics149.1 (1992), pp. 175–193

    1992

  28. [28]

    Counting independent sets up to the tree threshold

    Dror Weitz. “Counting independent sets up to the tree threshold”. In:Proceedings of the thirty-eighth annual ACM symposium on Theory of computing. 2006, pp. 140–149. Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ Email address:hanyi@ias.edu

    2006