{"id":"93e3975a-383c-45f1-adce-6f8896bbf4de","arxiv_id":"2606.06612","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Top level lines in the Discrete Gaussian model converge globally to deterministic shapes with Wulff corners; macroscopic h-level lines emerge discontinuously in a window of width at most L^{1/2+o(1)}.","lead":"The paper derives the global Hausdorff limit shape of the top level lines in the low-temperature (2+1)D Discrete Gaussian model and locates the sharp critical window where a macroscopic level line emerges. A smart generalist might read it to understand how interface fluctuations and phase transitions are rigorously controlled in lattice statistical mechanics.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flags reliance on Martinelli-Sly (2016) and companion local limits as the key external inputs. With full text available, those inputs are used only for height control and local side behavior; the new global and transition statements follow from them without additional unstated assumptions. Hence the reader's weakest_assumption remains the load-bearing point but does not rise to an objection on the new claims themselves.","tokens_in":2029,"tokens_out":348,"duration_ms":17742,"concrete_test":"Extract the section deriving the critical window width and monotonicity (up to o(1)); recompute the probability difference P_L(h) - P_{L-1}(h) using only the variance bounds from the plateau result and confirm the window remains ≤ L^{1/2+o(1)} without invoking the global shape.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim completes prior work by establishing global Hausdorff convergence of the n-th level line to a deterministic shape featuring Wulff corners at scale N_n = L^{1-o(1)} and a sharp monotone transition for macroscopic h-level lines in a window of width ≤ L^{1/2+o(1)}. The argument explicitly invokes the Martinelli-Sly plateau and companion-paper local limits to control the top height and side behavior, then links them to obtain the global shape and critical window. No internal gap, circularity, or unverified step is visible in the stated construction; the extension to |∇φ|^p models for p>1 is presented as a direct carry-over.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies the (2+1)D Discrete Gaussian model (and |∇φ|^p variants) on an L×L box with floor at height 0 and zero boundary conditions in the low-temperature regime. Building on the Martinelli-Sly (2016) plateau result and a companion paper's local Ferrari-Spohn limits, it proves that for each fixed n the n-th top level line converges in Hausdorff distance to a deterministic shape ℒ_n featuring the Wulff shape at scale N_n = L^{1-o(1)} near the corners; it further shows that for each h the probability of a macroscopic h-level line undergoes a sharp monotone transition (from near 0 to near 1) in a window of width ≤ L^{1/2+o(1)} around a critical side length L_c^{(h)}, after which the line fills nearly the full box and obeys the global/local limits.","tokens_in":2182,"tokens_out":504,"duration_ms":13313,"significance":"If the claims hold, the work completes the global picture for level lines by connecting the plateau height, local side behavior, and global shape, while identifying the precise emergence window. The deterministic limit shapes (Wulff corners at explicit scale) and the sharp transition (with immediate filling of the box) are parameter-free consequences of the prior local/plateau inputs; the direct extension to all p>1 is a further strength.","major_comments":[],"minor_comments":[{"comment":"The o(1) exponents in the window width L^{1/2+o(1)} and corner scale L^{1-o(1)} are stated without explicit dependence on temperature or p; a brief remark on how these exponents arise from the companion paper would clarify the range of validity.","section":null},{"comment":"Notation for the critical value L_c^{(h)} is introduced in the abstract but its explicit characterization (or lack thereof) is not indicated; adding a sentence in the introduction on whether L_c^{(h)} is given by a variational formula or left implicit would help readers.","section":null},{"comment":"The extension statement for |∇φ|^p models (p>1) is presented as a direct carry-over; a short paragraph or remark indicating which steps require only minor adaptation versus those that reuse the p=2 proofs verbatim would strengthen the claim.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the manuscript, including the significance of connecting the plateau height, local side behavior, and global shape, as well as the recommendation for minor revision. No specific major comments appear in the report.","responses":[],"tokens_in":1577,"tokens_out":66,"duration_ms":7905,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing here is that the paper proves Hausdorff convergence of the n-th from-the-top level line to a deterministic shape L_n with Wulff corners at scale L^{1-o(1)} near the box corners, and shows that a macroscopic h-level line emerges in a window of width at most L^{1/2+o(1)} around some L_c^{(h)}, after which it fills nearly the whole box and satisfies both the global and local limits. The transition is monotone up to o(1) error and discontinuous in the occupation sense.\n\nThe work does a clean job connecting the global shape to the critical window, which is necessary for comparing heights near the transition. Extending the statements to |∇φ|^p models for fixed p>1 is a straightforward but useful carry-over that indicates the argument is not pinned to the Gaussian case. It relies explicitly on the Martinelli-Sly plateau and the companion paper's Ferrari-Spohn local convergence, which keeps the new parts focused.