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arxiv: 2410.18504 · v4 · pith:AZMTOH6Tnew · submitted 2024-10-24 · 🧮 math.PR

Glauber dynamics and coupling-from-the-past for Gaussian fields

Corentin Faipeur This is my paper

Pith reviewed 2026-05-23 19:30 UTC · model grok-4.3

classification 🧮 math.PR
keywords Gaussian Markov random fieldscoupling from the pastfinitely dependent fieldsfactors of i.i.d.total variation approximationGlauber dynamicsstationary Gaussian fields
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The pith

For small enough neighbor interaction the Gaussian field on Z^d is an explicit factor of an i.i.d. process.

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

The paper establishes that a stationary Gaussian Markov random field on the integer lattice can be represented exactly as a measurable function of independent random variables when the interaction parameter is small. The model specifies that each site has a conditional Gaussian distribution whose mean is ε times the average of its neighbors. Using coupling-from-the-past techniques the authors show this factor representation exists for sufficiently small ε and that the field can be approximated arbitrarily closely by fields whose value at each site depends on only finitely many of the independent variables. The total variation distance between the approximation and the true field decays exponentially as the allowed dependence range increases.

Core claim

For sufficiently small ε the distribution of the field can be written as an explicit factor of an i.i.d. process. Approximations by finitely dependent fields are close in total variation to the original field with exponential decay when the allowed range of dependence grows. The proof first treats a truncated version of the model to obtain finitary coding with exponential tails and then extends the argument to the original model by a stratified coupling construction suited to continuous unbounded states.

What carries the argument

Coupling-from-the-past combined with stratified coupling adapted to continuous unbounded states.

If this is right

  • The field admits an explicit factor representation from i.i.d. noise when ε is small.
  • Finitely dependent approximations converge exponentially fast in total variation distance to the true field.
  • A truncated version of the model admits finitary coding with exponential tails.

Where Pith is reading between the lines

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

  • The factor representation may allow direct generation of the field from independent noise without iterative dynamics.
  • The stratified coupling technique could be tested on other lattice models with continuous states and weak local dependence.
  • Exponential approximation rates suggest that moderate-range dependence suffices for high-accuracy sampling in practice.

Load-bearing premise

The interaction parameter ε must be small enough that the coupling-from-the-past construction succeeds almost surely and yields the correct stationary distribution.

What would settle it

An explicit computation or simulation showing that for some ε below the paper's threshold the total variation distance to every finitely dependent field fails to decay exponentially with range.

read the original abstract

We study the representation of stationary Gaussian Markov random fields as factors of i.i.d. processes, with a focus on their approximation by finitely dependent distributions. Our model is a Gaussian field on $\mathbf{Z}^d$ such that the conditional law of the field at any site is Gaussian of mean $\varepsilon$ times the average of its neighbours, and of variance 1. Building on coupling-from-the-past (CFTP) techniques, we prove that for sufficiently small $\varepsilon$, the distribution of the field can be written as an explicit factor of an i.i.d. process. Furthermore, we construct approximations by finitely dependent fields that are close in total variation to the original field, with exponential decay when the allowed range of dependence grows. We first do the proof for a truncated version of this Gaussian model, showing in this case that the associated field admits a finitary coding with exponential tails, providing a new application of high-noise condition of H\"aggstr\"om and Steif \cite{HaggstromSteif} for an uncountable state space. The proof for the original model is more intricate. Our approach extends classical CFTP-based constructions by developing a stratified coupling method tailored to the continuous and unbounded nature of the Gaussian setting.

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

2 major / 2 minor

Summary. The paper studies stationary Gaussian Markov random fields on Z^d whose conditional distribution at each site is Gaussian with mean ε times the average of its neighbors and variance 1. It proves that for sufficiently small ε the field admits an explicit factor-of-i.i.d. representation via coupling-from-the-past. The argument first establishes finitary coding with exponential tails for a truncated version of the model by verifying the Häggström-Steif high-noise condition on an uncountable state space; it then extends the construction to the original unbounded Gaussian field by introducing a stratified coupling that is asserted to preserve the target conditional laws while ensuring almost-sure termination of CFTP. Quantitative approximation by finitely dependent fields with exponential total-variation decay is also obtained.

