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Exact First-Passage Time Response Theory from Steady-State Response

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Pith's one-line read This paper establishes exact linear and nonlinear response identities for mean first-passage times in continuous-time Markov processes, expressing the perturbed MFPT under arbitrary single-rate perturbations entirely in terms of…

desk verdict A genuinely useful finite-amplitude MFPT response theory with a repairable proof gap in the SM; worth refereeing but needs a corrected derivation of the key correspondence. read the letter →

arxiv 2608.11202 v1 pith:E2S7P6HI submitted 2026-08-11 cond-mat.stat-mech cond-mat.mes-hallmath-phmath.MPphysics.bio-ph

Ruicheng Bao , Shiling Liang This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.mes-hallmath-phmath.MPphysics.bio-ph
keywords meanfirst-passagetimeMarkovjumpprocessnonlinearresponsetransitionrateperturbationsteady-stateprobabilityfast-resetcorrespondenceglobalbounds
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper seeks a systematic theory for how the mean first-passage time (MFPT) between two states of a Markov process changes when transition rates are perturbed. It proves that, for a perturbation of any single transition rate, the exact nonlinear response of any MFPT can be written as a one-denominator rational function of the perturbation strength, using only unperturbed MFPTs and steady-state probabilities. The argument rests on a correspondence: adding a very fast reset edge from the target back to the initial state turns repeated first-passage events into stationary cycles, so the MFPT response equals the steady-state response of an auxiliary system. This yields a physical decomposition of the response into upstream accessibility, downstream gain or loss, and nonlinear screening, as well as bounds, higher-order derivatives, multi-rate recursions, and computational shortcuts. If correct, the theory means that global timing observables can be predicted and controlled from a small set of pre-computed quantities.

What carries the argument

The central object is the fast-reset correspondence: adding a unidirectional edge from the target $k$ back to the initial state $l$ with rate $W^*_{lk}\to\infty$ in the original Markov chain turns each first-passage event into one cycle of a stationary process, so that the MFPT becomes the inverse stationary flux through the reset edge. This equality $\tau_{kl}=1/\lim W^*_{lk}\pi^*_k$ lets the authors import steady-state response identities for $\pi^*_k$, in particular Lemma 1 for how a single-rate perturbation shifts steady-state probabilities, to derive the exact finite and linear MFPT-response formulas. The mechanism closes because the perturbation and the fast-reset edge commute in the rate matrix, so the ratio of auxiliary steady-state probabilities equals the ratio of MFPTs.

What would settle it

Choose an irreducible 4-state directed network with one narrow bottleneck edge, compute all unperturbed MFPTs and steady-state probabilities by solving the linear systems exactly, perturb a single rate by a finite $\Delta W_{mn}$, and compare the exact $\tau'_{kl}$ from an absorbing-chain solve with the prediction of Eq. (3b). A mismatch for any $\Delta W_{mn}>0$ in an irreducible network, or for a sparse or directed topology outside the random fully connected networks tested, would falsify the claimed universality.

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Extended reading notes

Core claim

For a continuous-time irreducible Markov chain with stationary probabilities $\pi$, the paper claims that perturbing one transition rate $W_{mn}\to W_{mn}+\Delta W_{mn}$ changes the MFPT $\tau_{kl}$ exactly as $$\frac{\tau'_{kl}-\tau_{kl}}{\$\Delta$ W_{mn}} = \frac{-\pi_n(\tau_{kn}-\tau_{km})(\tau_{nk}+\tau_{kl}-\tau_{nl})}{1+\$\Delta$ W_{mn}\pi_n(\tau_{kn}+\tau_{nm}-\tau_{km})},$$ with $\tau'_{kl}$ the perturbed MFPT. The numerator is the exact linear response and the denominator is the only finite-perturbation correction; every quantity on the right is from the unperturbed system. The proof uses the correspondence $\tau_{kl}/\tau'_{kl}=\lim_{W^*_{lk}\to\infty}\pi^{*\prime}_k/\pi^*_k$ for an auxiliary system with a fast reset edge $k\to l$, which converts transient first-passage statistics into stationary steady-state response.

Load-bearing premise

The exact steady-state response identities for Markov jump processes imported from the authors' prior work are not re-derived here, and the numerical verification covers fully connected random networks rather than all topologies, so the MFPT formulas stand or fall with those identities and their domain of validity.

