{"id":"3fd64b53-a7d6-466e-b4a4-9d76b9124ae3","arxiv_id":"2412.08267","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Random grids plus Dijkstra's algorithm find deterministic minimum-resistance paths on energy landscapes without the angle-dependent bias of regular grids.","lead":"This paper proposes finding reaction paths by running Dijkstra's algorithm on a graph built from randomly placed grid points, making the search deterministic and immune to local optima. The authors show random grids avoid the angle-dependent bias of regular grids and reproduce minimum-resistance paths on a three-hole model potential.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Discrete-to-continuum convergence of the Dijkstra minimizer is asserted, not proven; the 1.049 factor bounds only the length bias for f=1, so the claim that the graph path is the true continuous minimum-resistance path remains unsupported.","rationale":"The reader's CONDITIONAL verdict rests on the missing convergence proof; my stress test agrees that this is the load-bearing point. The proposed numerical test directly checks whether the graph optimum approaches the continuum optimum; if it passes, the central claim is empirically supported even absent a theorem, and the paper's 'globally optimal up to resolution' language is defensible for practical purposes. If it fails, the method's output is not the claimed optimum and the comparison to MaxFlux is anecdotal. I do not see an internal inconsistency that would justify REJECT: the random-grid idea is plausible, the distance-bias numerics are clean, and the method is a reasonable discretization. The missing proof and the Eq. S2 typo are addressable in revision. Hence the reader's CONDITIONAL verdict stands unchanged. Agreement: the reader's weakest_assumption identifies exactly this convergence concern.","tokens_in":12083,"tokens_out":10200,"duration_ms":115730,"concrete_test":"Construct a 2D landscape with a known analytic minimum-resistance path, e.g., f(x,y)=exp(β(x^2−y^2)) on a disk, whose minimizing curve is known from calculus of variations; run the proposed random-grid Dijkstra algorithm at densities ρ=10^3, 10^4, 10^5, 10^6, and 10^7, using at least 10 independent random grid realizations per density. For each realization, record (i) the relative cost gap C_graph/C_continuum − 1 and (ii) the Hausdorff distance between the graph path and the analytic path. If, as ρ increases, the median cost gap does not decrease monotonically to zero or the path deviation does not shrink, the asserted discrete-to-continuum convergence (Sections 2.3 and 3.2) fails. The same measurement on the three-hole potential against a carefully converged MaxFlux reference would be a useful supplement but cannot replace the analytic test.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central assertion is that Dijkstra's algorithm on random Delaunay grids finds a globally optimal path 'up to the resolution of random grids' (Section 3.2) and 'approximately minimizes the line integral' (Section 2.3). For this to be true, the graph shortest path must converge to the continuous minimizer of ∫ f ds as grid density grows. No proof or error bound is given; the only quantitative support is the 1.049 factor (Eq. 19), which is a heuristic bound on the expected Euclidean length overestimate for a straight line with f=1, assuming a mean coordination number of six and a uniform angular error of at most π/6. It says nothing about the convergence of the minimizing path when f varies by orders of magnitude (as exp βU does), nor about the cost gap between the graph optimum and the continuum optimum. The paper's comparisons with MaxFlux at ρ=10^4, 4×10^4, and 10^6 are purely visual; no cost differences, path deviations, or convergence rates are reported. If the graph geodesic does not Γ-converge to the weighted-length functional—for example because the midpoint edge cost (Eq. 6) fails to resolve narrow features of f, or because boundary effects of the finite Delaunay triangulation bias the graph—the central methodological claim is unsupported. A secondary but concrete weakness: the 3D consistency estimate in Eq. (S2) integrates 2π sinθ/cosθ over [-π/6, π/6], which is identically zero; the intended conical average yields 1.073, so the analytical support in 3D is mis-specified. This does not invalidate the numerical results, but it reduces confidence in the heuristic derivation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a deterministic path-search method for free-energy or potential-energy landscapes. Random points are placed uniformly in the coordinate domain, a Delaunay triangulation turns them into a graph, edge costs are set to the midpoint-weighted length (Eq. 6), and Dijkstra's algorithm finds the minimum-total-cost path. The authors argue that this approximates the minimization of a line-integral objective such as the minimum-resistance functional of Berkowitz et al. and MaxFlux. They test the distance-representation error in 2D and 3D, reporting a direction-independent overestimate of about 4% and 6%, respectively, and they compare minimum-resistance paths on a three-hole potential with MaxFlux and square-grid Dijkstra results at several temperatures.","tokens_in":12561,"tokens_out":5937,"duration_ms":62512,"significance":"If validated, the method would be a useful deterministic