{"id":"7ae33a16-a5a1-4e24-90da-33d2ce83c1c0","arxiv_id":"2507.05955","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under general resetting, stochastic functionals of random walks become deterministic at long times if reset times have a finite first moment, and obey new U-shaped, W-shaped, or inverted-U limiting distributions for power-law resetting.","lead":"By treating resetting as a renewal process, this paper derives exact formulas for the statistics of any time-integrated functional of a random walk when the walk is restarted from its origin at random times drawn from a general distribution. The main results are a universal ergodic phase when reset intervals have a finite mean, and new non-ergodic distribution shapes for heavy-tailed reset intervals, verified by Monte Carlo simulation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The U/W/∩ phase diagram rests on an unproven interchange of the ϵ→0 limit and the z→0,1 boundaries in Eqs. (50)–(53); the paper asserts uniform validity of the bulk Laplace inversion without proof.","rationale":"The paper's most distinctive claim is the non-ergodic limiting distribution Eq. (53) and the resulting shape transitions of P(T+/t) as a function of α and γ. The derivation of Eq. (53) from the bulk Laplace approximation is physically plausible and is supported by Monte Carlo simulations for three different resetting distributions, which show excellent agreement with the numerical integration of Eq. (53). I found no algebraic error in the renewal equation (11), the moment calculations (15)–(16) and (39)–(42), or the ergodic delta result for finite-moment resetting for functionals with power-law growing means. The main weakness is the unjustified uniformity of the ε→0 limit and the Tauberian expansion near z=0,1. This is precisely the assumption needed to turn the bulk formula (53) into boundary exponents (57)–(59), which in turn fix the U/W/∩ classification and the phase diagram in Fig. 5. Because the boundary exponents are asserted rather than proved, the phase diagram is not fully established on the basis of the analytic derivation alone; the numerical simulations provide the current support. This does not change the reader's conditional verdict, since the gap is addressable and the central claims are likely correct, but it is the single most load-bearing concern. The reader identified the same weakest assumption, so my assessment agrees.","tokens_in":19709,"tokens_out":25125,"duration_ms":282386,"concrete_test":"Evaluate Eq. (51) directly by numerical double-Laplace inversion of Eq. (49) for the Brownian case (arcsine f). For a fixed small s (e.g., s=10^{-6}) and a grid of χ=p/s, compute Q(p,s) = (1/s)g(χ) from Eq. (50), then perform a numerical inverse Laplace transform (e.g., fixed-Talbot) to obtain P(z) for z=10^{-4}, 10^{-3}, 10^{-2}, and similarly near z=1-10^{-4}. Compare the logarithmic slope d log P(z)/d log z against Eq. (57), which predicts α−1/2. If the slope differs for any α∈(0,1), Eq. (53) is not valid at the boundaries and the transition α_c is not established. Independently, derive P(T+/t < ϵ) for small ϵ directly from the renewal equation (11) without passing through Eqs. (50)–(53), and check whether the leading ϵ exponent matches α+γ/2−1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central new result is Eq. (53), the claimed limiting PDF of T+/t for resetting exponent 0<α<1. Its derivation from Eq. (49) uses the Godrèche–Luck inversion formula (51), which evaluates g(χ) on the branch cut χ=-1/z-i0. To compute the integrals in Eq. (50) in the limit ϵ→0, the paper splits the u-integral at u=z, assigning θ→π on [0,z] and θ→2π on [z,1], yielding the A_β, B_β expressions and finally Eq. (53). This is a standard branch-cut calculation, but its validity requires that (i) the small-argument Tauberian expansion (46) holds uniformly for all u∈[0,1] in the bulk limit; (ii) the limit ϵ→0 can be interchanged with the u-integration in Eq. (52), including for z arbitrarily close to 0 and 1; and (iii) the resulting improper integrals C_{α-1}(z) and C_{α-1}(1-z) (which diverge as z^{α+γ/2-1} when α+γ/2<1) indeed dominate the boundary behavior. The paper asserts this uniformity in Section V.B without proof. The boundary exponents (57) and (59) then determine the U/W/∩ classification and the transition lines α_c≈0.269 and α*=1−γ/2. If the bulk inversion is not uniform near the boundaries, the exponent α+γ/2−1 for P(z) could be modified, and the phase diagram would shift. The Monte Carlo data are consistent with the predicted shapes, so this is currently a gap in proof rather than a contradiction, but it is the load-bearing assumption for the paper's most distinctive claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives a renewal equation for the characteristic function of a stochastic functional of a random walk under general resetting, and uses it to obtain long-time results: a delta-function limit when resetting has finite moments, moment scaling for power-law resetting, and for the half-occupation time a universal limiting PDF (Eq. 53) with a phase diagram of shapes (U, W, inverted U). The theoretical results are compared with Monte Carlo simulations for Brownian and subdiffusive walkers under power-law, Mittag-Leffler, and Lévy resetting.","tokens_in":19929,"tokens_out":13504,"duration_ms":138364,"significance":"If the main results hold, the paper establishes a remarkably universal statement: for resetting with finite moments, all positive functionals become ergodic in the long-time limit; and for heavy-tailed resetting with exponent α<1, the limiting distribution of the half-occupation time depends only on the tail exponent α and the walker exponent γ, not on the detailed form of the resetting distribution. This universality is supported by simulations with three different resetting distributions. The renewal equation (Eq. 11) and the moment formulas (Eqs. 15–16) are clean and correctly reduce to the Poissonian limit (Eq. 12). The paper is also honest about its unproved expectations in Section VI.","major_comments":[{"comment":"The derivation of the limiting PDF (Eq. 53) from Eq. (49) via the Godrèche–Luck inversion (Eq. 51) requires that the small-argument Tauberian expansion (Eq. 46) holds uniformly for arguments s+pu with u in [0,1], and that the limit ε→0 in Eq. (51) can be interchanged with the u-integration defining C_β and D_β in Eq. (52), including near z=0 and z=1. This uniformity is asserted but not proved. The boundary exponents (57) and (59), and hence the U/W/∩ classification and the transition lines α_c and α*, are derived precisely from this boundary behavior. Please provide a proof of the needed uniformity or state explicitly the conditions under which Eq. (53) is proven, and discuss how the phase diagram could be affected if the interchange fails.","section":"Section V.B (Eqs. 46–53)"},{"comment":"The right half of Table I, intended for functionals with 1<μ≤2, lists only the cases 0<α≤1 and 1<α≤μ. For μ<α≤2, the asymptotic analysis of Eqs. (15)–(16) yields a different behavior: L[φ⟨Z⟩0] is then O(1) while L[φ*⟨Z⟩0] contributes subdominantly, giving ⟨Z⟩_r ∼ t and ⟨Z^2⟩_r ∼ t^2 rather than the power laws shown in the table. This range is missing from the table, so the claim that the table gives the temporal scaling for any stochastic functional with 1<μ≤2 is not supported as stated. Please add the missing row or restrict the claim accordingly.","section":"Section IV.B, Table I"}],"minor_comments":[{"comment":"There is an index typo: the last term in the sum should be τ_{N+1}, and the notation for the subscript r on Z(t|x0)_r is introduced before its definition; please clarify.","section":"Eq. (7)"},{"comment":"The sentence 'The transition between W and ∩ shapes is attained at α = αc' appears to be a typo; from the preceding classification, the ∪/W transition is at α_c and the W/∩ transition is at α* = 1 − γ/2.","section":"Section V.B.2, text after Eq. (59)"},{"comment":"The statement that the condition can be relaxed to require only a finite first moment is proved for T+ in Section V.B (Eq. 48) but is worded in V.A as if it were a general result; consider adding a cross-reference to Section V.B to avoid ambiguity.","section":"Section V.A, paragraph after Eq. (40)"}],"recommendation":"major_revision","confidential_remarks":"The paper's most distinctive prediction, Eq. (53), rests on an unproven uniformity assumption in the double Laplace inversion; this is the main obstacle to acceptance. The moment-scaling table also has an omitted parameter range. Both issues are fixable within the manuscript's scope, so I do not recommend rejection. The paper is otherwise well-written and the simulation comparisons are convincing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version. The paper's genuinely new content is the half-occupation-time analysis for power-law resetting: the ergodicity-breaking parameter, the limiting distribution Eq. (53), and the U/W/inverted-U shape transitions. The renewal equation (11) itself is a clean generalization of the Poissonian result to arbitrary resetting PDFs, and the ergodic delta theorem for finite-moment resetting is proved under a stated condition. The Monte Carlo evidence across power-law, Mittag-Leffler, and Levy resetting is convincing, and the claim that the long-time distribution depends only on the tail exponent alpha is well supported.