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arxiv: 2605.02433 · v1 · submitted 2026-05-04 · ❄️ cond-mat.stat-mech · math.PR

Recognition: 4 theorem links

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Aging Record Statistics in Saturating Self-Interacting Random Walks

J. Br\'emont , R. Voituriez , O. B\'enichou
Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:32 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math.PR
keywords record statisticsself-interacting random walksagingnon-Markovian processesextreme-value statisticsasymptotic distributionmemory effectsrandom walk geometry
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The pith

Saturating self-interacting random walks have an exact asymptotic record-age distribution that splits into a geometry-controlled regime and a memory-corrected regime.

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

The paper derives the exact asymptotic distribution of the record age tau_k, the waiting time between successive records, in saturating self-interacting random walks. This class of non-Markovian processes carries memory through an interaction that saturates with the number of visits to each site. The distribution depends on k and reveals two regimes separated by the scale k squared: below this scale the statistics are fixed by the shape of the visited region, while above it the memory contribution reduces to a k-dependent prefactor. A reader would care because the result converts scaling arguments about aging into a precise analytic formula for how memory alters extreme-value statistics.

Core claim

We derive the exact asymptotic distribution of tau_k for saturating self-interacting random walks. We uncover two asymptotic regimes, in agreement with recent scaling predictions: at short times (tau much smaller than k squared), record statistics are governed by the geometry of the explored region, while at long times (tau much larger than k squared), memory effects become subdominant and reduce to nontrivial prefactor corrections. Our exact result provides a rare analytic window beyond scaling theory and extends to a framework that fully quantifies aging dynamics in the presence of saturating self-interaction.

What carries the argument

The saturating self-interacting random walk, whose memory arises from a bounded interaction with visited sites, together with the asymptotic separation of its record-age distribution into a short-time geometric regime and a long-time prefactor-corrected regime.

If this is right

  • Record ages tau_k depend on the record index k, so the process exhibits aging.
  • For tau much smaller than k squared the distribution is determined solely by the geometry of the set of visited sites.
  • For tau much larger than k squared the non-Markovian memory appears only as a multiplicative k-dependent prefactor.
  • The two-regime structure holds for the broad class of saturating self-interacting walks and matches prior scaling predictions.
  • The derivation supplies an exact analytic description that goes beyond scaling arguments.

Where Pith is reading between the lines

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

  • The same separation into geometry and prefactor regimes may hold for other non-Markovian models whose memory saturates after a finite number of visits.
  • The geometric regime could be linked to known properties of the range and shape of random-walk trajectories in different dimensions.
  • Numerical verification of the exact distributions would provide a direct test of the regime boundary at tau approximately k squared.
  • The framework could be used to predict aging signatures in physical or biological systems that display saturating self-interaction.

Load-bearing premise

The memory generated by saturating self-interaction can be captured exactly by an asymptotic analysis that cleanly divides the distribution into a geometry-dominated part and a memory-only prefactor part without higher-order corrections that would change the functional form.

What would settle it

High-precision simulations of the walks for large k that show the empirical distribution of tau_k failing to match the predicted geometric expression for tau much less than k squared or the prefactor-corrected expression for tau much greater than k squared.

Figures

Figures reproduced from arXiv: 2605.02433 by J. Br\'emont, O. B\'enichou, R. Voituriez.

Figure 2
Figure 2. Figure 2: FIG. 2 view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 view at source ↗
read the original abstract

The record age tau_k, defined as the time between the k-th and k+1-st record-breaking events, is a central observable of extreme-value statistics. In Markovian processes, the absence of memory makes tau_k independent of k. How memory breaks this invariance and induces aging, meaning a dependence of tau_k on k, remains a fundamental question, closely connected to widely observed aging phenomena in non-Markovian dynamics. In this Letter, we derive the exact asymptotic distribution of tau_k for saturating self-interacting random walks, a broad class of non-Markovian processes. We uncover two asymptotic regimes, in agreement with recent scaling predictions: at short times (tau much smaller than k squared), record statistics are governed by the geometry of the explored region, while at long times (tau much larger than k squared), memory effects become subdominant and reduce to nontrivial prefactor corrections. Our exact result provides a rare analytic window beyond scaling theory and extends to a framework that fully quantifies aging dynamics in the presence of saturating self-interaction.

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

0 major / 3 minor

Summary. The paper derives the exact asymptotic distribution of the record age τ_k (time between the k-th and (k+1)-st record) for saturating self-interacting random walks. It identifies two regimes in agreement with scaling predictions: for τ ≪ k² record statistics are governed by the geometry of the explored region, while for τ ≫ k² memory effects are subdominant and appear only as nontrivial prefactor corrections. The result is presented as parameter-free and extends to a general framework for aging in non-Markovian dynamics with saturating self-interaction.

Significance. If the derivation is correct, the work supplies a rare exact analytic expression for aging record statistics beyond scaling theory, cleanly separating geometric and memory contributions. This strengthens the link between extreme-value observables and non-Markovian memory while providing a concrete, falsifiable prediction for the distribution of τ_k in a broad class of processes.

minor comments (3)
  1. [Abstract] The abstract states an 'exact' derivation but supplies no equations or outline of the steps; the full manuscript should include at least the key intermediate expressions (e.g., the form of the generating function or the integral representation used for the asymptotic analysis) so that the separation into geometry-dominated and memory-corrected regimes can be verified directly.
  2. [Introduction] Notation for the saturating self-interaction (e.g., the precise form of the interaction kernel or the saturation threshold) is used without an explicit definition in the opening paragraphs; add a short paragraph or equation in the introduction that fixes the model before the asymptotic analysis begins.
  3. [Results] The claim that memory effects 'reduce to nontrivial prefactor corrections' at long times should be accompanied by an explicit statement of what those prefactors are (or at least their leading k-dependence) rather than leaving them implicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. The referee's summary accurately reflects the scope and main findings of the manuscript regarding the exact asymptotic distribution of record ages τ_k in saturating self-interacting random walks, including the separation into geometry-dominated and memory-subdominant regimes.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper claims an exact asymptotic derivation of the record age distribution tau_k for saturating self-interacting random walks, cleanly separating short-time geometry-dominated regime (tau << k^2) from long-time memory-corrected regime (tau >> k^2) with only prefactor adjustments. This is presented as independent first-principles analysis that agrees with but does not derive from prior scaling predictions. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations reducing the central result to inputs, or ansatz smuggling are identifiable from the stated claims or abstract. The derivation is self-contained against external benchmarks and does not reduce by construction to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; full text would be required to audit any that support the derivation.

pith-pipeline@v0.9.0 · 5494 in / 1141 out tokens · 45799 ms · 2026-05-08T18:32:50.319016+00:00 · methodology

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Reference graph

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