alternative to nonlinear path optimization methods, avoiding local-minimum issues and the systematic angular bias of regular grids. Its strengths are its simplicity, the single free parameter (grid density), and the clear numerical demonstration that random grids eliminate the angle-dependent length error of regular grids. The agreement with MaxFlux on the three-hole model is encouraging, and the method requires no fitted parameters in its core construction. However, the general claim that the graph shortest path converges to the true continuous minimum of the line integral is not supported by proof or by quantitative convergence tests, and a supporting analytical estimate in the supplemental material contains a mathematical error.","major_comments":[{"comment":"The central methodological claim—that the graph shortest path “approximately minimizes the line integral” and is “globally optimal up to the resolution of random grids”—is asserted rather than established. The text gives no convergence theorem or error bound for the discrete-to-continuum limit of the weighted-length functional on random Delaunay graphs, and the heuristic estimate in Eq. (19) concerns the expected length overestimate for a straight segment with f=1 only. For f=exp(βU), whose values vary by orders of magnitude, the midpoint edge cost in Eq. (6) could in principle under-resolve narrow features or be biased by boundary effects. I recommend either proving a Γ-convergence statement or, at minimum, adding a quantitative numerical convergence study for the resistance paths: report C_total as a function of ρ_grids, compare it with a high-resolution reference resistance, and measure the deviation between the computed path and the MaxFlux path. As it stands, the general validity of the method is not supported beyond the distance test and the visual comparison in Figs. 6 and 7.","section":"§2.3 and §3.2"},{"comment":"The 3D consistency estimate in Eq. (S2) is incorrect as written. The integrand 2π sinθ/cosθ is odd on [-π/6, π/6], so the numerator is zero; the denominator ∫ 2π sinθ dθ over the same symmetric interval is also zero, making the ratio undefined. The value 1.073 is obtained if the integration is performed over [0, π/6]. This calculation therefore needs to be corrected before it can be cited as supporting the 1.06 overestimation factor reported in Table S1.","section":"Supplemental Eq. (S2)"},{"comment":"The agreement between the proposed random-grid paths and the MaxFlux paths is assessed only by visual inspection. No numerical path-cost differences, path deviations, or grid-density convergence rates are reported for the three-hole model. Since the paper's conclusion that the method “succeeds in selecting the same path” is based on these figures, quantitative metrics (e.g., the resistance C_total of each path, the mean distance between the two paths, or the change in the path with ρ_grids) should be provided.","section":"§3.2, Figs. 6 and 7"}],"minor_comments":[{"comment":"The paper does not state how the fixed endpoints x_R and x_P are merged with the randomly generated grids before Delaunay triangulation; please clarify this implementation detail.","section":"§2.4 and §2.5"},{"comment":"The word “lansdacpe” should be “landscape”.","section":"§1"},{"comment":"The phrase “we examin” should be “we examine”.","section":"§2.5"},{"comment":"The caption says “Radial dependence” while the text describes angular dependence; the intended meaning (the radius in the polar plot is the path length) should be stated explicitly.","section":"§3.1, Fig. 4 caption"},{"comment":"The sentence “Thus, our method is rigorous” is too strong given the heuristic error analysis; I suggest rephrasing to indicate that the method is deterministic and does not rely on nonlinear optimization.","section":"§3.2"},{"comment":"In the annealing description, the term 𝒞_B appears to be a typo for 𝒞_R, which is defined in Eq. (15); please check and correct.","section":"§2.5.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a reasonable methods paper with a clear empirical demonstration, but the missing convergence analysis and the erroneous Eq. (S2) are load-bearing enough to require a major revision. I do not see evidence of inappropriate citation or novelty problems; the issue is technical support for the central claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea is simple and worth knowing: replace regular grids with random Delaunay grids in Dijkstra-based path search on energy landscapes, and you remove the systematic angle-dependent length overestimation that regular grids cause. The paper shows empirically that path lengths on random grids are direction-independent with ~4% (2D) and ~6% (3D) overestimation, and that minimum-resistance paths on the three-hole model match MaxFlux at several temperatures while square-grid paths deviate, especially at β = 3.3. That is a real, useful result, and it is new as far as I can tell from the cited literature.\n\nWhat the paper does well: it is clearly written, the test suites are sensible (distance to points on circle/sphere, three-hole potential), and the comparison with MaxFlux is honest, including the regular-grid artifacts. The algorithm is parameter-light and deterministic, which is a genuine advantage for users who do not want to tune NEB/string parameters.