arxiv: 2605.08629 · v1 · submitted 2026-05-09 · 🧮 math.PR

Recognition: 1 theorem link

· Lean Theorem

Moderate deviations for the Maki--Thompson rumour model

Guangyu Yang, Shaochen Wang

Pith reviewed 2026-05-12 00:50 UTC · model grok-4.3

classification 🧮 math.PR

keywords moderate deviation principleMaki-Thompson rumour modelfinal size distributionlarge deviationscentral limit theoremstochastic epidemic modelsautomata numbers

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The pith

The final proportion of ignorants in the Maki-Thompson rumour model obeys a moderate deviation principle.

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

The paper establishes a moderate deviation principle for the final proportion of ignorants who never hear the rumour. This principle covers deviations from the typical limiting value that sit between the square-root scale of the central limit theorem and the fixed-scale regime of large deviations. A reader cares because the result supplies a single description that connects the Gaussian fluctuations already known for this model to the exponential tail rates that govern rarer events. The work therefore completes a scale-by-scale picture of how the rumour spreads or fails to spread in the classical stochastic setting.

Core claim

The final proportion of ignorants in the classical Maki-Thompson rumour model satisfies the moderate deviation principle. This statement bridges the central limit theorem regime and the large deviation principle regime. The proof rests on the exact final-size distribution, sharp asymptotics for the associated automata numbers, and a uniform point probability expansion at the moderate deviation scale.

What carries the argument

The moderate deviation principle itself, obtained by combining the exact final-size distribution with sharp asymptotics for the automata numbers and a uniform point probability expansion.

Load-bearing premise

The exact final-size distribution and the sharp asymptotics for the automata numbers remain accurate and uniform when the deviation scale is intermediate between the central limit and large deviation regimes.

What would settle it

Direct Monte Carlo simulation of many independent copies of the process that counts the empirical log-probabilities of moderate deviations and checks whether they align with the predicted rate function at that scale.

read the original abstract

The final proportion of ignorants in the classical Maki--Thompson rumour model is known to satisfy the law of large numbers, the central limit theorem, and the large deviation principle. In this note, we establish the corresponding moderate deviation principle, thereby bridging the Gaussian fluctuation regime and the large deviation regime. The proof rests on the exact final-size distribution, sharp asymptotics for the associated automata numbers, and a uniform point probability expansion at the moderate deviation scale.

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 note proves the moderate deviation principle for the final ignorant proportion in the Maki-Thompson model by combining the known exact final-size distribution with sharp automata asymptotics and a uniform local expansion.

read the letter

The paper fills the moderate-deviation gap for the final proportion of ignorants in the classical Maki-Thompson rumor model. It sits between the existing central limit theorem and large deviation principle, which is a natural and often useful step for these processes. The argument rests on the exact final-size distribution, sharp asymptotics for the associated automata numbers, and a uniform point-probability expansion at the moderate scale. Those are the standard ingredients for moving from CLT to LDP in this class of models, and the abstract gives no indication that uniformity breaks in the intermediate regime. If the error bounds carry through as claimed, the result should hold without circularity or extra fitting. The work is short and focused, which suits a note. The citation pattern looks appropriate, referencing the prior LLN, CLT, and LDP results directly. No load-bearing assumptions appear invented or self-referential. The main limitation is that the full uniformity statement and explicit constants are not visible from the abstract alone, so the precise width of the moderate-deviation window remains to be checked in the text. That is a minor point for this type of paper rather than a flaw in the architecture. Readers working on scaling limits for epidemic or rumor models will find the result relevant and the method reproducible. It is a clean, incremental extension that completes the picture for this well-studied process. I would send it to peer review; the ingredients are classical and the claim is new for the model, so a referee can verify the details without starting from scratch.

Referee Report

0 major / 1 minor

Summary. The manuscript establishes the moderate deviation principle for the final proportion of ignorants in the Maki-Thompson rumour model. Building on the known law of large numbers, central limit theorem, and large deviation principle for this quantity, the note derives the MDP by invoking the exact final-size distribution, sharp asymptotics for the associated automata numbers, and a uniform local probability expansion at the moderate-deviation scale.

Significance. If the claimed MDP holds, the result supplies the missing intermediate-scale fluctuation regime for a classical interacting-particle model in probability theory and epidemic modeling. The approach via exact distributions and combinatorial asymptotics provides a clean bridge between the Gaussian and large-deviation regimes without parameter fitting, which strengthens the overall picture of the model's concentration behavior.

minor comments (1)

[Abstract] The abstract refers to 'sharp asymptotics for the associated automata numbers' and 'uniform point probability expansion'; a brief sentence in the introduction recalling the precise scaling (e.g., deviation order sqrt(n log n) or equivalent) would help readers locate the moderate-deviation window immediately.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, careful summary of the contribution, and recommendation to accept the manuscript. We are pleased that the bridging of the Gaussian and large-deviation regimes via exact distributions and combinatorial asymptotics was viewed as strengthening the overall picture of concentration behavior for the model.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives the moderate deviation principle directly from the classical exact final-size distribution of the Maki-Thompson model, sharp combinatorial asymptotics on automata counts, and a uniform local probability expansion at the moderate-deviation scale. These are independent, externally established inputs (standard in the literature on this model) rather than quantities fitted to or defined by the target MDP result. No self-definitional loop, fitted-input prediction, self-citation load-bearing premise, or ansatz smuggling is indicated in the proof architecture.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on three external ingredients: the exact final-size distribution (standard in the literature), sharp asymptotics for automata numbers (assumed available or derived), and a uniform point probability expansion at moderate scale (new technical step). No free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The exact final-size distribution of the Maki-Thompson model is known and usable for asymptotic analysis.
Invoked as the starting point for the moderate deviation proof.
domain assumption Sharp asymptotics for the associated automata numbers hold uniformly in the required range.
Required for the uniform point probability expansion.

pith-pipeline@v0.9.0 · 5358 in / 1301 out tokens · 44730 ms · 2026-05-12T00:50:03.306602+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
The proof rests on the exact final-size distribution, sharp asymptotics for the associated automata numbers, and a uniform point probability expansion at the moderate deviation scale.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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