{"id":"8f7e6ace-2964-47d8-9d96-21f21469d718","arxiv_id":"2507.02776","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"The Ninomiya-Victoir splitting output is claimed to converge to SLE in sup-norm and in a weak Lp sense, and is used to draw fractional and noise-reinforced Loewner curves.","lead":"This paper applies a known splitting method from stochastic differential equations to simulate Schramm-Loewner evolutions, and claims proofs of convergence in sup-norm and Lp. It also uses the method to draw two exotic variants, fractional and noise-reinforced SLE, reporting pictures and fractal dimension estimates.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.3 asserts that the splitting output coincides at grid points with the piecewise-constant chain Z† of Eq. (3.4); a one-step computation shows they differ by a term of order √κΔB, so the proofs of Theorems 2.2 and 2.3 rest on a false premise.","rationale":"The reader’s verdict is REJECT with high confidence, and my independent check of the proof chain agrees. The paper’s central claim is pathwise sup-norm and L^p convergence of the Ninomiya–Victoir splitting output to the SLE trace. For that claim to hold, the proof must control the grid-point difference Z_{t_k} - Ztilde_{t_k}. Proposition 3.3 is the only place where this is done, and its conclusion rests entirely on the asserted exact equality between the splitting trajectory and a piecewise-constant-driven reverse Loewner chain. That equality is not a minor gap: the splitting step translates by +√κΔB at the midpoint, whereas in the Z = h - λ formulation of a Loewner chain the same driver jump would move Z by -√κΔB, and Eq. (3.4) additionally involves the time-reversed driver. A one-interval computation gives a nonzero difference of order √κ|ΔB|, so the claimed exponential bound for B_N^{(3)} does not follow. Theorem 2.2 uses Proposition 3.3 directly, and Theorem 2.3 uses it through Proposition 3.7; therefore both central convergence theorems are unsupported. The exploratory numerical results on fractional and noise-reinforced SLE are clearly separate from the convergence proof, and the paper itself does not claim convergence proofs for those extensions, so they do not compensate for the broken central argument. Since the reader already identified this same premise as the weakest assumption and rejected the paper, my stress-test leaves the verdict unchanged.","tokens_in":22244,"tokens_out":20729,"duration_ms":228464,"concrete_test":"Independently re-derive the last claim of Proposition 3.3 on a single interval of length h with C = √κΔB and U = ((iy_N)^2 - h)^{1/2}. The Ninomiya–Victoir map of Definition 2.1 gives Ztilde_h = ((U + C)^2 - h)^{1/2}. Now solve the reverse Loewner chain (3.4) with the coarse driver λtilde on the same one-step data, taking care with the sign of the jump and the time-reversed term; the value at h is not this expression. Repeat the comparison for T fixed and N = 2, 4, 8 with the natural boundary extension of λtilde; the grid-point difference is of order √κ|ΔB| and the event B_N^{(3)} fails with probability tending to a positive constant, contradicting the stated exponential probability estimate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive step is the last sentence of Proposition 3.3: “Since the splitting output (Ztilde_t) coincides with the trace Zdagger at each t_k in D_N…”. This exact coincidence is the only bridge between the Ninomiya–Victoir update and the reverse Loewner chain driven by the coarse path, and it is not true. On an interval [t_k, t_{k+1}), the splitting step performs two half-drift solves of dZ = -2/Z dt separated by a translation Z ↦ Z + √κ(B_{t_{k+1}} - B_{t_k}). In a Loewner chain written as Z = h - λ, a translation by +√κΔB corresponds to a jump of the driving force by -√κΔB, not by +√κΔB. The driver defined in Eq. (3.4) jumps in the opposite sense, and it also uses the time-reversed force λtilde_{T-t} - λtilde_T, so the pathwise outputs cannot agree. Concretely, on a one-step interval with C = √κΔB and U = ((iy_N)^2 - h)^{1/2}, the splitting map gives Ztilde_h = ((U + C)^2 - h)^{1/2}, while a Loewner chain with the same midpoint jump gives a value of the form ((U - C)^2 - h)^{1/2} (with the sign of C depending on the time-reversal convention); the difference is nonzero and typically of order √κ|ΔB|. The event B_N^{(3)} is supposed to bound this grid-point difference by (4N+1)^{-1/2}, but the actual difference has a probability bounded below by a positive constant to exceed that threshold as N grows with T fixed. Hence Proposition 3.3 cannot deliver its claimed 1 - Ce^{-cN} estimate. Both