\n\nWhere it is soft: the central convergence claim is asserted, not proved. The paper says Dijkstra on random grids gives \"globally optimal path up to the resolution of random grids\" but there is no Γ-convergence or error-bound argument connecting the discrete graph optimum to the continuous line-integral minimizer when f varies exponentially. The 1.049 factor is a heuristic bound for f = 1, and the 3D version in Eq. S2 is actually mis-specified: the integral as written over −π/6 to π/6 is zero because sin θ is odd; the intended cap average over 0 to π/6 yields ~1.073. That is a minor but concrete slip in the analytic support. Also, no code or data are shipped, and the path comparisons with MaxFlux are visual only—no cost differences or path deviations are reported. None of this kills the method; the numerical evidence is decent. But the paper should either add a convergence argument (even for a restricted class of f) or lower its claim to \"heuristic method with numerically demonstrated accuracy.\"\n\nBottom line: this is a useful contribution for practitioners who want a quick, deterministic path-finding alternative. A serious referee can push on the convergence issue and the analytic estimates. I would send it to review; with the Eq. S2 fix and a more careful statement of guarantees it would be solid. I would cite the random-grid angle-bias result if I worked on reaction path methods.","headline":"A useful, parameter-light path-search trick—random Delaunay grids kill the angle bias of regular grids—but the convergence claims need to be toned down or proven.","tokens_in":12981,"tokens_out":2635,"would_cite":true,"duration_ms":26866,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that random grids used as graph nodes for Dijkstra's algorithm eliminate the angle-dependent error of regular grids and find minimum-resistance paths matching MaxFlux.","keywords":["reaction path","free-energy landscape","minimum resistance path","Dijkstra's algorithm","random grids","Delaunay triangulation","MaxFlux method","path-length error"],"falsifier":"Take a two-dimensional potential whose exact minimum-resistance path is known analytically, run the random-grid Dijkstra method with successively finer grids, and check whether the resistance of the returned path converges to the exact value; if the error does not shrink as grid density grows, the continuum-limit assumption is false.","tokens_in":11930,"feed_emoji":"🧭","tokens_out":10543,"duration_ms":103851,"temperature":0.7,"pith_summary":"Reaction paths on energy or free-energy landscapes are usually found by nonlinear optimization, which can settle into suboptimal solutions. This paper proposes converting the landscape into a weighted graph and using Dijkstra's algorithm, which deterministically returns the cheapest path on the graph. The paper's key move is to place the graph nodes on random grids instead of regular grids: square and triangular grids systematically overestimate diagonal moves by up to 41%, an error that does not go away with finer spacing, whereas random grids show no direction-dependent error and overestimate path length by only 4% in two dimensions and 6% in three dimensions. On a three-hole potential, the random-grid Dijkstra paths match MaxFlux minimum-resistance paths at all tested temperatures, including a case where square grids select the wrong route. If the continuum-limit assumption holds, the method offers a parameter-light, local-minimum-free route to reaction-path search.","feed_headline":"Random grids fix a built-in bias in reaction-path search","feed_subtitle":"Randomly placed grid nodes remove angle-dependent path-length error, leaving a constant 4% overestimate in 2D, 6% in 3D.","key_machinery":"The central object is a random Delaunay graph: grid points placed uniformly at random in the coordinate space, with Delaunay triangulation deciding which points count as neighbors. Because the edges point in all directions with no privileged lattice direction, the graph has no systematic bias of the kind that makes diagonal moves on square or triangular grids artificially expensive. The edge cost uses a trapezoidal approximation of the landscape function between neighboring points, so the total cost along a path approximates the line integral of that function; Dijkstra's algorithm then returns the minimum-cost path from reactant to product. For the resistance objective, $f=\\exp(\\beta U)/D$ is positive, so the nonnegative-edge condition for Dijkstra's algorithm is satisfied automatically.","core_discovery":"The paper's central claim is that random grids eliminate the angle-dependent systematic error in path length that is inevitable with regular grids, and that a Dijkstra search on the resulting random Delaunay graph yields the minimum-resistance path. The edge cost $C_{i,j}^{\\mathrm{edge}} = (f(\\mathbf{x}_i)+f(\\mathbf{x}_j))\\,d_{i,j}/2$ makes the total cost approximate the line integral $\\int f(\\mathbf{x})\\,ds$, so the shortest graph path approximately minimizes that integral. With $f=\\exp(\\beta U_{\\mathrm{TH}})$, the minimum-resistance paths on the three-hole potential agree with the MaxFlux method at $\\beta=1.0,2.0,3.3,4.0$, while square grids fail at $\\beta=3.3$ by missing the intermediate basin. The paper reports an angle-independent path-length overestimate of 4% in 2D and 6% in 3D, consistent with a simple average