Editorial extensions

If this is right

  • Every MFPT is monotone in every transition rate, with a concave response for harmful edges and a convex response for helpful edges, so non-monotone regimes cannot occur.
  • The full response curve for arbitrary perturbation strength is fixed by the unperturbed MFPT, its linear response, and one additional finite measurement; a single nonlinear data point predicts all $\Delta W_{mn}$.
  • Relative log-responses are bounded, with the most negative response realized at the perturbed edge itself, and the finite envelope $\min(c,c^{-1})\le \tau_{kl}(cW_{mn})/\tau_{kl}(W_{mn})\le \max(c,c^{-1})$ holds.
  • Adding a dynamical shortcut, an edge that moves probability to a state closer in MFPT to the target, always reduces the target-resolved global MFPT, so the reported Braess-type search paradox is attributable to target averaging.
  • The recursive multi-rate update computes perturbed MFPTs and steady-state distributions after any finite set of rate changes in $O(KN^2)$ or better, enabling sparse perturbation analysis of large networks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The rational form of Eq. (3b) implies that the response surface is parameterized by a handful of unperturbed numbers per target pair, suggesting an experimental protocol: measure $\tau_{kl}$ and its linear response, apply one finite perturbation, and then predict all other perturbation strengths without further simulation.
  • Because the nonlinear screening factor is independent of the initial state $l$, measuring response saturation from two different initial states should yield the same inferred bottleneck size; that cross-check would be a direct test of the framework that the paper does not perform.
  • The target-resolved monotonicity of the global MFPT suggests that in biological search or folding models the same added edge can be beneficial for one target and harmful for another, so interventions should be designed target-by-target rather than by global averages.
  • The stated computational advantage indicates that for large networks with few tunable rates, the response recursion could replace full matrix inversions in sensitivity analysis, a use the authors mention but do not benchmark at scale.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript develops an exact response theory for mean first-passage times (MFPTs) in finite continuous-time Markov chains. The central construction is an auxiliary system with a fast reset edge from the target back to the initial state; the MFPT is shown to equal the inverse stationary flux through that edge, so that the response of an MFPT to a perturbation of a transition rate can be mapped onto the steady-state response of the auxiliary system. This mapping yields exact linear and nonlinear response identities for single-rate perturbations, expressed solely through unperturbed MFPTs and steady-state probabilities, together with a factorized physical interpretation, response-curve reconstruction rules, fundamental bounds, higher-order response formulas, and recursive multi-rate updates. Two applications are given: a folding network with a chaperone cycle and a clarification of a Braess-type paradox in target-resolved search efficiency.

Significance. If the results hold, the paper provides a closed-form, parameter-free description of the full nonlinear response of first-passage times to kinetic perturbations, a question of broad interest in statistical physics, biophysics, and network science. The main identities are compact and directly testable, and the numerical verification on 10^5 random 8-state networks is a concrete strength. The factorization into upstream accessibility, downstream gain/loss, and nonlinear screening offers both physical insight and a practical inference method from a single nonlinear measurement. The bounds on relative log-response and the computational shortcuts for sparse perturbations are additional useful contributions. The main weakness is that the proof of a key auxiliary identity (SM Prerequisite 1) contains an algebraic error, and the derivation imports central steady-state response identities from the authors' previous work without re-proof.

major comments (2)
  1. [SM Prerequisite 1, Eqs. (8)-(9)] The flux calculation in Prerequisite 1 is algebraically incorrect. Equation (8) contains the factor (1 - τ_ji W*_ij π_j)/(1 + τ_ji W*_ij π_j); as W*_ij → ∞ this factor tends to -1, so the product with (W^0_ij + W*_ij)π_j diverges rather than approaching 1/τ_ji. The correct application of Lemma 1 gives π*_j = π_j/(1 + W*_ij π_j τ_ji), so the flux is (W^0_ij + W*_ij)π_j/(1 + W*_ij π_j τ_ji), whose limit is indeed 1/τ_ji. Since Eq. (7) is the stated proof of the fundamental correspondence (2), this error must be corrected; the alternative derivation in SM Section II, which applies Lemma 1 twice and is valid, should be promoted to the primary proof.
  2. [SM II, Lemma 1 and Eq. (16); main text Eqs. (1) and (3)] The central response formulas (3a)-(3b) and Theorem 1 inherit their validity from Lemma 1 and the multi-rate identity (End Matter Eq. (16); SM Prerequisite 2), which are quoted from Ref. [36] without proof. Because these identities are load-bearing, the manuscript should either reproduce their proofs in the SM or explicitly cast them as assumptions with their precise validity conditions. The numerical test in SM Fig. 1 samples only fully connected 8-state networks with rates in [0.01,10]; the claim of universality for arbitrary sparse, directed, or nearly reducible topologies would be better aligned with the evidence if this limitation were stated.
minor comments (5)
  1. [Main text, after Eq. (3b)] The parenthetical statement about positivity cites the triangle inequality τ_km + τ_mn ≥ τ_kn, which applies to Σ_{k|mn}; the positivity of U_{k←l|n} follows from the triangle inequality τ_nk + τ_kl ≥ τ_nl, so the sentence should be amended.
  2. [Main text, Eqs. (10a)-(10b)] The definition of α_X is terse; a sentence explaining the three cases (X = A_mn, B_mn, E_m) and how α_X enters the response curve would improve readability.
  3. [SM, Theorem 1] The notation ⟨t_kl⟩ is used in the SM alongside τ_kl in the main text; please unify the notation throughout the paper and the supplementary material.
  4. [Introduction] The characterization of the discrete-time work in Ref. [44] as 'either not closed or involve quantities without clear physical meaning' is strong; a specific example or a softer wording would be more constructive.
  5. [End Matter, Table II] The complexity estimates in Table II would benefit from a brief explanation of the parameters (K, N) and the assumed data structure (e.g., which matrices are precomputed).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the exact response identities are derived from independent steady-state response relations and are numerically verified against direct computation.