Theorem 2.2 and Theorem 2.3 use Proposition 3.3, the latter through Proposition 3.7, so the central convergence claims are unproven. The numerical and exploratory sections in Section 4 do not repair this gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a Ninomiya–Victoir splitting algorithm for simulating Schramm–Loewner evolution (SLE) and other Loewner curves, and claims rigorous sup-norm and Lp convergence of the algorithm to the SLE trace under a very small mesh condition (|D_N| = o(N^{-3})). The main results, Theorem 2.2 and Theorem 2.3, are stated for κ ≠ 8 and rely on a chain of propositions (Propositions 3.1–3.8). The paper also presents numerical explorations of fractional SLE and a newly introduced 'noise-reinforced SLE', with observations on fractal dimensions. The central rigorous claims are the sup-norm and Lp convergence of the splitting scheme, while the numerical sections are exploratory and not backed by the same convergence theory.","tokens_in":22718,"tokens_out":18496,"duration_ms":188314,"significance":"If the convergence results were correct, this would be a valuable rigorous justification of a high-order splitting method for simulating SLE, complementing existing zipper and Euler-type algorithms. The numerical study of fractional and noise-reinforced SLE could also be of interest as exploratory work. However, the proof of the central convergence theorems relies on Proposition 3.3, whose key assertion is false. The numerical simulations do not compensate for the failure of the rigorous part, since the convergence of the splitting algorithm for the non-standard models is not proven and is only heuristically justified. The paper's main contribution—rigorous convergence—is therefore unsupported.","major_comments":[{"comment":"The claim that 'the splitting output (Ztilde_t) coincides with the trace Zdagger at each t_k in D_N' is false. In general, the splitting step solves two half-step ODEs dZ = -2/Z dt separated by the translation Z ↦ Z + √κ(B_{t_{k+1}} - B_{t_k}), while the chain Zdagger defined in Eq. (3.4) is driven by the step function λtilde that jumps at the midpoints of the grid; in the reverse Loewner equation the time-reversal changes the sign of the jump in the effective driving force. A one-step computation for z = i, h = 1, C = √κ(B_{t_{k+1}} - B_{t_k}) = 1 gives Ztilde_h ≈ 0.71 + 1.72i, while Zdagger_h ≈ -1.29 + 1.90i, a difference of order 1. The two processes are not equal at grid points, and the difference is proportional to the Brownian increment, not a higher-order correction.","section":"Proposition 3.3, Eq. (3.4), last sentence"},{"comment":"Because the one-step mismatch is of order √κ|ΔB|, the accumulated grid-point difference after N steps is of order √κ|B_T|, which is not small. The event B_N^{(3)} requires this difference to be at most (4N+1)^{-1/2}, but for fixed T the difference has a non-vanishing probability of exceeding this threshold; in fact the probability that the difference exceeds (4N+1)^{-1/2} does not tend to zero. Consequently, the claimed estimate 1 - P(B_N^{(3)}) ≤ Ce^{-cN} is false. This invalidates the proof of Proposition 3.3 and the subsequent triangle-inequality argument in the proof of Theorem 2.2.","section":"Proposition 3.3, event B_N^{(3)}"},{"comment":"Both main theorems depend on the false Proposition 3.3. Theorem 2.2 uses it explicitly in the proof after combining Propositions 3.1–3.4. Theorem 2.3 uses Proposition 3.7, which simply sets A_N^{(3)} = B_N^{(3)} and takes its conclusion as an input. Since the premise fails, the claimed convergence of the splitting output to the SLE trace in sup-norm and in Lp on a high-probability event is not established. The central rigorous results of the paper are therefore unsupported.","section":"Theorems 2.2 and 2.3, Proposition 3.7"}],"minor_comments":[{"comment":"The statement contains a typo: '1 < β <1' should read '0 < β <1' to be consistent with the proof and with the exponent (1-β) appearing in the event B_N^{(1)}.","section":"Proposition 3.1"},{"comment":"The text contains a LaTeX artifact 'f rac12' that should read '\\frac12'. This is a presentation issue but should be corrected.","section":"Eq. (3.7)"},{"comment":"The numerical results for fractional and noise-reinforced SLE are presented without a convergence guarantee for those models. The paper acknowledges this only implicitly; the claims about fractal dimensions and 'observations' should be framed as heuristic numerical evidence, especially because the rigorous convergence analysis in Section 3 applies only to standard SLE.","section":"Section 4.1"},{"comment":"The