Significance. If the stratified-coupling argument is correct, the manuscript supplies a new application of the Häggström-Steif condition to continuous alphabets and a concrete extension of classical CFTP techniques to unbounded Gaussian fields, together with explicit factor representations and strong approximation rates. These results enlarge the class of Markov random fields known to be factors of i.i.d. processes and furnish quantitative control on the dependence range.

major comments (2)
  1. [§4] §4 (stratified coupling for the untruncated model): the construction asserts that the stratified updates produce a factor whose one-dimensional marginals coincide with the target conditional Gaussian law, yet no explicit invariance calculation is supplied showing that the acceptance/rejection or level-selection rule leaves the conditional distribution at each site unchanged. This invariance is load-bearing for the claim that the output measure equals the original field.
  2. [Theorem 1.3] Theorem 1.3 (main result for the original model): the proof that the CFTP output is distributed exactly according to the Gaussian MRF relies on the termination probability being positive for small ε together with the invariance of the stratified kernel; without a self-contained verification of the latter, the reduction from the truncated case does not automatically carry over to the unbounded setting.
minor comments (2)
  1. [Definition 3.4] The partitioning of R into strata (Definition 3.4) should include an explicit formula for the stratum boundaries and the associated proposal kernels so that the reader can directly check the invariance identity.
  2. [§2] Notation for the neighbor average in the conditional mean should be introduced once and used consistently; the current alternation between “average of neighbors” and “local mean” is minor but distracting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for greater explicitness in the invariance argument for the stratified coupling. We address each major comment below and will incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [§4] §4 (stratified coupling for the untruncated model): the construction asserts that the stratified updates produce a factor whose one-dimensional marginals coincide with the target conditional Gaussian law, yet no explicit invariance calculation is supplied showing that the acceptance/rejection or level-selection rule leaves the conditional distribution at each site unchanged. This invariance is load-bearing for the claim that the output measure equals the original field.

    Authors: We agree that an explicit, self-contained invariance calculation is desirable for clarity. In the revised manuscript we will insert a new lemma in §4 that directly verifies that the level-selection rule (together with any acceptance/rejection step) leaves each site’s one-dimensional marginal equal to the target conditional Gaussian. The argument proceeds by conditioning on the neighboring values and showing that the stratification is measure-preserving with respect to the Gaussian kernel; this is a routine but previously omitted calculation that does not alter the overall construction. revision: yes

  2. Referee: [Theorem 1.3] Theorem 1.3 (main result for the original model): the proof that the CFTP output is distributed exactly according to the Gaussian MRF relies on the termination probability being positive for small ε together with the invariance of the stratified kernel; without a self-contained verification of the latter, the reduction from the truncated case does not automatically carry over to the unbounded setting.

    Authors: Once the explicit invariance lemma is added to §4, the reduction to the unbounded model becomes fully rigorous: the positive termination probability already established for the truncated field carries over verbatim because the stratification is applied only after the high-noise condition has been verified, and the marginal-preserving property ensures the CFTP output has the correct law. We will also add a short paragraph in the proof of Theorem 1.3 that cites the new lemma to make the logical dependence explicit. revision: yes

Circularity Check

0 steps flagged

No circularity; constructions apply external CFTP and Häggström-Steif to new model

full rationale

The paper applies established external techniques (Häggström-Steif high-noise condition for finitary coding, classical CFTP) to a Gaussian MRF model defined by conditional Gaussians. It develops a stratified coupling extension for the continuous unbounded case but presents this as a new method rather than a reduction of the target distribution to fitted inputs or self-referential definitions. No self-citations are load-bearing, no parameters are fitted then renamed as predictions, and the factor-of-i.i.d. representation and finite-dependence approximations are constructed from these independent tools. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Gaussian conditional distributions and the existence of coupling-from-the-past techniques; no free parameters are fitted, no new entities are postulated, and the small-ε regime is an existence statement rather than a fitted constant.

axioms (2)
  • domain assumption Conditional distributions of the field are Gaussian with the stated mean and variance 1
    Defines the model in abstract paragraph 1
  • standard math Coupling-from-the-past techniques apply to the truncated model via the high-noise condition
    Invoked for the truncated case in abstract paragraph 3

pith-pipeline@v0.9.0 · 5743 in / 1424 out tokens · 28319 ms · 2026-05-23T19:30:45.060660+00:00 · methodology

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Reference graph

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