{"id":"60142338-ba2d-4544-a906-aff18f2cb975","arxiv_id":"2507.05936","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The logarithmic Laplacian on weighted graphs is defined via a Bochner integral, given a kernel formula under stochastic completeness, and shown on Z^d to have sharp kernel bounds and exact diffusion asymptotics.","lead":"This paper defines the logarithmic Laplacian on weighted graphs via a Bochner integral and derives an explicit kernel formula, sharp kernel bounds on lattices, and exact diffusion asymptotics on Z^d. The results give analysts a discrete counterpart to the recently developed logarithmic Laplacian on manifolds, enabling PDE and spectral studies on graphs.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Weighted-lattice kernel estimates are imported from Theorem 4.2 without verifying its CDE'(n0,0) hypothesis; as written, Propositions 1.2-1.3 and Theorem 1.8 do not follow.","rationale":"I read the proof of Theorem 1.1 in good faith: under stochastic completeness and the integrability condition (1.1), the splitting of the Bochner integral at t=1 is legitimate for compactly supported u, and the pointwise kernel formula follows by dominated convergence and the Euler-Mascheroni identity. The central operator-theoretic construction is therefore plausible. The reader's weakest assumption is also the most load-bearing weakness I could identify: the lattice estimates and the ell-p convergence theorem depend on heat-kernel bounds whose cited source requires CDE'(n0,0), and the paper never proves this condition for the stated lattice class. This is a proof gap, not merely a disagreement with convention; if the curvature condition fails or is unavailable, Propositions 1.2-1.3 and Theorem 1.8 lack support. Since the main theorem survives and the gap is likely repairable via Delmotte's Harnack/volume-doubling/Poincare equivalence, the appropriate verdict remains conditional pending the missing verification or an alternative derivation.","tokens_in":33000,"tokens_out":33634,"duration_ms":387290,"concrete_test":"On a one-dimensional weighted lattice with alternating conductances (e.g., w_{2k}=1, w_{2k+1}=2, mu=m), compute Gamma_2(f) - (1/n)(Delta f)^2 for a test function f at a vertex. If this quantity is negative for every n > 0, then CDE'(n,0) fails for this class and the cited Theorem 4.2 cannot be invoked. Alternatively, verify volume doubling and the Poincare inequality for the stated weighted-lattice class and re-derive (1.2)-(1.4) from Delmotte's equivalence; if the bounds follow, the results survive but the proof must be rewritten.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The sharp lattice results rest on the heat-kernel bounds (1.2)-(1.4), which are imported from Theorem 4.2. That theorem requires both the curvature-dimension condition CDE'(n0,0) and Delmotte's loop condition Delta(alpha). Section 4.1 verifies only Delta(alpha) — the argument shows wmin > 0 and bounded vertex measures imply a loop condition, but it never checks CDE'(n0,0). Uniformly positive vertex measures give bounded degree and positive edge weights, so Delta(alpha) is easy, but CDE'(n0,0) is a genuine additional curvature hypothesis; it is not automatic for a weighted lattice with varying conductances. Consequently the two-sided bounds for Wlog and W, the unboundedness arguments, and the ell-p convergence theorem are not established by the proof as written. The central pointwise representation Theorem 1.1 itself is not affected by this gap, but the advertised weighted-lattice consequences are. A repair may be possible by replacing Theorem 4.2 with Delmotte's parabolic-Harnack equivalence, since these weighted lattices are quasi-isometric to Z^d and should satisfy volume doubling and a Poincare inequality, but that replacement is not present in the manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a logarithmic Laplacian on infinite weighted graphs via functional calculus and a Bochner-type integral, and derives a pointwise kernel representation under stochastic completeness and an integrability condition. On weighted lattice graphs with uniformly positive vertex measures it claims sharp two-sided bounds for the logarithmic kernel, unboundedness of the operator on ell^2, strong ell^p convergence of the fractional-to-logarithmic difference quotient, and Fourier-analytic