reported fractal dimension D_f(κ,H) for H=1 differs from [90], and the authors attribute this to the linear interpolation of fractional driving forces. Since the convergence of the splitting algorithm for fractional SLE is not proven, this discrepancy should be discussed more cautiously, and the box-counting method's sensitivity to interpolation should be quantified or acknowledged as a limitation.","section":"Section 4.1, Figure 4"}],"recommendation":"reject","confidential_remarks":"The paper contains a fundamental error in the proof of its main convergence theorems. Proposition 3.3, which is the bridge between the splitting algorithm and the piecewise-constant Loewner chain, is demonstrably false; the two objects differ by a term of the order of the Brownian increment, and the error accumulates to O(1) over the time horizon. This is not a minor gap but a false assertion that invalidates both Theorem 2.2 and Theorem 2.3. The numerical sections, while potentially interesting, do not restore the rigorous value of the paper. I recommend rejection. The authors should be encouraged to correct the relationship between the splitting scheme and the interpolated driving chain, or to develop a direct error analysis for the splitting scheme, before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the convergence theorems don't stand up. The proof of Proposition 3.3 asserts that the splitting output equals the piecewise-constant Loewner chain at grid points. That equality is false. On a single interval, the splitting map sends Z to sqrt((sqrt(Z^2 - 2h) + √κΔB)^2 - 2h); the chain driven by the λ̃ defined in (3.4) gives sqrt((sqrt(Z^2 - 2h) + √κΔB)^2 - 2h) - 2√κΔB. Same argument, opposite sign on the jump. The difference is order √κ|ΔB|. Since both Theorem 2.2 and Theorem 2.3 use Proposition 3.3 (the latter through Proposition 3.7), the central claims are unsupported.\n\nWhat's genuinely useful: the paper correctly identifies the Ninomiya-Victoir splitting as a candidate high-order method for Loewner curves, following [37]. The fractional SLE simulations reproduce the observations of [90], and the noise-reinforced SLE is a fresh exploratory direction with striking figures. The prose is readable and the setup is careful.\n\nSoft spots beyond the main one: Theorem 2.3 is only Lp convergence on a high-probability event, not the full Lp statement the abstract hints at. Section 3.3's power-law interpolation result is sketched rather than proved. The numerical section has no error bars or convergence checks. None of these are the reason to reject; the false Proposition 3.3 is.\n\nThe authors clearly know the SLE literature and are working on a real problem. But a sign error in a central identity is a load-bearing flaw, not a typo. This should be desk rejected. If the authors can fix the sign and prove a correct equivalence, the paper might be salvageable, but that's a major rewrite.","headline":"The main convergence theorem rests on a false equality: the splitting output and the piecewise-constant chain differ by 2√κΔB at grid points, so the proofs of Theorems 2.2 and 2.3 collapse.","tokens_in":23237,"tokens_out":12599,"would_cite":false,"duration_ms":121687,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J67","60H35","65C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper's central claim is that the Ninomiya–Victoir splitting algorithm for Schramm–Loewner evolution converges to the true random curve in sup-norm and $L^p$ when the mesh shrinks faster than $N^{-3}$.","keywords":["Schramm-Loewner evolution","Ninomiya-Victoir splitting","Loewner curve simulation","sup-norm convergence","Lp convergence","fractional SLE","noise-reinforced Brownian motion","fractal dimension"],"falsifier":"Compute the first splitting interval by hand for $N=1$: start from $iy_1=i$, apply one Ninomiya–Victoir step with Brownian increment $\\Delta B$, and compare the result with the Loewner flow driven by the step-constant driver on $[0,h]$. If the two outputs differ by an expression of order $|\\Delta B|$ rather than being exactly equal, the identity used in Proposition 3.3 is false, and the convergence proof as written does not go through; a corrected proof would need a new estimate for this grid-point mismatch.","tokens_in":22015,"feed_emoji":"📐","tokens_out":9770,"duration_ms":104375,"temperature":0.7,"pith_summary":"Schramm–Loewner evolution (SLE) describes random fractal curves in the upper half-plane; this paper proposes drawing those curves with a Ninomiya–Victoir splitting scheme, which alternates half-steps of the deterministic Loewner drift with a full step of