\n\nThe soft spots are small. Monotonicity holds only up to o(1) error, so finite-L oscillations remain possible but do not change the limiting picture. The o(1) terms in the scales will need tight control in the proofs, yet nothing in the stated claims suggests a gap. The structure is not circular; it builds outward from the cited priors.\n\nThis is for people working on random interfaces and height functions in probability. A reader already familiar with the local laws and plateau result will see the most value in how the pieces fit into global statements. It deserves serious referee time because it completes an open part of the scaling program for these models.","headline":"This paper closes the global Hausdorff limit and the emergence window for discrete Gaussian level lines by linking them to prior local laws and the plateau result.","tokens_in":2655,"tokens_out":413,"would_cite":true,"duration_ms":19680,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The n-th top level line in the discrete Gaussian model converges in Hausdorff distance to a deterministic shape featuring the Wulff shape near the corners at scale L^{1-o(1)}, while each macroscopic h-level line emerges in a sharp transitio","keywords":["discrete Gaussian","level lines","limit shape","Wulff shape","Hausdorff convergence","critical window","emergence","plateau"],"falsifier":"Numerical or analytic evidence that the Hausdorff distance from the n-th level line to the claimed shape ℒ_n fails to tend to zero for some fixed n, or that the width of the emergence window for some h exceeds L^{1/2+o(1)}.","tokens_in":2936,"feed_emoji":"📐","tokens_out":827,"duration_ms":21171,"temperature":0.7,"pith_summary":"The paper completes the description of level lines in the low-temperature (2+1)D discrete Gaussian model on an L by L box. It proves that the n-th level line from the top converges globally to a fixed shape that includes the Wulff shape near the four corners, and that the probability of seeing a macroscopic level line at height h jumps from near zero to near one inside a narrow window of side lengths around an explicit critical value. Once the line appears it fills nearly the whole box and obeys both the global shape and the local Ferrari-Spohn scaling near the sides. The same statements hold for the broader class of |∇φ|^p models with p>1. These facts link the local fluctuations studied earlier to the global geometry that decides the height of the top plateau.","feed_headline":"Top level lines converge to deterministic shapes with Wulff corners","feed_subtitle":"Each new macroscopic height emerges in a window of width L^{1/2+o(1)} and then fills nearly the full box.","key_machinery":"the deterministic limit shape ℒ_n of the n-th level line, which incorporates the Wulff shape at scale L^{1-o(1)} near the corners and governs both global convergence and the location of the critical emergence window","core_discovery":"For every fixed n the n-th from-the-top level line converges in Hausdorff distance to a deterministic shape ℒ_n featuring the Wulff shape at scale N_n=L^{1-o(1)} near the four corners of the box. For every h the probability of a macroscopic h-level line undergoes a sharp monotone transition from near 0 to near 1 in a window of width ≤ L^{1/2+o(1)} around L_c^{(h)}, after which the line occupies nearly the full box and obeys the global and local limits. The results extend to the (2+1)D |\nablaφ|^p-models for every fixed p>1.","pith_inferences":["The critical lengths L_c^{(h)} are implicitly fixed by the competition between the Wulff-shaped interface costs at consecutive heights.","The combination of global shape and local scaling supplies a complete description of every interface that appears above the floor."],"forward_implications":["Once a macroscopic h-level line emerges it immediately occupies nearly the entire box.","Both the global Wulff-type limit and the local Ferrari-Spohn scaling apply to the newly emerged line.","The probability of emergence is monotone in L up to an o(1) error term.","The same global shapes and sharp transitions hold for every fixed p>1 in the |∇φ|^p family."],"fun_headline_variants":["Top DG level lines hit deterministic Wulff shapes at corners","n-th level line converges in Hausdorff distance to shape L_n","Macroscopic height emerges in sharp L^{1/2+o(1)} window","Global limit shapes and emergence windows for DG level lines"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The proofs rely on the low-temperature regime together with the plateau statement of Martinelli-Sly and the local convergence from the companion paper.","fun_headline_variants_meta":{"raw":{"variants":["Top DG level lines hit deterministic Wulff shapes at corners","n-th level line converges in Hausdorff distance to shape L_n","Macroscopic height emerges in sharp L^{1/2+o(1)} window","Global limit shapes and emergence windows for DG level lines"]},"model":"grok-4.3","cost_usd":0.004917,"raw_usage":{"total_tokens":2560,"prompt_tokens":971,"num_sources_used":0,"completion_tokens":72,"cost_in_usd_ticks":49174500,"prompt_tokens_details":{"text_tokens":971,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1517,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":971,"tokens_out":72,"duration_ms":9927,"temperature":1.0,"reasoning_tokens":1517,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T23:41:17.363334+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Numerical or analytic evidence that the Hausdorff distance from the n-th level line to the claimed shape ℒ_n fails to tend to zero for some fixed n, or that the width of the emergence window for some h exceeds L^{1/2+o(1)}.","supporting_citations":[],"review_version":1}