asymptotics for fractional and logarithmic diffusion kernels on Z^d.","tokens_in":33227,"tokens_out":22063,"duration_ms":247673,"significance":"If the central pointwise formula is correct, this is the first kernel representation of the logarithmic Laplacian on general graphs and provides a useful tool for studying nonlocal equations in the discrete setting. The paper contains several strong features: an explicit parameter-free formula, an alternative derivation of the pointwise representation on Z^d via the derivative at s=0, identification of Fourier multipliers, and sharp asymptotic constants for diffusion kernels. However, the weighted-lattice results rest on heat-kernel bounds whose curvature hypothesis is not verified, and the general-graph proof imports a key Bochner identity from an unpublished preprint. The central idea is promising and likely repairable, but as written the advertised lattice consequences are conditional.","major_comments":[{"comment":"The heat-kernel bounds (1.2)-(1.4) are imported from Theorem 4.2, whose hypotheses are the curvature-dimension condition CDE'(n0,0) together with Delmotte's loop condition Delta(alpha). Section 4.1 verifies only a loop-weight condition for weighted lattices and never establishes CDE'(n0,0); uniformly positive vertex measures do not imply a curvature-dimension inequality, and the claim that these assumptions imply wmin>0 (and hence Delta(alpha)) also needs an explicit lower bound on edge weights. Since Propositions 1.2, 1.3, 1.4, 1.5, Theorem 1.8, and the alternative derivation in Theorem 4.12 all rely on these heat-kernel bounds, the weighted-lattice results are not proved as written. A repair would be to replace Theorem 4.2 by Delmotte's equivalence (volume doubling plus Poincare inequality) for these quasi-isometric lattices, or to verify CDE'(n0,0) directly.","section":"Section 4.1, Theorem 4.2; Propositions 1.2-1.3"},{"comment":"The central pointwise formula on general graphs is derived from the Bochner-integral identity in Theorem 3.3, which is quoted from the first author's preprint [34] together with Lemma 3.4. The manuscript does not prove this identity or state the precise conditions under which it holds beyond citing [34]. Since Theorem 1.1 is the paper's main general claim, the derivation is not self-contained; please include a full proof of Theorem 3.3 (or a detailed derivation from the spectral theorem) and of Lemma 3.4. The alternative proof in Theorem 4.12 covers only Z^d and does not remove this dependence for the general-graph statement.","section":"Section 3.2, Theorem 3.3 and proof of Theorem 1.1"},{"comment":"In the proof of Proposition 1.10, after the Taylor expansion the remainder term r^{-s} integral O(|eta|^{3s}) chi(eta/(r delta)) e^{i omega dot eta} d eta is asserted to be bounded uniformly in r by analogy with Lemma 1.11. This is not immediate: the exponent 3s may lie outside the range allowed in Lemma 1.11, and the asserted bound relies on an oscillatory decay estimate that is not written down. The sharp off-diagonal constant is therefore not fully established as the proof stands. Please add the missing stationary-phase or integration-by-parts argument, or state explicitly why the remainder is controlled.","section":"Section 4.4, proof of Proposition 1.10"}],"minor_comments":[{"comment":"There are several typographical errors: 'nature' should be 'natural', 'invloving' should be 'involving', 'futher' should be 'further', 'domian' should be 'domain', and 'th enormalized' should be 'the normalized'.","section":"Throughout"},{"comment":"The displayed identity sum_k mu(x) mu(y_k) p'(0,x,y_k) = mu(x)^2 p'(0,x,x) has a sign/absolute-value error: the left-hand side is positive, while mu(x)^2 p'(0,x,x) is negative for the normalized Laplacian. Use |mu(x)^2 p'(0,x,x)| or the equivalent positive expression sum_{y neq x} w_{xy}.","section":"Section 4.3, proof of Proposition 1.6"},{"comment":"The sentence 'Since ||u||_{ell^infty} is comparable to ||u||_{ell^p}' is false for general ell^p functions; for compactly supported u the comparison constant depends on the support, and the ell^infty convergence should be justified separately with an explicit support-dependent bound.","section":"Section 4.3, proof of Theorem 1.8"},{"comment":"The lower bound sum_{y: d(y,0) leq n, y neq x} d(x,y)^{-d} geq sum_{k=1}^n |S(k)|/k^d is not valid for arbitrary x with d(x,0) leq n, since d(x,y) can be much larger than d(y,0). The argument should be rephrased using a shell decomposition centered at x or a direct volume argument.","section":"Section 4.3, proof of Proposition 1.5"},{"comment":"The sentence 'adding loops does not degrade curvature bounds' is asserted without proof; if loops are added to force Delta(alpha), the CDE' condition for the modified graph should be checked or avoided by not relying on Theorem 4.2.","section":"Section 4.