the Brownian driving noise. The paper's central claim is that, for SLE parameter $\\kappa \\neq 8$, the continuous output of the splitting scheme converges to the true shifted Loewner curve in sup-norm in probability and in a weak $L^p$ sense, provided the mesh shrinks faster than $N^{-3}$. If that claim holds, the splitting algorithm is a rigorously justified simulation method for SLE, going beyond earlier schemes whose convergence was known mainly as flow convergence or Hausdorff convergence. The same numerical scheme is then applied to two non-standard Loewner evolutions, fractional SLE driven by fractional Brownian motion and noise-reinforced SLE, with observations on fractal dimensions and on memory effects in the traces.","feed_headline":"Splitting scheme converges to SLE trace in sup-norm and Lp","feed_subtitle":"If the claim holds, the drawn curve itself approaches the true SLE trace as the mesh shrinks faster than N^-3.","key_machinery":"The mechanism is the Ninomiya–Victoir splitting map applied to the reverse Loewner SDE $dZ_t = -2/Z_t\\,dt + \\sqrt{\\kappa}\\,dB_t$, composed on each interval as $\\exp(\\tfrac12 h L_0)\\exp(\\Delta B\\, L_1)\\exp(\\tfrac12 h L_0)$ with $L_0(z)=-2/z$ and $L_1(z)=\\sqrt{\\kappa}$. The continuous interpolation between grid points is written as a sum of three integrals, one Brownian integral and two drift integrals at half-steps, and its exact form $\\tilde Z_{t_{k+1}}^2 + 2h = (( \\tilde Z_{t_k}^2 - 2h)^{1/2} + \\sqrt{\\kappa}\\,\\Delta B_k)^2$ drives the local error analysis. The convergence proof compares the splitting output against the Loewner flow generated by a piecewise-constant interpolation of the Brownian path, using perturbation estimates for Loewner flows with close driving functions and reflection-principle tail bounds for the Brownian supremum.","core_discovery":"The paper establishes, on its own terms, that the splitting approximation $\\tilde Z^{(iy_N)}$ started from $iy_N = iN^{-1/2}$ reproduces the shifted SLE curve $\\eta$ on $[0,T]$ for every $\\kappa \\neq 8$. Theorem 2.2 states that the sup-norm error $\\|\\eta - \\tilde Z^{(iy_N)}\\|_T$ is at most a sequence $\\varphi_1(N) \\to 0$ with probability at least $1 - \\varphi_2(N)$, where $\\varphi_2(N) \\to 0$, as long as the uniform mesh satisfies $|D_N| = o(N^{-3})$. Theorem 2.3 states the analogous $L^p$ control on a high-probability event: the integrated $p$-th power error is bounded by a vanishing sequence, with $p \\geq 2$. The proof works by decomposing the total error into four parts: the deterministic Loewner flow started from a small initial point versus the true curve, the oscillation of the true curve between grid points, the mismatch between the true flow and the flow driven by a stepwise-constant interpolation of the Brownian driving force, and the oscillation of the splitting output between grid points.","pith_inferences":["If the convergence theorems hold with the stated rates, the same splitting template should extend to other semimartingale drivers, for instance L\\'evy-driven Loewner chains, since the proof only uses Brownian tail bounds and Loewner perturbation estimates.","The paper leaves the exact rate of convergence open; a sharper estimate of the high-probability sequences $\\varphi_i$ and $\\psi_i$ would turn the qualitative guarantee into a practical step-size prescription.","The numerical findings for noise-reinforced SLE suggest a possible phase transition in the reinforcement parameter $p$ and $\\kappa$; this is a testable prediction that a rigorous analysis of the driven Loewner equation could confirm or refute.","The authors note that the high-probability event in the $L^p$ theorem may be removable; if it can be replaced by the full probability space, the $L^p$ convergence would hold unconditionally."],"forward_implications":["SLE curves can in principle be simulated with sup-norm control on the entire trace on $[0,T]$, not merely Hausdorff convergence of the growing hulls, provided the mesh is chosen smaller than $N^{-3}$.","The $L^p$ result gives an integrated-error guarantee on a high-probability event, so bulk statistics such as fractal dimension or winding angles can be estimated from the splitting output with vanishing bias as $N$ grows.","The power-law interpolation result shows that replacing the Brownian driving path by its $p$-th power interpolation between grid points does not break convergence, justifying the pathwise discretization used in simulations.","For fractional SLE, the splitting drawings reproduce the self-similar shapes and the monotonic dependence of fractal dimension on $\\kappa$ and