\n\nWhat the paper does well: the derivations are transparent and the moment table, apart from one missing row, reproduces the known special cases. The EB parameters (40) and (42) are exact and match simulations. The phase diagram for the subdiffusive case, with the transition at alpha* = 1 - gamma/2, ties nicely to the known first-passage transition in Ref. [16]. Credit is given to the earlier Poissonian and interval-occupation work.\n\nSoft spots. The stress-test concern about the uniformity of the Laplace inversion near z=0,1 is real but less damaging than it looks: the boundary exponents (57) and (59) are derived from the explicit formula (53) using the small-z behavior of C_beta(z), not by interchanging limits in the inversion itself. Still, the derivation of (53) from (49) via the Godreche-Luck formula is a standard but unproved branch-cut calculation for non-integer alpha; a rigorous referee would want that step justified. The conclusion's claim that the delta result should extend to functionals with mu<1 under only a finite first moment is explicitly labeled as unproved expectation - fine, but it should be flagged as a conjecture, not a result. Table I's right panel is missing the row for mu<alpha<=2 with 1<mu<=2, a minor oversight. No code or data are provided; the simulation descriptions are adequate but not complete.\n\nWho this is for: statistical physicists working on stochastic resetting and functionals. It deserves serious peer review; the weaknesses are addressable and the central claims are well supported. I'd send it to a referee, asking them to scrutinize the Laplace inversion and the boundary asymptotics.","headline":"Genuinely new half-occupation-time results for power-law resetting, with convincing simulations, but the Laplace-inversion step behind the phase diagram deserves closer scrutiny.","tokens_in":20639,"tokens_out":3288,"would_cite":true,"duration_ms":37173,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K40","82C31"],"pacs":["05.40.-a"],"model":"deepseek-v4-flash","headline":"For any random walker whose position is reset at random times, a single renewal equation gives the full statistics of every time-integrated observable, and the resetting tail exponent alone decides whether those statistics are…","keywords":["stochastic resetting","stochastic functionals","half-occupation time","ergodicity breaking","power-law resetting","Brownian motion","subdiffusion","renewal equation"],"falsifier":"Measure the half-occupation-time density for Brownian motion with power-law resetting at $\\alpha=0.4$ and times long enough that $P(z)$ has converged, then compare the measured $P(z)$ near $z=10^{-6}$ with the predicted $P(z)\\sim z^{-0.1}$; a clear disagreement in that boundary exponent would falsify the uniformity assumption behind Eq. (53). Alternatively, for a subdiffusive walker with $\\gamma=0.7$, test whether the W-to-∩ transition occurs at $\\alpha^*=0.65$; finding it elsewhere would falsify the phase-diagram claim.","tokens_in":19308,"feed_emoji":"🔄","tokens_out":13504,"duration_ms":141655,"temperature":0.7,"pith_summary":"The paper derives one renewal identity that gives the generating function of any stochastic functional of a random walk whose position is reset to the origin at random times drawn from an arbitrary distribution. It then shows that if the resetting distribution has finite moments, every such functional becomes ergodic in the long-time limit: its probability density collapses to a delta function at its mean. If the resetting distribution instead decays as a power law $t^{-1-\\alpha}$ with $0<\\alpha<1$, the paper obtains the limiting distribution of the half-occupation time $T_+$ (the time spent on the positive side) and shows it takes one of three shapes — ∪, W, or ∩ — depending on $\\alpha$ and, for subdiffusive walkers, on the walker exponent $\\gamma$. The practical claim is that only the tail exponent of the resetting distribution matters, not its detailed form; Monte Carlo checks with power-law, Mittag-Leffler, and Lévy resetting densities support this. If the paper is right, the ergodic or non-ergodic character of any such observable can be read off from the resetting tail alone.","feed_headline":"Resetting tail exponent alone fixes occupation-time statistics","feed_subtitle":"General renewal formula: finite-mean resets erase randomness; power-law resets yield U, W, or inverted-U half-time laws.","key_machinery":"The central object is the renewal equation (11) for the double Laplace transform of the functional's characteristic function under resetting, $\\tilde Q_r(p,s)=\\mathcal L[\\varphi^*(t)Q_0(p,t)]/(1-\\mathcal L[\\varphi(t)Q_0(p,t)])$, where $\\varphi$ is the resetting-time density, $\\varphi^*$ is the survival probability without reset, and $Q_0$ is the characteristic