arxiv: 2605.07341 · v1 · submitted 2026-05-08 · 🧮 math.PR

Recognition: 2 theorem links

· Lean Theorem

A Scaling Limit of Random Walks in the Rational Adeles

Rahul Rajkumar

Pith reviewed 2026-05-11 01:17 UTC · model grok-4.3

classification 🧮 math.PR

keywords random walksrational adelesLévy processesscaling limitsSkorokhod topologyweak convergencep-adic numberssurvival times

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The pith

Adele-valued random walks converge weakly to an adelic Lévy process in the J1 Skorokhod topology.

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

The paper constructs random walks taking values in the rational adeles by first defining them componentwise on the p-adic numbers for each prime and then lifting to the infinite product. Survival time analysis on the individual p-adic components is used to prove that the walks remain supported on the adeles almost surely at every time. These adelic walks are shown to differ only by a small perturbation from processes that live on finite products of path spaces. The central result is that, after suitable scaling, the walks converge weakly to an adelic Lévy process. A sympathetic reader would care because the result supplies a scaling-limit description of stochastic motion that respects the full adelic structure of the rationals.

Core claim

The paper proves that suitably scaled random walks on the rational adeles converge weakly to an adelic Lévy process in the J1 Skorokhod topology. The construction begins with random walks on each p-adic component, lifts them to the infinite product, and uses survival-time analysis to verify that the resulting process is almost surely adelic for all times. The adelic walks are then shown to be small perturbations of processes supported on finite products of path spaces, after which weak convergence to the adelic Lévy limit is established.

What carries the argument

Survival time analysis on the p-adic components, which establishes that the infinite-product random walk is almost surely adelic for all time and thereby allows reduction to finite-product approximations.

Load-bearing premise

The random walks defined on the infinite product of p-adic spaces remain almost surely adelic at every time, which is proved via survival time analysis on the individual p-adic components.

What would settle it

An explicit construction or simulation of a random walk on the product space whose total time spent outside any finite set of p-adic components is infinite with positive probability.

read the original abstract

This paper shows the convergence of adele-valued random walks to an adelic L\'evy process under scaling limits. We use random walks on the $p$-adic numbers to construct random walks initially on the infinite product space, and use survival time analysis to prove that the random walks are almost surely adelic for all time. The adelic random walks are shown to be small perturbations of processes that are supported on a finite product of path spaces. Weak convergence to an adelic L\'evy process is established in the $J_1$ Skorokhod topology.

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

The paper builds adele-valued walks from p-adic components, claims a survival-time argument keeps them adelic almost surely, and proves a scaling limit to an adelic Lévy process in J1 topology.

read the letter

The paper constructs random walks on the rational adeles by combining p-adic random walks on each component and then shows that a scaled version converges weakly to an adelic Lévy process in the J1 Skorokhod topology. It does this by first building the process on the infinite product and using survival time analysis to show it lands in the adeles almost surely at every time. The argument then reduces the infinite case to a perturbation of a finite-product process, which is easier to handle. This is a natural move from p-adic probability to the adelic setting, and the topology choice fits the cadlag paths one expects from Lévy processes. The potential issue is whether the survival probabilities really make the process adelic almost surely. For independent components, each p-adic walk has a positive chance of leaving Z_p by any fixed time, and if those probabilities do not sum to a finite number over all primes, then by Borel-Cantelli infinitely many components will have left by that time almost surely. The paper claims the survival time analysis fixes this, so the estimates on how the exit rate behaves for large p must be the key part to verify. If those estimates work out, the perturbation step and the convergence should follow from standard results on Skorokhod spaces. This kind of work is mainly for specialists in stochastic processes on non-Archimedean groups or in adelic analysis. Someone already familiar with p-adic Lévy processes would see the value quickly. It is worth sending to peer review. The construction is original and the claims are stated clearly enough that referees can check the details on the adelic property and the convergence.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes a scaling limit result for random walks taking values in the rational adeles: suitably scaled walks constructed from independent p-adic components converge weakly to an adelic Lévy process in the J1 Skorokhod topology. The argument proceeds by first building the walks on the infinite product space, invoking survival time analysis to conclude that the paths lie in the adeles almost surely for all time, representing the resulting processes as small perturbations of walks supported on finite products of path spaces, and then transferring standard weak-convergence results to the adelic setting.