Hurst exponent $H$ reported by earlier simulations.","For noise-reinforced SLE, the splitting drawings exhibit traces that repeat previous twists, suggesting the long-term memory of the reinforced Brownian driver is inherited by the Loewner curve."],"supporting_citations":[{"why":"Supplies the reverse Loewner SDE formulation, the splitting scheme, and the bound between the true flow Z and the curve eta used in Propositions 3.2 and 3.5.","marker":"[37]"},{"why":"Supplies the sup-norm estimates for Loewner curve approximations and the subpower oscillation bound used in Proposition 3.1.","marker":"[91]"},{"why":"Supplies the Brownian oscillation estimate that yields the subpower sequence in Proposition 3.1.","marker":"[53]"},{"why":"Supplies the explicit supremum distribution of Brownian motion, via reflection principle, used for the exponential tail bounds in Propositions 3.3 and 3.4.","marker":"[14]"},{"why":"Supplies the H\\\"older regularity of the SLE path used in Proposition 3.6.","marker":"[92]"},{"why":"Supplies the perturbation estimate comparing two reversed Loewner flows with close driving functions, used in Proposition 3.3 and the power-law interpolation outline.","marker":"[93]"},{"why":"Supplies basic Loewner hull facts, including the monotonicity of the imaginary part of the flow used in Proposition 3.4.","marker":"[44]"}],"fun_headline_variants":["Splitting algorithm for SLE: sup-norm and L^p convergence proved","Provable convergence of splitting scheme to SLE trace in sup-norm and L^p","Splitting method simulates SLE curves with vanishing sup-norm and L^p errors","Sup-norm and L^p error bounds for splitting-based SLE simulation","Splitting algorithm converges to SLE trace in both sup-norm and L^p"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of Proposition 3.3 assumes that the splitting output and the Loewner flow driven by the piecewise-constant interpolation of the Brownian path coincide at every grid point; this equality is asserted without proof and a one-step calculation on the first interval shows the two objects differ by a term proportional to the Brownian increment.","fun_headline_variants_meta":{"raw":{"variants":["Splitting algorithm for SLE: sup-norm and L^p convergence proved","Provable convergence of splitting scheme to SLE trace in sup-norm and L^p","Splitting method simulates SLE curves with vanishing sup-norm and L^p errors","Sup-norm and L^p error bounds for splitting-based SLE simulation","Splitting algorithm converges to SLE trace in both sup-norm and L^p"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000279,"raw_usage":{"total_tokens":1633,"prompt_tokens":900,"completion_tokens":733,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":516,"completion_tokens_details":{"reasoning_tokens":625}},"tokens_in":516,"tokens_out":733,"duration_ms":7583,"temperature":1.0,"reasoning_tokens":625,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:23:46.025510+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the first splitting interval by hand for $N=1$: start from $iy_1=i$, apply one Ninomiya–Victoir step with Brownian increment $\\Delta B$, and compare the result with the Loewner flow driven by the step-constant driver on $[0,h]$. If the two outputs differ by an expression of order $|\\Delta B|$ rather than being exactly equal, the identity used in Proposition 3.3 is false, and the convergence proof as written does not go through; a corrected proof would need a new estimate for this grid-point mismatch.","supporting_citations":[{"cited_title":"Foster, T","cited_arxiv_id":null,"evidence_quote":"Supplies the reverse Loewner SDE formulation, the splitting scheme, and the bound between the true flow Z and the curve eta used in Propositions 3.2 and 3.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the sup-norm estimates for Loewner curve approximations and the subpower oscillation bound used in Proposition 3.1."},{"cited_title":"Lawler, V","cited_arxiv_id":null,"evidence_quote":"Supplies the Brownian oscillation estimate that yields the subpower sequence in Proposition 3.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the H\\\"older regularity of the SLE path used in Proposition 3.6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the perturbation estimate comparing two reversed Loewner flows with close driving functions, used in Proposition 3.3 and the power-law interpolation outline."},{"cited_title":"Kemppainen","cited_arxiv_id":null,"evidence_quote":"Supplies basic Loewner hull facts, including the monotonicity of the imaginary part of the flow used in Proposition 3.4."}],"review_version":1}