{"id":"c9a989c1-ab36-46cf-bddc-a1413f161ad0","arxiv_id":"2502.08568","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For dynamical percolation in Z^d, the paper derives the e^{-2λ} term in the speed expansion and proves that on the critical curve μ²=p(1-p), d≥2, the speed is eventually increasing in the bias.","lead":"Using the environment seen from the walker, this paper computes the second-order correction to the speed of a biased random walk on edges that randomly open and close, with an explicit constant in one dimension. It also proves that on the critical curve in higher dimensions the speed is eventually increasing in the bias, resolving an open question.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sign of the second-order coefficient is not pinned down: Lemma 5.6 and Eqs. (53)/(56) use opposite signs for C↑↓, and Lemma 5.3's proof is omitted.","rationale":"The reader's CONDITIONAL verdict is appropriate. I found no fatal flaw in the d=1 expansion, the environment-process construction, or the overall coupling strategy. However, the specific step producing the positive second-order coefficient is the most load-bearing point for Theorem 1.4, and it is not currently established. The sign conflict in Lemma 5.6 and Eqs. (53)/(56) is an internal inconsistency rather than merely a missing proof: the lemma statement, the lemma proof, and the main text disagree about whether C↑↓ enters with a plus or minus sign. Since the conclusion is exactly a sign statement, this must be resolved before the proof can be accepted. Lemma 2.2/Lemma 2.3 is also a genuine gap, but it is plausibly a mutatis mutandis transfer from [1]; the sign conflict is present in this paper itself. The physical direction of the coupling suggests the corrected sign is negative, which would preserve the desired positivity and keep the theorem true, so I do not recommend rejection. The verdict remains conditional pending the repairs and verifications described above.","tokens_in":30707,"tokens_out":20863,"duration_ms":204834,"concrete_test":"Re-derive the conditional displacement difference on E↑↓ from the coupling in Lemma 5.6: track X and \\tildeY between the two o-points T1 and T2, apply Proposition 2.4 at T2, and compute the sign of E[Xτk − \\tildeYτk | E↑↓]. If the sign is negative, correct Lemma 5.6 and Eq. (53) to use minus, and verify that the aggregate constant C1b∪2o in Eq. (57) remains strictly positive. In parallel, supply the omitted proof of Lemma 5.3 and check that C1b > 0 with the claimed O(e^{-λ}) error.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.4 requires the coefficient of e^{-2λ} in Theorem 1.3 to be strictly positive, since monotonicity is a sign statement about this coefficient. In the proof, the coefficient is assembled in Eq. (57) from C1b, C2o\\↑↓, C↑↓, and Cμ,p, and its positivity is asserted as C1b∪2o > 0. This assertion is not secure. Lemma 5.3, which supplies C1b > 0, has its proof omitted ('we leave the details to the reader'). More seriously, Lemma 5.6 is internally inconsistent: its statement reads |E[Xτk − \\tildeYτk | E↑↓] − C↑↓| ≤ δ + O(e^{-λ}) with C↑↓ > 0, so X is ahead of \\tildeY, while its proof establishes P[Xt ≤ \\tildeYt] = 1 and derives E[XT2 − \\tildeYT2 ] = −C↑↓ + O(e^{-λ}), so X is behind. The main text then uses the plus sign in Eq. (53) and the minus sign in Eq. (56). Only one sign can be correct, and the eventual positivity of C1b∪2o — hence the sign of c2 — depends on it. As written, the proof does not determine whether the e^{-2λ} correction is positive, negative, or zero.