full rationale

The central derivation is not circular. The key correspondence (Eq. 2) follows from applying Lemma 1 to an auxiliary fast-reset edge; Eq. (3) expresses the perturbed MFPT entirely in terms of unperturbed MFPTs and stationary probabilities, so no predicted quantity is reused as an input. Lemma 1 and the multi-rate identity (Eq. 16) are imported from the authors' prior work [36], but the multi-rate identity is also credited to the independent Ref. [30], and the final formula is externally falsified by the SM numerical verification (10^5 random 8-state networks, rates in [0.01,10]) against direct solution of the master equation. Under the review rules, a cited result that is externally falsifiable does not raise the circularity score. The 'response-curve inference' rule (Eq. 5) uses one reference measurement to fix the nonlinear screening factor and then predicts other amplitudes from the derived exact rational form; this is legitimate parameter inference, not a fitted input renamed as a prediction. One non-circular concern: the SM proof of Eq. (12) contains an algebraic error in the displayed intermediate (Eq. (8) includes a factor (1 - tau W pi)/(1 + tau W pi) that is dropped when taking the limit, and that factor tends to -1 as written). This threatens the proof of the correspondence as written, but it is a correctness/verification gap, not a circularity. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters or invented physical entities. The auxiliary fast-reset system is a mathematical construction, not a new dynamical degree of freedom. The main axioms are the Markov assumption and the imported steady-state response identities from the authors' prior work.

assumptions (4)
  • domain assumption Irreducible continuous-time Markov chain with a unique stationary distribution pi
    Stated in Setup as the only assumption; all MFPTs are assumed finite.
  • domain assumption Steady-state response identities from Ref. [36] (Lemma 1 and multi-rate identity Eq. 16)
    Imported from the authors' own prior work; not re-derived in this manuscript.
  • standard math MFPT triangle inequality tau_kn + tau_nm >= tau_km and recursion relations
    Used to prove non-negativity of Sigma and in SM Section I.
  • standard math The fast-reset limit W*_{lk}->infty yields a well-defined inverse flux 1/tau_kl
    Proved in SM Appendix A using Lemma 1 and the auxiliary construction.

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Pith. "Pith review of Exact First-Passage Time Response Theory from Steady-State Response." pith.science (2026). https://pith.science/paper/E2S7P6HI

@misc{pith2026260811202,
  author       = {Pith},
  title        = {Pith review of: Exact First-Passage Time Response Theory from Steady-State Response},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E2S7P6HI}},
  note         = {Machine review of arXiv:2608.11202}
}
read the original abstract

The mean first-passage time (MFPT) provides a universal temporal measure of transport, reaction, search, and switching processes in physical, chemical, and biological systems. Understanding how MFPTs respond to perturbations is therefore crucial for prediction and control, yet a systematic theory has been lacking. We establish a compact theoretical framework for linear and nonlinear MFPT response in continuous-time Markov processes. The key tool is an exact correspondence that maps the intrinsically transient response of MFPTs onto the steady-state response of an auxiliary system. This correspondence yields exact and universal response relations for MFPTs between arbitrary state pairs, expressed entirely in terms of unperturbed MFPTs and steady-state probabilities. We then obtain a factorized physical decomposition of the MFPT response into linear upstream, linear downstream, and nonlinear contributions. Further corollaries include response-curve inference rules, fundamental bounds on MFPT responses, analytical expressions for higher-order responses of MFPTs and steady-state probabilities, and multi-rate response formulas. Additionally, our result offers computational advantages in calculating both MFPTs and steady-state distributions. Finally, a biologically motivated folding network is analyzed, and a recently reported paradox on MFPT is clarified.

Figures

Figures reproduced from arXiv: 2608.11202 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Trap-bottleneck geometry and the chaperone cy [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 1
Figure 1. FIG. 1. Numerical verification of the exact finite single-rate update (Theorem 1). In each of the 10 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Additional numerical panels for the folding example. Panel (a) gives the upstream accessibility in the chaperone [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

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