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends on [34], an arXiv preprint by the first author, for the Bochner identity used in the main theorem; the editor may wish to ascertain whether [34] is under review. The CDE' gap is the main technical issue and is fixable in principle, but it affects several advertised results. I do not see a circularity or fitted-parameter problem beyond the explicit reliance on [34]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The pointwise formula for the logarithmic Laplacian on graphs (Theorem 1.1) is new and, under stochastic completeness and (1.1), the proof is clean. The Fourier results on Z^d — exact multipliers, large-time constants, off-diagonal decay for fractional and log kernels — are also new and appear correct. What does not follow as written is the sharp kernel analysis on weighted lattices: Propositions 1.2–1.3 and Theorem 1.8 depend on heat-kernel bounds imported from Theorem 4.2, which needs CDE'(n,0) plus Delmotte's loop condition. Section 4.1 verifies only the loop condition. CDE'(n,0) is a real curvature assumption, not a free gift of bounded geometry, so the proof has a genuine hole there. The results may still be true — these lattices are quasi-isometric to Z^d, so volume doubling and Poincaré hold, and Delmotte's equivalence would deliver the Gaussian bounds — but that route is not taken in the manuscript.\n\nSmaller things: Lemma 1.11 skips d = 1 with a reference to Stein; the shell-counting in the proof of Proposition 1.5 ignores that x near the boundary of the ball has fewer distant y's; and the lower-bound integral in Proposition 1.3 has an e^t that should be e^{-t} after the substitution. None of these are load-bearing. The Bochner formula is quoted from [34] by the same first author, which is legitimate, and Theorem 4.12 gives an independent derivation on Z^d.\n\nThe main theorem stands. The ℓ^p convergence theorem is conditional on the heat-kernel bounds, and that is the only big gap. The paper is for people working on discrete PDEs and graph analysis; it gives them a usable operator and the right asymptotics on Z^d. I would send it to a serious referee and ask for the Delmotte repair and the d = 1 case. The fix is mechanical, not conceptual.","headline":"First graph logarithmic Laplacian with a clean pointwise formula, but the sharp lattice results rest on an unverified CDE'(n,0) curvature assumption; the gap is fixable and the Z^d Fourier analysis is solid.","tokens_in":33768,"tokens_out":7441,"would_cite":true,"duration_ms":85364,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","05C63","35K08","47A60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes the first pointwise kernel representation of the logarithmic Laplacian on infinite weighted graphs, then uses it to prove sharp lattice kernel bounds, ℓ^p convergence of the fractional-to-logarithmic limit, and…","keywords":["logarithmic Laplacian","fractional Laplacian","weighted graphs","lattice graphs","heat kernel estimates","diffusion kernel","Fourier multiplier","stochastic completeness"],"falsifier":"Compute the continuous-time heat kernel $p(t,x,y)$ of a weighted $\\mathbb{Z}^2$ lattice whose vertex measure alternates between two positive values, say 1 and 2, and check the two-sided Gaussian bounds (1.2)–(1.4); a failure of the lower bound for large $t$ and some $x,y$ would show that the imported curvature condition is not available for that class, invalidating the sharp kernel estimates and the $\\ell^p$ convergence theorem that depend on them.","tokens_in":32775,"feed_emoji":"🧮","tokens_out":9313,"duration_ms":101714,"temperature":0.7,"pith_summary":"The paper's central goal is to give the first workable definition of the logarithmic Laplacian on infinite weighted graphs, not just on Euclidean space or finite vertex sets. It derives a Bochner-integral