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the speed of a λ-biased random walk on dynamical percolation on Z^d. For d=1 it proves an explicit second-order expansion v(λ) = μp/(μ+1-p) - μp/(μ+1-p) C_{μ,p} e^{-2λ} + O(e^{-4λ}) with an explicit positive constant C_{μ,p}. For d≥2, on the critical curve μ^2 = p(1-p), it claims a second-order expansion v(λ) = μp/(μ+1-p) - C_{μ,p,d} e^{-2λ} + O(e^{-3λ}) with C_{μ,p,d}>0, and derives from this, together with a derivative expansion, that the speed is eventually strictly increasing in λ. The methods are the environment seen from the walker, a stationary measure Q constructed via regeneration times, couplings to totally asymmetric walks, and event decompositions by backward and orthogonal jump attempts.","tokens_in":31107,"tokens_out":11521,"duration_ms":108039,"significance":"If completed, the paper resolves an open question from Andres et al. [1] on the critical curve and provides a quantitative two-term expansion that is stronger than the qualitative dichotomy known off the critical curve. The d=1 part is a genuine contribution: it gives an explicit constant, uses a clean invariant-measure construction, and contains a detailed mixing argument (Lemma 4.2). The paper also ships explicit formulas rather than fitted constants, which is a strength. However, the d≥2 proof currently rests on several omitted or sketched arguments and contains a sign inconsistency in a load-bearing lemma, so the central monotonicity claim is not yet secure as written.","major_comments":[{"comment":"The statement and proof of Lemma 5.6 assign opposite signs to C_{↑↓}. The statement (45) says |E[X_{τ_k} - \\tilde Y_{τ_k} | E_{↑↓}] - C_{↑↓}| ≤ δ + O(e^{-λ}) with C_{↑↓} > 0, which means X is ahead of \\tilde Y; the proof, however, derives E[X_{T_2} - \\tilde Y_{T_2}] = -C_{↑↓} + O(e^{-λ}) and maintains P[X_t ≤ \\tilde Y_t] = 1, which means X is behind. The main text then uses the plus sign in Eq. (53) and the minus sign in Eqs. (54) and (56). Only one sign can be correct, and the sign of the e^{-2λ} correction in Theorem 1.3 depends on this term. As written, the proof does not determine whether the second-order coefficient is positive, negative, or zero.","section":"Section 5.3, Lemma 5.6 and Eqs. (53)-(56)"},{"comment":"Lemma 5.3, which supplies the positive constant C_{1b}, has its proof omitted: the authors write 'we leave the details to the reader'. This is not a cosmetic gap, because Eq. (57) assembles the decisive positive constant C_{1b∪2o} from C_{1b}, C_{2o\\↑↓}, C_{↑↓}, and the event probabilities, and Theorem 1.3 requires C_{1b∪2o} > 0. The d=1 computation in Theorem 1.2 does not directly cover the d≥2 situation where o-points interact with the backward-jump event. In addition, Eq. (50) uses the correction -C_{1b} P(E_{1b}), while Eq. (57) uses - μp/(μ+1-p) C_{μ,p} P(E_{1b}); the substitution of C_{1b} by this expression is unexplained and needs justification. Similar sketched positivity assertions appear for C_{2o\\↑↓} in Proposition 5.4(3) and C_{2o''} in Lemma 5.5(2).","section":"Section 5.1, Lemma 5.3 and Eq. (57)"},{"comment":"The k-dependence of C_{1b∪2o} is not tracked consistently. Eq. (57) allows C_{1b∪2o} to depend on k, but Eq. (58) writes E[X_{τ_k}] = k v_Y e^{1/μ} - C_{1b∪2o} e^{1/μ} e^{-2λ} + ... without indicating whether the constant absorbs a factor of k. After dividing by E[τ_k] = k e^{1/μ}, the correction term would vanish as k → ∞ unless C_{1b∪2o} is O(k), which is not shown. Since Lemma 4.1 permits any k, the proof must either fix a sufficiently large k and control all constants for that k, or explicitly track the k-dependence through Eqs. (58)-(61). As written, the limiting argument leading to Eq. (61) is not justified.","section":"Section 5.4, Eqs. (57)-(61)"},{"comment":"The proof of the derivative expansion in Lemma 2.2, which is essential for Theorem 1.4, relies on Lemma 2.3. The proof of Lemma 2.3 is omitted, with only the remark that it 'follows mutatis mutandis from [1, Lemma 4.10]'. This is load-bearing: the conclusions c_1 = 0 and c_2 > 0 in the proof of Theorem 1.4 depend on the λ,ε-independence of the functions f_{I,J} and g_{I,J}. Moreover, the displayed formula (10) contains exponents such as 1-k and k-|I|-|J|-1 that appear to be