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arxiv: 2601.07169 · v2 · pith:P26K3JVInew · submitted 2026-01-12 · 🧮 math.PR · cond-mat.stat-mech· cs.DM· math.CO· math.ST· stat.TH

Approximate FKG inequalities for phase-bound spin systems, with applications to central limit theorems for exponential random graphs

Satyaki Mukherjee , Vilas Winstein This is my paper

Pith reviewed 2026-05-21 16:14 UTC · model grok-4.3

classification 🧮 math.PR cond-mat.stat-mechcs.DMmath.COmath.STstat.TH
keywords FKG inequalityexponential random graphscentral limit theoremsphase coexistencemetastable mixingspin systemsCurie-Weiss models
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The pith

Individual phases in exponential random graph models satisfy an approximate FKG inequality.

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

This paper shows that when a spin system is confined to one phase in a low-temperature regime with multiple coexisting phases, it still displays an approximate form of the FKG inequality, which gives positive correlations between increasing functions. A sympathetic reader would care because the standard FKG lattice condition fails inside phases, leaving open whether the positive correlations seen in the overall mixture arise inside each phase or only from the choice between phases. The authors derive the approximate inequality from assumptions on metastable mixing and apply it to complete proofs of central limit theorems for subgraph counts inside each phase of ERGMs, resolving a question from prior work. They frame the ERGM result as a special case of a more general theorem expected to apply to other phase-bound systems.

Core claim

The individual phases in ERGMs satisfy an approximate form of the FKG inequality internally. This is used to finish the proof of various CLTs within each individual phase in the phase-coexistence regime. The FKG inequality for ERGMs follows from a more general result that holds under certain inputs related to metastable mixing, with the general result also applying to a class of generalized higher-order ferromagnetic Curie-Weiss models.

What carries the argument

Approximate FKG inequality for phase-bound spin systems derived from metastable mixing inputs.

Load-bearing premise

The phases must satisfy certain metastable mixing conditions for the approximate FKG inequality to hold.

What would settle it

A direct calculation or simulation that finds two coordinate-wise increasing functions whose covariance is negative when the system is conditioned to remain inside one fixed phase.

Figures

Figures reproduced from arXiv: 2601.07169 by Satyaki Mukherjee, Vilas Winstein.

Figure 1
Figure 1. Figure 1: Left: even if x and y are in the metastable well, x ∧ y or x ∨ y may not be. Right: to remedy this, we assume that x and y are almost ordered, which means that x ∧ y is close to either x or y, and similarly for x ∨ y. To make this argument work, we also need to restrict x and y to be a bit away from the boundary of the metastable well, so that nearby points remain in the well. Note that in these pictures, … view at source ↗
read the original abstract

The Fortuin-Kasteleyn-Ginibre (FKG) inequality is an invaluable tool in monotone spin systems satisfying the FKG lattice condition, which provides positive correlations for all coordinate-wise increasing functions of spins. This inequality has numerous applications and plays an integral role in the proof of various central limit theorems (CLTs), including recent work on ferromagnetic exponential random graph models (ERGMs) wherein a Hamiltonian tilt promotes the presence of small subgraphs like triangles. However, the FKG lattice condition fails to hold when confining a spin system to a particular phase in the low-temperature regime of parameters. Thus it is not a priori clear if each phase internally has positive correlations for increasing functions, or if the positive correlations in the overall model (which is a mixture of phases) arise primarily from the global choice of phase. In this article, we show that the individual phases in ERGMs do indeed satisfy an approximate form of the FKG inequality internally. We use this to finish the proof of various CLTs within each individual phase in the phase-coexistence regime, answering a question posed by Bianchi, Collet, and Magnanini. We present the FKG inequality for ERGMs as a consequence of a more general result which holds under certain inputs related to metastable mixing; we expect this general result to be widely applicable, and we devote a section to spelling out the details of its application to a class of generalized higher-order ferromagnetic Curie-Weiss models where the necessary inputs are relatively transparent.

