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arxiv: 1907.03632 · v1 · pith:EFO2IDXVnew · submitted 2019-07-08 · ❄️ cond-mat.stat-mech

Survival probability of stochastic processes beyond persistence exponents

N. Levernier , M. Dolgushev , O. B\'enichou , R. Voituriez , T. Gu\'erin This is my paper

Pith reviewed 2026-05-25 00:56 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords survival probabilitypersistence exponentrandom walkfirst-passage timestochastic processesnon-Markovian dynamics
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0 comments X

The pith

The survival probability prefactor S0 for compact random walks equals an expression involving the mean first-passage time in a large confining volume.

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

This paper derives explicit expressions for the prefactor S0 in the algebraic decay S(t) ~ S0/t^θ of the survival probability for compact random walks in unbounded space. The derivation uses an analytic relation that connects S0 directly to the mean first-passage time of the identical process inside a large confining volume. The relation is shown to hold for non-Markovian processes and produces results that match numerical simulations, including for fractional Brownian motion.

Core claim

For compact random walks, explicit expressions for the survival probability prefactor S0 in unbounded space are obtained by establishing an analytic relation with the mean first-passage time of the same walk in a large confining volume; the resulting formulas apply even to strongly correlated non-Markovian processes and agree with simulations.

What carries the argument

The analytic relation that equates the unbounded-space survival prefactor S0 to the mean first-passage time measured in a large confining volume.

If this is right

  • Explicit formulas for S0 become available for any compact process whose confined mean first-passage time can be computed.
  • The formulas extend to non-Markovian cases such as fractional Brownian motion and remain accurate in simulations.
  • Quantitative predictions for the statistics of the longest first-passage events in unbounded space follow directly from the confined mean first-passage time.

Where Pith is reading between the lines

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

  • The same relation could be used to obtain S0 for other recurrent processes once their confined mean first-passage time is known.
  • It offers a practical route to prefactors in problems where direct unbounded-space simulation of rare long-time survival is expensive.
  • The approach may generalize to other algebraic tails whose prefactors have resisted analytic treatment.

Load-bearing premise

The analytic relation between the unbounded survival prefactor S0 and the confined mean first-passage time holds for the full class of compact recurrent processes, including non-Markovian ones.

What would settle it

A numerical simulation that extracts the long-time prefactor S0 from the survival probability of a compact random walk in unbounded space and compares it to the value predicted from the confined mean first-passage time; mismatch at sufficiently long times would refute the relation.

Figures

Figures reproduced from arXiv: 1907.03632 by M. Dolgushev, N. Levernier, O. B\'enichou, R. Voituriez, T. Gu\'erin.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
read the original abstract

For many stochastic processes, the probability $S(t)$ of not-having reached a target in unbounded space up to time $t$ follows a slow algebraic decay at long times, $S(t)\sim S_0/t^\theta$. This is typically the case of symmetric compact (i.e. recurrent) random walks. While the persistence exponent $\theta$ has been studied at length, the prefactor $S_0$, which is quantitatively essential, remains poorly characterized, especially for non-Markovian processes. Here we derive explicit expressions for $S_0$ for a compact random walk in unbounded space by establishing an analytic relation with the mean first-passage time of the same random walk in a large confining volume. Our analytical results for $S_0$ are in good agreement with numerical simulations, even for strongly correlated processes such as Fractional Brownian Motion, and thus provide a refined understanding of the statistics of longest first-passage events in unbounded space.

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

2 major / 2 minor

Summary. The manuscript derives explicit expressions for the prefactor S0 in the algebraic decay S(t) ~ S0/t^θ of the survival probability for compact (recurrent) random walks in unbounded space. The central step is an analytic relation linking S0 to the mean first-passage time of the same process inside a large confining volume; the resulting formulas are tested numerically against direct simulations, including for fractional Brownian motion.

Significance. If the relation holds without hidden fitting parameters or circularity, the work supplies a concrete route to the amplitude S0 for non-Markovian compact processes where direct analytic access has been limited. This is quantitatively useful for longest-first-passage statistics and extends the literature beyond the persistence exponent θ alone. The reported numerical agreement for FBM is a positive indicator of practical utility.

major comments (2)
  1. [§2–3] The derivation of the key analytic relation between the unbounded-space prefactor S0 and the confined-volume MFPT is the load-bearing step. The manuscript must make explicit whether this relation follows from first principles for the full class of compact processes (including non-Markovian) or whether any auxiliary assumptions (e.g., on the form of the propagator or boundary conditions) are introduced; a self-contained proof or clear reference to an earlier result is required in §2 or §3.
  2. [Numerical results section / Table 1] Table or figure reporting the numerical comparison (presumably the one showing agreement for FBM) should include the precise definition of the confining volume size, the fitting window for the MFPT, and the statistical uncertainty on the extracted S0; without these, it is impossible to judge whether the reported agreement is within the expected error for the claimed relation.
minor comments (2)
  1. [Introduction] Notation for the survival probability and the persistence exponent should be introduced once and used consistently; the symbol S0 is introduced in the abstract but its precise definition (including any normalization) should be restated at the beginning of the main text.
  2. [Discussion] The manuscript should state the precise range of Hurst parameters or correlation strengths for which the relation is claimed to hold, even if only numerically verified for a subset.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. The comments are constructive and help strengthen the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [§2–3] The derivation of the key analytic relation between the unbounded-space prefactor S0 and the confined-volume MFPT is the load-bearing step. The manuscript must make explicit whether this relation follows from first principles for the full class of compact processes (including non-Markovian) or whether any auxiliary assumptions (e.g., on the form of the propagator or boundary conditions) are introduced; a self-contained proof or clear reference to an earlier result is required in §2 or §3.

    Authors: The relation is obtained from first principles via renewal theory applied to the survival probability and the long-time asymptotics of the MFPT in a large confining domain; it holds for any compact (recurrent) process without further assumptions on the propagator beyond the recurrence property itself, and is therefore applicable to non-Markovian cases such as FBM. To make this fully transparent, the revised manuscript will expand the derivation in §2 into a self-contained proof that explicitly lists every step and states the minimal assumptions. revision: yes

  2. Referee: [Numerical results section / Table 1] Table or figure reporting the numerical comparison (presumably the one showing agreement for FBM) should include the precise definition of the confining volume size, the fitting window for the MFPT, and the statistical uncertainty on the extracted S0; without these, it is impossible to judge whether the reported agreement is within the expected error for the claimed relation.

    Authors: We agree that these parameters are required for a rigorous assessment of the numerical agreement. The revised version will augment the numerical-results section and the corresponding table/figure with the exact confining-volume definition (linear size or number of sites), the fitting interval used to extract the MFPT, and the statistical uncertainties obtained from the ensemble of trajectories. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation links distinct quantities via analytic relation

full rationale

The central claim derives explicit S0 expressions for compact random walks in unbounded space by establishing an analytic relation to the mean first-passage time in a large confining volume. This relation is presented as independently derived and then applied, with direct numerical validation for processes including fractional Brownian motion. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the two quantities (unbounded survival prefactor and confined MFPT) are distinct, and the paper treats the relation as a derived bridge rather than an ansatz or renaming. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields limited visibility into parameters or axioms; no free parameters, invented entities, or explicit axioms are stated in the provided text.

pith-pipeline@v0.9.0 · 5713 in / 1042 out tokens · 17158 ms · 2026-05-25T00:56:58.404352+00:00 · methodology

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Reference graph

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