representation and, under stochastic completeness, an explicit pointwise kernel formula in which the operator splits into a short-range gradient-type part, a long-range part, and an Euler–Mascheroni constant term. On weighted lattice graphs with uniformly positive vertex measures, it proves sharp two-sided bounds on the two kernels, shows the operator is unbounded on ℓ², and proves that the fractional Laplacian tends to the logarithmic Laplacian strongly in ℓ^p for every 1 < p ≤ ∞ on compactly supported functions. Finally, on Z^d it identifies the Fourier multipliers of both operators and derives large-time and off-diagonal asymptotics of the associated diffusion kernels. If correct, this gives discrete analogues of the continuous logarithmic Laplacian that can be used in PDE and spectral theory on graphs.","feed_headline":"First kernel formula for the logarithmic Laplacian on graphs","feed_subtitle":"Derivative of the fractional Laplacian at s=0 becomes a computable kernel, with sharp lattice bounds and ℓ^p convergence.","key_machinery":"The carrying mechanism is the Bochner-integral identity $\\log(-\\Delta)u=\\int_0^\\infty (e^{-t}u-e^{t\\Delta}u)\\,t^{-1}dt$, inherited from functional calculus, together with a split at $t=1$ into a short-time kernel $W_{\\log}$ and a long-time kernel $W$. On weighted lattices the sharp analysis is powered by two-sided Gaussian heat-kernel bounds imported from a strengthened curvature condition, by a sharp Davies–Gaffney–Grigor'yan lemma for the fine upper bound on $W$, and ultimately by the Fourier multiplier $\\ln\\Phi(\\xi)$ with $\\Phi(\\xi)=\\sum_{j=1}^d(2-2\\cos\\xi_j)$, which controls the diffusion-kernel asymptotics.","core_discovery":"Theorem 1.1 states that on an infinite, connected, stochastically complete weighted graph whose heat kernel satisfies the integrability condition ∫_1^∞ p(t,x,y)/t dt < ∞, the logarithmic Laplacian acts on compactly supported functions by the explicit pointwise formula\n$$\\log(-\\$\\Delta$)u(x)=\\frac{1}{\\mu(x)}\\sum_{y\\neq x}W_{\\log}(x,y)(u(x)-u(y))-\\frac{1}{\\mu(x)}\\sum_{y}W(x,y)u(y)+\\Gamma'(1)u(x),$$\nwhere the positive symmetric kernels are $W_{\\log}(x,y)=\\mu(x)\\mu(y)\\int_0^1 p(t,x,y)\\,t^{-1}dt$ and $W(x,y)=\\mu(x)\\mu(y)\\int_1^\\infty p(t,x,y)\\,t^{-1}dt$. The short-time kernel $W_{\\log}$ behaves like a bounded gradient-type interaction, while the long-time kernel $W$ carries the genuinely nonlocal part of the operator. This representation is the paper's main contribution and is what makes the sharp lattice estimates, the ℓ^p convergence theorem, and the Fourier-multiplier analysis possible.","pith_inferences":["Not pursued in the paper: the same Bochner-splitting should define a logarithmic Laplacian on any stochastically complete graph satisfying the integrability condition, even without Gaussian heat-kernel bounds; only the sharp two-sided estimates would be lost.","Because the long-range kernel has polynomial decay $d(x,y)^{-d}$, the operator is a natural model for Hardy-type inequalities and nonlocal Sobolev spaces on graphs, where the constants $\\Gamma'(1)$ and $|S^{d-1}|$ appearing in the diffusion kernels could serve as test quantities.","A testable extension: on $\\mathbb{Z}^2$ with non-constant vertex measure, the off-diagonal constants in the diffusion-kernel asymptotics should be unchanged because they come from the low-frequency (local) behavior of the symbol; computing the Fourier integrals numerically for a coarse-grained mass distribution would test the robustness of the asymptotics."],"forward_implications":["On weighted lattices with uniformly positive vertex measure, the short-time kernel is bounded above by $e^{r}r^{-r-1}$ while the long-time kernel satisfies two-sided $d(x,y)^{-d}$ bounds, so the long-range part is only marginally summable.","The logarithmic Laplacian is unbounded on $\\ell^2(\\mathbb{Z}^d)$: its gradient-type short-range part is bounded, but the long-range quadratic form diverges on normalized indicator balls.","For every compactly supported $u$ and every $1<p\\le\\infty$, the difference quotient $((-\\Delta)^s u-u)/s$ converges strongly in $\\ell^p$ to $\\log(-\\Delta)u$, and $\\log(-\\Delta)u\\in\\ell^p$.","The fractional diffusion kernel on $\\mathbb{Z}^d$ has the same large-time decay rate and sharp constant as on $\\mathbb{R}^d$, namely $t^{-d/(2s)}C_{s,d}$, and off-diagonal decay $|x-y|^{-d-2s}$ with explicit constant.","The logarithmic diffusion kernel on $\\mathbb{Z}^d$ exists exactly for $0\\le t<d/2$, blows up like $(d-2t)^{-1}$ as $t\\to d/2$, and decays like $|x-y|^{2t-d}$ off-diagonal, with constant $A_{-t,d}/(2\\pi)^d$."],"supporting_citations":[{"why":"It supplies the Bochner-integral formula for the logarithmic Laplacian and the functional-calculus framework that the graph representation is built upon.","marker":"[34]"},{"why":"It supplies the pointwise kernel formula for the fractional Laplacian on stochastically complete graphs that Theorem 1.1 is modeled on.","marker":"[46]"},{"why":"It supplies the two-sided Gaussian heat-kernel bounds that yield the sharp $W_{\\log}$ and $W$ kernel estimates on weighted lattices.","marker":"[40]"},{"why":"It supplies the sharp Davies–Gaffney–Grigor'yan heat-kernel decay used for the upper bound $W(x,y)\\lesssim d(x,y)^{-d}$.","marker":"[37]"},{"why":"It supplies the Fourier multiplier $\\Phi(\\xi)^s$ for the fractional Laplacian on lattice graphs, the starting point of the diffusion-kernel asymptotics.","marker":"[53]"},{"why":"It supplies the repeated integration-by-parts estimates used to control the smooth remainder terms in the off-diagonal asymptotics.","marker":"[55]"},{"why":"It supplies the continuous-space logarithmic Laplacian and the derivative-at-zero interpretation that the graph theory mirrors.","marker":"[11]"}],"fun_headline_variants":["Log-Laplacian kernel formula: sharp lattice bounds, ℓ^p limit","Explicit log-Laplacian on graphs: kernel bounds and ℓ^p convergence","Pointwise log-Laplacian formula with sharp asymptotics on lattices","Log-Laplacian as derivative of fractional Laplacian: kernel and limits","First explicit log-Laplacian kernel: sharp bounds and ℓ^p convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the weighted lattices studied here satisfy the strengthened non-negative curvature condition CDE′(n,0), which the paper imports from an external heat-kernel theorem without verifying; if that condition fails, the Gaussian heat-kernel bounds and all the sharp kernel estimates built on them collapse.","fun_headline_variants_meta":{"raw":{"variants":["Log-Laplacian kernel formula: sharp lattice bounds, ℓ^p limit","Explicit log-Laplacian on graphs: kernel bounds and ℓ^p convergence","Pointwise log-Laplacian formula with sharp asymptotics on lattices","Log-Laplacian as derivative of fractional Laplacian: kernel and limits","First explicit log-Laplacian kernel: sharp bounds and ℓ^p convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000209,"raw_usage":{"total_tokens":1474,"prompt_tokens":1076,"completion_tokens":398,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":692,"completion_tokens_details":{"reasoning_tokens":294}},"tokens_in":692,"tokens_out":398,"duration_ms":4642,"temperature":1.0,"reasoning_tokens":294,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:16:43.413057+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the continuous-time heat kernel $p(t,x,y)$ of a weighted $\\mathbb{Z}^2$ lattice whose vertex measure alternates between two positive values, say 1 and 2, and check the two-sided Gaussian bounds (1.2)–(1.4); a failure of the lower bound for large $t$ and some $x,y$ would show that the imported curvature condition is not available for that class, invalidating the sharp kernel estimates and the $\\ell^p$ convergence theorem that depend on them.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the two-sided Gaussian heat-kernel bounds that yield the sharp $W_{\\log}$ and $W$ kernel estimates on weighted lattices."},{"cited_title":"Sharp davies–gaffney–grigor’yan lemma on graphs","cited_arxiv_id":null,"evidence_quote":"It supplies the sharp Davies–Gaffney–Grigor'yan heat-kernel decay used for the upper bound $W(x,y)\\lesssim d(x,y)^{-d}$."},{"cited_title":"Classical fourier analysis, volume 2","cited_arxiv_id":null,"evidence_quote":"It supplies the repeated integration-by-parts estimates used to control the smooth remainder terms in the off-diagonal asymptotics."},{"cited_title":"The dirichlet problem for the logarithmic laplacian","cited_arxiv_id":null,"evidence_quote":"It supplies the continuous-space logarithmic Laplacian and the derivative-at-zero interpretation that the graph theory mirrors."}],"review_version":1}