typos, and the Taylor expansion in that line is hard to verify as written. The authors should either supply the proof of Lemma 2.3 or state precisely which statements in [1] cover the present setting.","section":"Section 2.2, Lemma 2.3 and Eq. (10)"}],"minor_comments":[{"comment":"The integral of the derivative is computed as c_1 e^{-s} + 2c_2 e^{-2s} + O(e^{-3s}), but ∫_s^{2s} v'(t) dt = c_1 e^{-s} + (c_2/2 - c_1) e^{-2s} + O(e^{-3s}). Since c_1 = 0 on the critical curve, the monotonicity conclusion is unaffected, but the displayed coefficient should be corrected.","section":"Section 2.2, Proof of Theorem 1.4"},{"comment":"The integrals in the first proof are written over (-∞, ∞), whereas the exponential and Gamma densities vanish on the negative half-line; the integrals should be over [0, ∞).","section":"Lemma 3.8"},{"comment":"In the proof, the display 'E[X_{θ_n}]' should refer to the TARWDP Y, not X; the notation should be made consistent with the statement of the lemma.","section":"Lemma 4.4"},{"comment":"The sentence describing the rates of P^f, P^b, and P^o swaps the labels for backward and orthogonal jumps: the f-rate is e^λ Z_λ^{-1}, the o-rate is (2d-2)Z_λ^{-1}, and the b-rate is e^{-λ}Z_λ^{-1}.","section":"Section 5.1"},{"comment":"The exponent '1-k' in the factor (1 - e^{λ+ε}Z_{λ+ε}^{-1} + e^λ Z_λ^{-1})^{1-k} seems suspect and may be a typo; the formula should be checked carefully, as the surrounding argument depends on the polynomial growth in k of the coefficients.","section":"Eq. (10)"}],"recommendation":"major_revision","confidential_remarks":"The d=1 result appears sound and is likely acceptable after polishing. The d≥2 part is the main obstacle: the sign inconsistency in Lemma 5.6 is serious, and the omitted proofs of Lemma 2.3 and Lemma 5.3 are not merely organizational issues. Given the heavy reliance on [1], including co-author overlap, I would recommend that the d≥2 proof be checked by an independent expert before acceptance. The paper is potentially a good fit for math.PR if these issues are resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a real step toward completing the phase diagram for biased random walks on dynamical percolation. The d=1 result (Theorem 1.2) gives an explicit second-order coefficient C_{μ,p}, and the environment-process computation that backs it is careful and largely self-contained. The d≥2 theorem on the critical curve is exactly the open question from Andres et al., and the overall strategy—comparing the walk to a one-dimensional TARWDP via couplings, then assembling the e^{-2λ} coefficient—is plausible.\n\nBut the d≥2 proof has a serious internal inconsistency. Lemma 5.6's statement says E[X−Y|E↑↓] ≈ C↑↓ with C↑↓ > 0, meaning X is ahead. Its proof, two lines later, says X is slower than Y, and derives E[XT2−YT2] = −C↑↓ + O(e^{-λ}). The main text then uses the plus sign in Eq. (53) and the minus sign in Eq. (56). Both cannot be right. Since the sign of this term feeds into the constant C1b∪2o, which must be positive for Theorem 1.4, this is not a cosmetic typo. The proof, as written, does not establish the claim.\n\nThere are also two omitted proofs that the reader should treat as load-bearing rather than routine: Lemma 2.3 (the independence of f_{I,J} of λ, ε) underlies the derivative expansion in Lemma 2.2, and Lemma 5.3 (positivity of C1b) is asserted without details. The d=1 part does not suffer from these problems; the computation of E[S|A], E[S|B], E[S|C] is explicit and the mixing argument in Lemma 4.2 is sound.\n\nIf the sign of C↑↓ is a typo and the proof is correct, the paper likely goes through after a careful rewrite. But as it stands, a referee cannot accept it. It deserves a serious referee—the result is important and the d=1 part is a genuine contribution—but the author response should be major revision, not minor.\n\nBring it to reading group if you want a case study in how a sign error can destabilize an otherwise competent argument.