function without reset. It reduces the problem to two input ingredients: the resetting density and the reset-free functional statistics. The long-time analysis is carried by two tools: the small-$s$ Tauberian expansion of the resetting Laplace transform ($1-b_\\alpha s^{\\alpha}$ for $0<\\alpha<1$, $1-\\langle t\\rangle_R s$ for $1<\\alpha<2$) and, for the limiting density, the bulk inversion formula $P(z)_r = -\\frac{1}{\\pi z}\\lim_{\\epsilon\\to 0}\\operatorname{Im} g_\\alpha(-1/z+i\\epsilon)$, with $g_\\alpha$ the scaling function built from the ratio of integrals over the reset-free density $f$. The shape classification follows from the boundary exponents $P(z)_r\\sim z^{\\alpha+\\gamma/2-1}$, where $\\gamma=1$ for Brownian motion.","core_discovery":"The central result is that the double Laplace transform of the characteristic function of any functional under general resetting satisfies $\\tilde Q_r(p,s)=\\mathcal L[\\varphi^*(t)Q_0(p,t)]/(1-\\mathcal L[\\varphi(t)Q_0(p,t)])$, where $\\varphi$ is the resetting-time density, $\\varphi^*$ is the no-reset survival probability, and $Q_0$ is the characteristic function without resetting. From this renewal equation, finite-moment resetting implies the long-time distribution is $\\delta(Z-\\langle Z\\rangle_r)$ and the ergodicity-breaking parameter vanishes. For power-law resetting with $0<\\alpha<1$, the half-occupation-time density is given by Eq. (53) in terms of integrals $C_\\beta(z)$ over the reset-free limiting density $f(u)=P(T_+/t)$; for an isotropic random walk this formula depends on the resetting distribution only through $\\alpha$. Specializing $f$ to the Lévy arcsine law (Brownian motion) yields boundary exponents $P(z)_r\\sim z^{\\alpha-1/2}$, so the limiting density is ∪ for $\\alpha<\\alpha_c\\simeq0.269$, W for $\\alpha_c<\\alpha<1/2$, and ∩ for $\\alpha>1/2$; specializing $f$ to the Lamperti distribution (subdiffusion with exponent $\\gamma$) gives boundary exponents $z^{\\alpha+\\gamma/2-1}$ and a W-to-∩ transition at $\\alpha^*=1-\\gamma/2$.","pith_inferences":["A natural extension the paper leaves implicit is that the tail-only universality in Eq. (53) should hold for any renewal process with power-law inter-event times, not just positional resetting; a numerical sweep of other functionals of the same walk under Mittag-Leffler resets would test this without new analytic work.","The sharp ergodic transition at $\\alpha=1$ could be used in reverse: estimating whether trajectory-to-trajectory variance of $T_+/t$ vanishes gives a direct empirical test that the resetting mechanism has a finite mean reset time.","The boundary exponents are the most exposed part of the derivation because the paper's uniformity assumption is strained as $z\\to0$ and $z\\to1$; a dedicated simulation at very small $z$ would either confirm the predicted $z^{\\alpha-1/2}$ scaling or locate exactly where the inversion limit fails.","If the paper's conjecture on relaxing the finite-moment condition to a finite first moment for functionals with $\\mu<1$ is correct, the ergodic phase becomes much larger than the theorem proves, and the same delta-function limit should be observed for resetting densities with infinite second moments but finite means."],"forward_implications":["If the resetting-time distribution has finite moments, the long-time distribution of any time-integrated observable is a delta function at its mean, so trajectory-to-trajectory fluctuations of time averages vanish.","For power-law resetting with $0<\\alpha<1$, the limiting half-occupation-time distribution is universal in the tail exponent: the same Eq. (53) fits simulations for power-law, Mittag-Leffler, and Lévy resetting densities.","The ergodicity-breaking parameter of $T_+$ is $(1-\\alpha)(2-\\gamma)/2$ in the non-ergodic phase and zero for $\\alpha>1$, so $\\alpha=1$ is a sharp boundary between random and deterministic occupation times.","For subdiffusive walkers the W-to-∩ transition lies at $\\alpha^*=1-\\gamma/2$, tying the shape of the occupation-time distribution to the mean of the first-passage time to the origin.","The same machinery applies to any functional whose no-reset limiting density has the scaling form $t^{-1}f(Z/t)$, including observables such as the time-averaged position or the area under the trajectory."],"supporting_citations":[{"why":"Supplies the renewal-equation structure for resetting that Eq. (11) generalizes to arbitrary resetting-time densities.","marker":"[21]"},{"why":"Provides the earlier Poisson-resetting characteristic function, the special case Eq. (12), used to check the general formula.","marker":"[22]"},{"why":"Gives the occupation-time-in-an-interval results for Brownian motion under power-law resetting that