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

1 major / 2 minor

Summary. The paper establishes an approximate FKG inequality that holds inside individual phases of spin systems in the phase-coexistence regime, under quantitative metastable mixing assumptions. This is derived from a general theorem and applied to complete the proofs of central limit theorems for ferromagnetic exponential random graph models (ERGMs) within each phase, answering a question of Bianchi, Collet, and Magnanini. The general result is illustrated in detail for generalized higher-order ferromagnetic Curie-Weiss models, where the mixing inputs are more transparent.

Significance. If the metastable mixing hypotheses are verified with uniform quantitative bounds inside the coexistence window, the approximate FKG provides a useful tool for controlling correlations within phases and thereby obtaining CLTs where the global mixture prevents direct application of the classical FKG lattice condition. The general framework and its explicit Curie-Weiss treatment could extend to other metastable spin systems.

major comments (1)
  1. [Application to ERGMs] Application to ERGMs section: the claim that the metastable mixing inputs hold for ERGMs in the phase-coexistence regime is stated without a self-contained verification that the required correlation-decay or mixing-time bounds remain uniform near the coexistence boundary; if these rates degrade, the error term in the approximate FKG may fail to be o(1) on the scale needed for the CLT.
minor comments (2)
  1. [General result] The precise quantitative form of the approximate FKG (including the dependence of the error on the mixing parameters) should be stated explicitly in the statement of the general theorem.
  2. Notation for the phase indicator and the restricted measure should be introduced earlier and used consistently when passing from the general theorem to the ERGM application.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the recognition of the potential utility of the approximate FKG inequality for metastable spin systems and its role in completing the CLT proofs for ERGMs. We address the major comment below.

read point-by-point responses

  1. Referee: Application to ERGMs section: the claim that the metastable mixing inputs hold for ERGMs in the phase-coexistence regime is stated without a self-contained verification that the required correlation-decay or mixing-time bounds remain uniform near the coexistence boundary; if these rates degrade, the error term in the approximate FKG may fail to be o(1) on the scale needed for the CLT.

    Authors: We thank the referee for highlighting this point. Our general theorem takes the metastable mixing assumptions (including quantitative correlation decay and mixing-time bounds) as inputs, and the ERGM application invokes these from the existing literature on phase transitions and large deviations for exponential random graphs. We agree that the manuscript would benefit from greater transparency on the uniformity of these bounds near the coexistence boundary to confirm that the resulting error in the approximate FKG remains o(1) at the scale required for the CLT. In the revised version, we will add a clarifying remark or short subsection in the ERGM application section that explicitly cites and summarizes the relevant uniform quantitative estimates from the literature, ensuring the error control is rigorous. This does not alter the main theorems but strengthens the presentation. revision: yes

Circularity Check

0 steps flagged

Derivation applies general metastable mixing theorem to obtain approximate FKG for phases; inputs treated as independent and verified for Curie-Weiss class

full rationale

The paper derives the approximate FKG inequality for individual ERGM phases as a direct consequence of a more general theorem that assumes certain metastable mixing conditions (quantitative mixing times or correlation decay inside phases). These inputs are presented as external to the target FKG statement and are spelled out explicitly for generalized Curie-Weiss models in a dedicated section, where they are relatively transparent. No step in the provided abstract or description reduces the claimed FKG or the subsequent CLT completion to a fitted parameter, self-definition, or load-bearing self-citation chain; the central result remains an instantiation of the general theorem rather than a tautological renaming or construction from the ERGM data itself. The derivation is therefore self-contained against the stated external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on inputs related to metastable mixing for the general result and on standard definitions of phases and spin systems from prior literature. No free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • standard math Standard probability axioms and definitions of spin systems and phases.
    The work builds on the FKG lattice condition and phase coexistence concepts from earlier papers.

pith-pipeline@v0.9.0 · 5823 in / 1245 out tokens · 51293 ms · 2026-05-21T16:14:24.714373+00:00 · methodology

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