\n\nBest","headline":"Important gap-filling result with a solid d=1 expansion, but the d≥2 proof has a sign inconsistency in the central second-order coefficient that must be fixed before the monotonicity claim is established.","tokens_in":31575,"tokens_out":5981,"would_cite":true,"duration_ms":55467,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60K37"],"pacs":[],"model":"deepseek-v4-flash","headline":"For $d \\ge 2$, the speed of a biased random walk on dynamical percolation with $\\mu = \\sqrt{p(1-p)}$ is eventually strictly increasing in the bias $\\lambda$, closing the critical-curve case left open in [1].","keywords":["dynamical percolation","biased random walk","asymptotic speed","environment process","critical curve","monotonicity","regeneration times"],"falsifier":"Compute or simulate the coefficient of $e^{-2\\lambda}$ in (61) for $d=2$, $p=1/2$, $\\mu=1/2$: if the displayed constant $(2d-2)^2p^2/(\\mu+1-p) + \\mu^2 p/(\\mu+1-p)^2 + C_{1b\\cup 2o}$ were zero or negative, the speed would approach its limit from above and eventual increase would fail. More directly, a numerical evaluation of $v'(\\lambda)$ at large $\\lambda$, for example by finite differences of regeneration-time averages, showing a non-positive derivative would refute Theorem 1.4.","tokens_in":30505,"feed_emoji":"🎲","tokens_out":10532,"duration_ms":85641,"temperature":0.7,"pith_summary":"Biased random walks on dynamical percolation move on a lattice where each edge refreshes at rate $\\mu$ and is open with probability $p$, while the walker tries to jump along the $e_1$ axis with strength $e^{\\lambda}$ relative to backward strength $e^{-\\lambda}$. An earlier paper [1] proved that for $d \\ge 2$ the asymptotic speed is eventually increasing when $\\mu^2 > p(1-p)$ and eventually decreasing when $\\mu^2 < p(1-p)$, leaving the critical curve $\\mu^2 = p(1-p)$ open. This paper closes that gap: on the critical curve the speed has the second-order expansion $v(\\lambda) = \\mu p/(\\mu+1-p) - C_{\\mu,p,d} e^{-2\\lambda} + O(e^{-3\\lambda})$ with $C_{\\mu,p,d} > 0$, and therefore $v(\\lambda)$ is strictly increasing for all sufficiently large $\\lambda$. The proof combines the environment seen from the walker, regeneration times, and several couplings, and also gives an explicit one-dimensional second-order expansion. The result matters because it completes the phase diagram: even at the critical refresh rate, the walk still accelerates as the bias grows, approaching its infinite-bias limit from below.","feed_headline":"Speed of biased walk on critical percolation eventually increases","feed_subtitle":"At the critical refresh rate, the second-order correction makes the speed eventually rise with bias.","key_machinery":"The central object is the environment seen from the walker, the configuration of open and closed edges shifted so that the walker sits at the origin. For the totally asymmetric walk that only attempts right jumps, the paper constructs an invariant measure $Q$ on this environment, shows it is extremal invariant and equivalent to the Bernoulli-$p$ product measure, and computes the stationary law of the projection onto the two edges adjacent to the walker. This stationary law gives the acceptance probability $\\mu p/(\\mu+1-p)$ and the expected time shift $C_{\\mu,p}$ in one dimension. For $d \\ge 2$, the argument decomposes jump attempts into forward, backward, and orthogonal points, couples the biased walk to reduced walkers and one-dimensional totally asymmetric walkers, and uses a monotone coupling (Proposition 2.4) to show that each backward or orthogonal excursion costs a fixed positive number of forward steps. Lemma 2.2, the exponential Taylor expansion of $v'(\\lambda)$, then converts the second-order speed expansion into a sign for the derivative.","core_discovery":"On its own terms, the paper's central claim is