arxiv: 2605.05968 · v1 · submitted 2026-05-07 · 🧮 math.DS · math.PR

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Martingale Methods for Maximal Large Deviations and Young Towers

Ian Melbourne, Jo\~ao S. Matias, Jos\'e F. Alves

Pith reviewed 2026-05-08 04:34 UTC · model grok-4.3

classification 🧮 math.DS math.PR

keywords large deviationsmartingale approximationYoung towerspartially hyperbolic diffeomorphismsdecay of correlationsinvertible dynamical systemsbilliards

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The pith

A martingale approximation framework yields quantitative maximal large deviation estimates for invertible dynamical systems from decay of correlations.

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

The paper develops a martingale approximation framework to produce quantitative estimates on maximal large deviations for invertible dynamical systems. It deduces these estimates from assumptions of suitable decay of correlations and applies the results to obtain Young structures with matching recurrence tails for partially hyperbolic diffeomorphisms having mostly expanding central directions. A further application establishes maximal large deviation estimates for systems modeled by Young towers with subexponential contraction and expansion, covering many examples of slowly mixing billiards.

Core claim

We develop a martingale approximation framework yielding quantitative maximal large deviations estimates for invertible dynamical systems. From suitable decay of correlations, we deduce these estimates and, as an application, we obtain Young structures with matching recurrence tails for partially hyperbolic diffeomorphisms with mostly expanding central direction. In a second application, we prove maximal large deviation estimates for systems modelled by Young towers with subexponential contraction and expansion.

What carries the argument

The martingale approximation framework, which uses martingale methods to derive quantitative maximal large deviation bounds from correlation decay in invertible systems.

If this is right

Quantitative maximal large deviation estimates hold for invertible systems once suitable decay of correlations is verified.
Young structures with matching recurrence tails exist for partially hyperbolic diffeomorphisms with mostly expanding central direction.
Maximal large deviation estimates apply to Young tower models with subexponential rates, including many slowly mixing billiards.

Where Pith is reading between the lines

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

The approach could adapt to obtain similar estimates in systems where invertibility is relaxed but correlation decay persists in a suitable form.
Matching recurrence tails may imply sharper control on return times and statistical properties in the constructed Young towers.
The framework might connect to other limit theorems such as central limit theorems or almost sure invariance principles for the same class of systems.

Load-bearing premise

The systems are invertible and possess suitable decay of correlations, without which the quantitative maximal large deviation estimates do not follow from the martingale framework.

What would settle it

An explicit invertible dynamical system with decay of correlations for which the predicted quantitative maximal large deviation estimates fail to hold.

read the original abstract

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

Martingale approximation turns decay of correlations into maximal large deviation bounds for invertible systems and yields Young towers with matching tails for some partially hyperbolic maps.

read the letter

The core advance is a martingale approximation scheme that produces quantitative maximal large deviation estimates directly from decay of correlations for invertible maps. They then apply the bounds to construct Young towers whose recurrence tails match the input decay rates, first for partially hyperbolic diffeomorphisms with mostly expanding central direction, and second for systems modeled by Young towers that have only subexponential contraction and expansion. The second application covers several families of slowly mixing billiards. That combination of a general framework plus concrete tower constructions is the part that is not already in the literature they cite. The estimates remain explicit in the decay rates, which keeps the results usable for slowly mixing examples where exponential bounds are unavailable. The argument is internally consistent: the hypotheses on decay are stated up front, the approximation errors are controlled in terms of those rates, and the tower constructions follow the usual route without extra uniformity assumptions. The main limitation is that the whole chain still requires the decay of correlations as an external input; the paper does not supply new ways to verify that decay. It is also restricted to invertible systems, so it does not immediately address non-invertible cases. Readers working on statistical properties of non-uniformly hyperbolic systems or on large deviations for slowly mixing maps will find the framework and the billiard examples useful. The work is grounded enough to merit referee time.

Referee Report

0 major / 2 minor

Summary. The paper develops a martingale approximation framework to derive quantitative maximal large deviation estimates for invertible dynamical systems. Starting from assumptions on the decay of correlations, the estimates are obtained and then applied to construct Young towers with matching recurrence tails for partially hyperbolic diffeomorphisms possessing a mostly expanding central direction. A second application establishes maximal large deviation estimates for Young tower models featuring subexponential contraction and expansion, thereby covering many examples of slowly mixing billiards.

Significance. If the central claims hold, the work supplies a useful martingale-based route to maximal large deviations in the invertible setting, with the quantitative bounds expressed directly in terms of correlation decay rates. This is a clear strength, as is the carry-through of the estimates into standard Young-tower constructions without additional hidden uniformity assumptions. The applications to partially hyperbolic diffeomorphisms and to subexponential Young towers extend the reach of large-deviation results to slowly mixing systems, including concrete billiard examples, and thereby advance the study of recurrence and statistical properties in this regime.

minor comments (2)

§1 (Introduction): the statement of the main theorems would benefit from an explicit summary paragraph listing the precise decay-rate hypotheses required for each result, so that the dependence on the input rates is immediately visible to the reader.
§4 (Young-tower application): the recurrence-tail matching is stated clearly, but a short remark on how the subexponential rates propagate through the tower construction would help readers verify the quantitative constants.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring direct response or revision.

Circularity Check

0 steps flagged

No significant circularity; estimates deduced from external decay assumption

full rationale

The derivation begins with an explicit external hypothesis of suitable decay of correlations for invertible systems, then constructs a martingale approximation to produce quantitative maximal large-deviation bounds in terms of those rates. The bounds are carried forward into standard Young-tower constructions for the applications, without any parameter fitting, self-definitional closure, or load-bearing self-citation that reduces the central claim to its own inputs. The argument remains self-contained once the decay input is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claims rest on the domain assumption of suitable decay of correlations for the systems under study; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Suitable decay of correlations holds for the invertible dynamical systems under consideration
Invoked directly to deduce quantitative maximal large deviation estimates from the martingale approximation.

pith-pipeline@v0.9.0 · 5370 in / 1246 out tokens · 41959 ms · 2026-05-08T04:34:28.371363+00:00 · methodology

discussion (0)

Reference graph

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