of $1/\\cos\\theta$ over the $\\pm\\pi/6$ angle spread of a six-coordinated random grid.","pith_inferences":["The construction is tied only to the line-integral form of the cost, so the same algorithm should handle any positive weight $f(\\mathbf{x})$; testing it on a case with a known analytic optimum would separate discretization error from intrinsic approximation error.","Because the overestimate appears angle-independent, dividing the reported total resistance by 1.04 in 2D or 1.06 in 3D might give a better estimate of the continuum resistance, although the paper does not state this.","Higher-dimensional behavior is untested: extending the random-Delaunay construction to $d>3$ would reveal whether the favorable angle-averaging property persists when the coordination number of random Delaunay triangulations grows.","Averaging over several independently drawn random grids could give a practical estimate of discretization uncertainty for a single path, since each realization is one random graph sample."],"forward_implications":["For a fixed set of random grids, Dijkstra's algorithm returns the globally cheapest graph path, so the search cannot be trapped in a local optimum.","Temperature dependence is inherited from the Boltzmann weight: at high temperature the method favors the shorter direct route, and at low temperature the longer route through lower saddles, matching MaxFlux at every temperature tested.","Position-dependent diffusion can be included through $f(\\mathbf{x})=\\exp(\\beta U(\\mathbf{x}))/D(\\mathbf{x})$, extending the method beyond constant-diffusion Smoluchowski dynamics.","The only tunable parameter is grid density, and even at $\\rho_{\\mathrm{grids}}=10^4$ the recovered paths remain consistent with MaxFlux, indicating robustness to resolution."],"supporting_citations":[{"why":"Supplies Dijkstra's algorithm, the deterministic shortest-path engine that avoids nonlinear optimization.","marker":"[20]"},{"why":"Defines the minimum-resistance objective as a line integral of the exponential weight divided by diffusivity, which the random-grid edge cost is built to approximate.","marker":"[22]"},{"why":"Provides the MaxFlux method and the three-hole potential model used as the benchmark and comparison baseline.","marker":"[23]"},{"why":"Defines Delaunay triangulation, the rule used to connect neighboring random grid points into a graph.","marker":"[34]"},{"why":"Supplies the Mersenne Twister pseudorandom generator used to place the random grids.","marker":"[35]"},{"why":"Supplies Qhull, the numerical implementation used for Delaunay triangulation in the tests.","marker":"[36]"}],"fun_headline_variants":["Random grids remove angle bias in reaction-path search","Dijkstra on random grids finds min-resistance reaction paths","Random grids yield constant path-length error: 4% in 2D, 6% in 3D","Angle-independent path error achieved with random grids","Random grids: 4% error in 2D, 6% in 3D path search"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The path found by connecting random grid points is assumed to approach the true optimal path of the continuous landscape as the grid gets finer, but the paper gives no proof or error bound for that convergence.","fun_headline_variants_meta":{"raw":{"variants":["Random grids remove angle bias in reaction-path search","Dijkstra on random grids finds min-resistance reaction paths","Random grids yield constant path-length error: 4% in 2D, 6% in 3D","Angle-independent path error achieved with random grids","Random grids: 4% error in 2D, 6% in 3D path search"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000971,"raw_usage":{"total_tokens":4129,"prompt_tokens":946,"completion_tokens":3183,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":3084}},"tokens_in":562,"tokens_out":3183,"duration_ms":24060,"temperature":1.0,"reasoning_tokens":3084,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T18:00:38.807701+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a two-dimensional potential whose exact minimum-resistance path is known analytically, run the random-grid Dijkstra method with successively finer grids, and check whether the resistance of the returned path converges to the exact value; if the error does not shrink as grid density grows, the continuum-limit assumption is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Dijkstra's algorithm, the deterministic shortest-path engine that avoids nonlinear optimization."},{"cited_title":"Berkowitz, J","cited_arxiv_id":null,"evidence_quote":"Defines the minimum-resistance objective as a line integral of the exponential weight divided by diffusivity, which the random-grid edge cost is built to approximate."},{"cited_title":"Huo, and J","cited_arxiv_id":null,"evidence_quote":"Provides the MaxFlux method and the three-hole potential model used as the benchmark and comparison baseline."},{"cited_title":"Cheng, T","cited_arxiv_id":null,"evidence_quote":"Defines Delaunay triangulation, the rule used to connect neighboring random grid points into a graph."},{"cited_title":"Matsumoto, and T","cited_arxiv_id":null,"evidence_quote":"Supplies the Mersenne Twister pseudorandom generator used to place the random grids."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Qhull, the numerical implementation used for Delaunay triangulation in the tests."}],"review_version":1}