Theorem 1.4: for $d \\ge 2$ and $p \\in (0,1)$, with $\\mu = \\sqrt{p(1-p)}$, there exists $\\lambda_0(p,d)$ such that the speed $v_{\\mu,p}(\\lambda)$ of the $\\lambda$-biased random walk on dynamical percolation is strictly increasing for all $\\lambda \\ge \\lambda_0$. The quantitative content behind it is Theorem 1.3, an asymptotic expansion $$v(\\$\\lambda$) = \\frac{\\mu p}{\\mu+1-p} - C_{\\mu,p,d} $e^{{-2\\lambda}}$ + O($e^{{-3\\lambda}}$)$$ with $C_{\\mu,p,d} > 0$, valid for $\\lambda \\ge \\lambda_0$. Combined with a regularity lemma stating that the derivative $v'(\\lambda)$ admits an exponential Taylor expansion $v'(\\lambda) = \\sum_{i=1}^n c_i e^{-i\\lambda} + O(e^{-(n+1)\\lambda})$, the vanishing of the $e^{-\\lambda}$ coefficient exactly on the critical curve and the strict positivity of the $e^{-2\\lambda}$ coefficient force $v'(\\lambda) = 2c_2 e^{-2\\lambda} + O(e^{-3\\lambda}) > 0$ for large $\\lambda$. In one dimension the paper also derives an explicit expansion with the closed-form constant $C_{\\mu,p}$ given in (2).","pith_inferences":["Editorial extension: the same mechanism suggests that the sign of the $e^{-\\lambda}$ coefficient, which vanishes precisely on the critical curve, is what flips the eventual monotonicity between $\\mu^2 > p(1-p)$ and $\\mu^2 < p(1-p)$; one could test whether the speed-versus-$\\lambda$ curve is in fact monotone for all $\\lambda$, with the large-$\\lambda$ sign already fixed.","Editorial extension: the environment-process stationary measure $Q$ and the explicit projection on the two adjacent edges could be used to derive higher-order corrections or fluctuation bounds, such as variance scaling, for the totally asymmetric walk rather than only its speed.","Editorial extension: a direct numerical check of Theorem 1.3 for $d=2$ is feasible by simulating the speed at several large $\\lambda$ values at $\\mu = \\sqrt{p(1-p)}$; the predicted sign of the $e^{-2\\lambda}$ coefficient is robust because the constant in (61) is a sum of manifestly positive terms plus $C_{1b\\cup 2o}$."],"forward_implications":["The critical curve $\\mu^2 = p(1-p)$ joins the increasing regime: for $d \\ge 2$, $v(\\lambda)$ is eventually strictly increasing, so Question 1.6 of [1] is resolved.","For large $\\lambda$ on the critical curve, $v(\\lambda)$ approaches its limiting value $\\mu p/(\\mu+1-p)$ from below at rate $e^{-2\\lambda}$, and the linear-order correction vanishes exactly on this curve.","In dimension one, the explicit expansion (1) holds for every $\\mu > 0$ and $p \\in (0,1)$ with constant $C_{\\mu,p} > 0$; since $C_{\\mu,p} \\to 2$ as $\\mu \\to \\infty$, the fast-refreshment limit recovers the speed of a biased walk that accepts every move with probability $p$.","The derivative admits an exponential Taylor expansion $v'(\\lambda) = \\sum_{i=1}^n c_i e^{-i\\lambda} + O(e^{-(n+1)\\lambda})$, which together with the second-order speed expansion upgrades asymptotic proximity into a monotonicity statement."],"supporting_citations":[{"why":"Supplies the model, the existence and differentiability of the speed, the regeneration-time framework, and the dichotomy that leaves the critical curve as an open question that this paper resolves.","marker":"[1]"},{"why":"Provides the infected-set comparison principle and the lemma on regeneration times with exponential tails used throughout the derivative-regularity and time-shift arguments.","marker":"[14]"},{"why":"Introduced the infected-set construction that defines the regeneration times on which the speed formula and coupling arguments rely.","marker":"[23]"},{"why":"Supplies the general theory of invariant measures for Feller processes, including compactness and extremality criteria, used to construct and characterize the environment-process measure $Q$.","marker":"[17]"},{"why":"Provides the tagged-particle ergodic method used to identify the speed of the totally asymmetric