Significance. If the central claims are correct, the result would be a meaningful extension of scaling-limit theory to the adelic setting, combining Archimedean and non-Archimedean components in a single stochastic process. The technical contributions—survival-time control on countably many p-adic walks and a perturbation reduction to finite products—are potentially reusable in other infinite-product constructions. The manuscript receives credit for explicitly identifying the J1 topology and for outlining a perturbation strategy that avoids direct analysis of the infinite product.

major comments (2)

[Abstract] Abstract (and the survival-time section): the claim that survival-time analysis yields almost-sure adelicity for all time is load-bearing. The p-adic components are independent, so each has exit probability 1-q_p > 0 from Z_p. Without an explicit decay rate on 1-q_p such that sum_p (1-q_p) < ∞, the Borel-Cantelli lemma implies that infinitely many components exit Z_p almost surely, so the path fails to be adelic with probability 1. The manuscript must supply the necessary tail estimates or dependence structure to close this gap.
[Perturbation argument] The perturbation argument (paragraph following the survival-time claim): the statement that the adelic walks are 'small perturbations' of finite-product processes requires a precise quantification of the perturbation size in the J1 metric on the infinite product space. It is not clear whether the perturbation vanishes in the scaling limit or merely remains bounded, which directly affects whether the limit object remains adelic.

minor comments (2)

The abstract refers to 'rational adeles' while the title uses 'Rational Adeles'; adopt a single capitalization convention.
The definition of the target adelic Lévy process should be stated explicitly before the convergence theorem, rather than deferred to the final section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The points raised identify areas where additional explicit details will strengthen the presentation. We address each major comment below and will incorporate the necessary revisions.

read point-by-point responses

Referee: [Abstract] Abstract (and the survival-time section): the claim that survival-time analysis yields almost-sure adelicity for all time is load-bearing. The p-adic components are independent, so each has exit probability 1-q_p > 0 from Z_p. Without an explicit decay rate on 1-q_p such that sum_p (1-q_p) < ∞, the Borel-Cantelli lemma implies that infinitely many components exit Z_p almost surely, so the path fails to be adelic with probability 1. The manuscript must supply the necessary tail estimates or dependence structure to close this gap.

Authors: The referee correctly identifies that the current text does not supply explicit tail estimates on the exit probabilities. We will revise the survival-time section to introduce a concrete decay condition (e.g., 1-q_p = O(p^{-2})) ensuring ∑_p (1-q_p) < ∞. With this condition and the independence of the p-adic components, the Borel-Cantelli lemma directly yields that only finitely many components exit Z_p almost surely, so the paths remain adelic for all time with probability 1. The revised manuscript will state the assumption on q_p explicitly and include the lemma application. revision: yes
Referee: [Perturbation argument] The perturbation argument (paragraph following the survival-time claim): the statement that the adelic walks are 'small perturbations' of finite-product processes requires a precise quantification of the perturbation size in the J1 metric on the infinite product space. It is not clear whether the perturbation vanishes in the scaling limit or merely remains bounded, which directly affects whether the limit object remains adelic.

Authors: We agree that a precise quantification of the perturbation in the J1 metric is required. We will add a lemma that bounds the J1 distance between the adelic process and its finite-product truncation. Under the scaling, this distance tends to zero in probability as the scaling parameter tends to infinity (for any fixed truncation level, with the tail controlled uniformly). This ensures the weak limit remains an adelic Lévy process. The revised argument will include the explicit metric estimates. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs random walks on the infinite product of p-adics, applies survival time analysis to establish the almost-sure adelic property for all time, then uses perturbation to finite products and proves weak convergence to an adelic Lévy process in the J1 Skorokhod topology. None of these steps reduces by construction to its own inputs, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation chain; the argument invokes standard external tools (Skorokhod topology, p-adic valuation properties, Borel-Cantelli lemmas) whose validity is independent of the target result. The derivation therefore remains non-circular even if the survival-time estimate itself requires verification against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard mathematical structures from number theory and probability; no free parameters or invented entities are apparent from the abstract.

axioms (1)

standard math Standard properties of p-adic numbers, adeles as restricted direct products, and the J1 Skorokhod topology on path spaces.
Invoked in the construction of the random walks and the convergence statement.

pith-pipeline@v0.9.0 · 5379 in / 1193 out tokens · 33845 ms · 2026-05-11T01:17:34.558346+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
We use random walks on the p-adic numbers to construct random walks initially on the infinite product space, and use survival time analysis to prove that the random walks are almost surely adelic for all time.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Weak convergence to an adelic Lévy process is established in the J1 Skorokhod topology.

Reference graph

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