benchmark the moment scalings in Table I.","marker":"[23]"},{"why":"Provides the occupation-time renewal methodology and the inversion formula used to turn the scaling characteristic function into the limiting density.","marker":"[20]"},{"why":"Supplies the Lévy arcsine law used as the no-reset density f(u) for Brownian half-occupation time.","marker":"[28]"},{"why":"Supplies the Lamperti distribution used as the no-reset density f(u) for subdiffusive half-occupation time.","marker":"[29]"},{"why":"Provides the Tauberian theorem behind the small-s expansion of the power-law resetting Laplace transform.","marker":"[30]"},{"why":"Gives the alpha=1-gamma/2 transition for the propagator under power-law resetting used to interpret the W-to-cap transition.","marker":"[16]"},{"why":"Provides the no-reset second moment of the subdiffusive half-occupation time used in Eq. (41).","marker":"[27]"}],"fun_headline_variants":["Resetting power-law exponent dictates three occupation-time shapes","General resetting formula: finite mean ergodic, power-law non-ergodic","Half-time law: U, W, or inverted-U from resetting tail","Universal renewal equation fixes functional statistics under resetting"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The U/W/∩ classification of the limiting density assumes that the bulk long-time limit can be taken uniformly for all $z\\in[0,1]$, including the boundaries $z\\to0$ and $z\\to1$; the paper states this uniformity rather than proving it, and separately only conjectures the finite-first-moment relaxation for other functionals.","fun_headline_variants_meta":{"raw":{"variants":["Resetting power-law exponent dictates three occupation-time shapes","General resetting formula: finite mean ergodic, power-law non-ergodic","Half-time law: U, W, or inverted-U from resetting tail","Universal renewal equation fixes functional statistics under resetting"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000171,"raw_usage":{"total_tokens":1315,"prompt_tokens":1031,"completion_tokens":284,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":647,"completion_tokens_details":{"reasoning_tokens":211}},"tokens_in":647,"tokens_out":284,"duration_ms":3735,"temperature":1.0,"reasoning_tokens":211,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:18:43.020362+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the half-occupation-time density for Brownian motion with power-law resetting at $\\alpha=0.4$ and times long enough that $P(z)$ has converged, then compare the measured $P(z)$ near $z=10^{-6}$ with the predicted $P(z)\\sim z^{-0.1}$; a clear disagreement in that boundary exponent would falsify the uniformity assumption behind Eq. (53). Alternatively, for a subdiffusive walker with $\\gamma=0.7$, test whether the W-to-∩ transition occurs at $\\alpha^*=0.65$; finding it elsewhere would falsify the phase-diagram claim.","supporting_citations":[{"cited_title":"Godreche and J.-M","cited_arxiv_id":null,"evidence_quote":"Provides the earlier Poisson-resetting characteristic function, the special case Eq. (12), used to check the general formula."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the occupation-time-in-an-interval results for Brownian motion under power-law resetting that benchmark the moment scalings in Table I."},{"cited_title":"Turgeman, S","cited_arxiv_id":null,"evidence_quote":"Provides the occupation-time renewal methodology and the inversion formula used to turn the scaling characteristic function into the limiting density."},{"cited_title":"Abramovitz and I","cited_arxiv_id":null,"evidence_quote":"Supplies the Lévy arcsine law used as the no-reset density f(u) for Brownian half-occupation time."},{"cited_title":"Occupation time statistics for non-Markovian random walks","cited_arxiv_id":"2412.05247","evidence_quote":"Supplies the Lamperti distribution used as the no-reset density f(u) for subdiffusive half-occupation time."},{"cited_title":"L´ evy, Sur certains processus stochastiques homog` enes, Compositio Mathematica 7, 283 (1940)","cited_arxiv_id":null,"evidence_quote":"Provides the Tauberian theorem behind the small-s expansion of the power-law resetting Laplace transform."},{"cited_title":"Mas´ o-Puigdellosas, D","cited_arxiv_id":null,"evidence_quote":"Gives the alpha=1-gamma/2 transition for the propagator under power-law resetting used to interpret the W-to-cap transition."},{"cited_title":"Kac, On Some Connections between Probability Theory and Differential and Integral Equations, in Second Berkeley Symposium on Mathematical Statistics and Probability , edited by J","cited_arxiv_id":null,"evidence_quote":"Provides the no-reset second moment of the subdiffusive half-occupation time used in Eq. (41)."}],"review_version":1}