walk through the environment seen from the walker.","marker":"[18]"},{"why":"Provides the Markov-chain convergence theorem used to bound the exponential convergence rate of the edge-open probability in the one-dimensional speed approximation.","marker":"[16]"}],"fun_headline_variants":["Critical percolation: biased walk speed eventually rises","At large bias, walk on critical percolation speeds up","Second-order term forces eventual speed boost for biased walk","Biased walk on critical curve accelerates at high bias"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole monotonicity result rests on an unproved regularity lemma, stated as Lemma 2.3 and cited as following from [1, Lemma 4.10], asserting that certain conditional expected crossing times are independent of the bias; if that lemma or the strict positivity of the $e^{-2\\lambda}$ coefficient in the derivative expansion failed, the speed could still fail to be eventually increasing.","fun_headline_variants_meta":{"raw":{"variants":["Critical percolation: biased walk speed eventually rises","At large bias, walk on critical percolation speeds up","Second-order term forces eventual speed boost for biased walk","Biased walk on critical curve accelerates at high bias"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00027,"raw_usage":{"total_tokens":1607,"prompt_tokens":910,"completion_tokens":697,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":632}},"tokens_in":526,"tokens_out":697,"duration_ms":7044,"temperature":1.0,"reasoning_tokens":632,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T04:36:48.309440+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute or simulate the coefficient of $e^{-2\\lambda}$ in (61) for $d=2$, $p=1/2$, $\\mu=1/2$: if the displayed constant $(2d-2)^2p^2/(\\mu+1-p) + \\mu^2 p/(\\mu+1-p)^2 + C_{1b\\cup 2o}$ were zero or negative, the speed would approach its limit from above and eventual increase would fail. More directly, a numerical evaluation of $v'(\\lambda)$ at large $\\lambda$, for example by finite differences of regeneration-time averages, showing a non-positive derivative would refute Theorem 1.4.","supporting_citations":[{"cited_title":"Biased random walk on dynamical percolation","cited_arxiv_id":null,"evidence_quote":"Supplies the model, the existence and differentiability of the speed, the regeneration-time framework, and the dichotomy that leaves the critical curve as an open question that this paper resolves."},{"cited_title":"A comparison principle for random walk on dynamical percolation","cited_arxiv_id":null,"evidence_quote":"Provides the infected-set comparison principle and the lemma on regeneration times with exponential tails used throughout the derivative-regularity and time-shift arguments."},{"cited_title":"Random walks on dynamical percolation: mixing times, mean squared displacement and hitting times.Probability Theory and Related Fields, 162(3):487–530, 2015","cited_arxiv_id":null,"evidence_quote":"Introduced the infected-set construction that defines the regeneration times on which the speed formula and coupling arguments rely."},{"cited_title":"Interacting particle systems, volume 2","cited_arxiv_id":null,"evidence_quote":"Supplies the general theory of invariant measures for Feller processes, including compactness and extremality criteria, used to construct and characterize the environment-process measure $Q$."},{"cited_title":"Stochastic interacting systems: contact, voter and exclusion processes, vol- ume 324","cited_arxiv_id":null,"evidence_quote":"Provides the tagged-particle ergodic method used to identify the speed of the totally asymmetric walk through the environment seen from the walker."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Markov-chain convergence theorem used to bound the exponential convergence rate of the edge-open probability in the one-dimensional speed approximation."}],"review_version":1}