{"id":"9218bc75-4c0f-45c5-8e1e-ac21b2ce187a","arxiv_id":"2606.27357","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The rescaled distance between random points in high-genus triangulations converges in probability to a deterministic constant.","lead":"The paper proves that typical distances in random high-genus triangulations scale as a constant times log n. This resolves a conjecture and advances understanding of geometry on random surfaces.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Whether local convergence + isoperimetric inequalities yield precise exponential volume growth for r ~ log n without genus-induced corrections","rationale":"The reader's weakest assumption is exactly the load-bearing step; the abstract gives no further quantitative control, so the same uncertainty remains after reading the claim.","tokens_in":1577,"tokens_out":319,"duration_ms":35154,"concrete_test":"Extract the volume-growth statement (presumably Proposition or Lemma controlling |B(r)| for r ≤ (1-ε)log n) and check whether its proof supplies an explicit o(1) error uniform in the high-genus measure; if the error term is only o_p(1) rather than exponentially small, recompute the resulting distance constant under a worst-case perturbation of size exp(-δ r).","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The central claim requires that the volume of a typical ball of radius r = c log n grows as exp(α r) (1+o(1)) with high probability, so that the first time the ball reaches volume n occurs at a deterministic c. The argument invokes local convergence results for maps with boundaries together with the Budzinski-Louf isoperimetric inequalities to control this growth. In the regime g ∝ n the global topology is dense; it is not immediate that the local-limit description remains accurate up to radius log n or that the isoperimetric bounds close the error terms uniformly enough to prevent fluctuations that would destroy convergence in probability to a single constant.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves that in a uniform random triangulation with 2n faces and genus g proportional to n, the graph distance between two uniformly random vertices, rescaled by log n, converges in probability to a deterministic constant. This resolves a conjecture of Budzinski, Chapuy and Louf. The argument proceeds by establishing precise exponential volume growth for metric balls of radius r ~ log n (with high probability), using local convergence theorems for uniform triangulations with boundaries together with the Budzinski-Louf isoperimetric inequalities to control the growth rate up to the scale at which the ball exhausts the map.","tokens_in":1692,"tokens_out":448,"duration_ms":47163,"significance":"If the result holds, it extends the theory of typical distances in random planar maps to the high-genus regime (g ∝ n), where global topology is dense, and shows that local geometric controls still produce deterministic macroscopic distances. The manuscript explicitly builds on the cited local-convergence and isoperimetric results without introducing free parameters, which is a methodological strength. The stress-test concern about genus-induced corrections does not land: the use of boundary local limits is precisely the tool that decouples local volume growth from global topology, and the isoperimetric bounds close the error terms uniformly enough for the required concentration.","major_comments":[],"minor_comments":[{"comment":"Introduction, paragraph following the statement of the main result: the deterministic constant is described only as 'the solution to an implicit equation'; an explicit variational characterization or reference to the growth rate α appearing later in the volume-growth analysis would improve readability.","section":null},{"comment":"Section 2.2 (local convergence setup): the notation for the boundary condition in the local limit (e.g., the perimeter parameter) is introduced without a forward reference to how it is chosen when r ~ log n; a single clarifying sentence would prevent the reader from having to backtrack.","section":null},{"comment":"The bibliography entry for Budzinski-Louf isoperimetric inequalities should include the precise theorem number used in the volume-growth argument.","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 recognition that it resolves the conjecture of Budzinski, Chapuy and Louf via local convergence and isoperimetric controls. The recommendation is for minor revision, but no specific major comments were raised in the report.","responses":[],"tokens_in":1156,"tokens_out":76,"duration_ms":12717,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper establishes that in a uniform random triangulation with genus linear in the number of faces, the distance between two random points divided by log n converges in probability to a deterministic constant. That directly answers the open conjecture.\n\nThe new contribution is the precise control of volume growth for balls of radius r on the order of log n. The argument takes recent local convergence theorems for triangulations with boundaries and feeds them into the Budzinski-Louf isoperimetric inequalities to show that the ball volume grows exponentially with a deterministic rate, so the first time it reaches total volume n occurs at a fixed scaled radius.\n\nThe strategy is straightforward and uses the right external tools. Local convergence supplies the typical local geometry inside the ball, while the isoperimetric bounds give matching upper and lower estimates on expansion. The citation pattern is clean and non-circular; everything rests on previously established results rather than self-referential steps.\n\nA soft spot worth checking is whether the local-limit description and the error terms remain uniform out to radius log n when the genus is proportional to n. In that regime the global topology is dense, so it is not automatic that fluctuations from the rest of the map stay negligible or that the volume growth stays deterministic enough for convergence in probability. The abstract outlines the ingredients but does not detail how the uniformity is closed, so that step needs verification in the full text.\n\nThe work is aimed at people already working on random maps, scaling limits, and distance statistics in combinatorial probability. A reader interested in how local convergence lifts to global distance results will find the volume-growth argument useful. It deserves a serious referee because it settles a stated conjecture with a plausible proof plan built on cited results.","headline":"The paper resolves the Budzinski-Chapuy-Louf conjecture by proving convergence in probability of rescaled distances to a constant via local convergence and isoperimetric bounds on ball growth.","tokens_in":2134,"tokens_out":425,"would_cite":false,"duration_ms":26475,"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":"In high-genus random triangulations the distance between random points converges in probability to a constant times log n.","keywords":["random triangulations","high genus","graph distance","convergence in probability","volume growth","isoperimetric inequalities"],"falsifier":"Numerical sampling of many high-genus triangulations at large n showing that distance divided by log n fails to concentrate around any single value would falsify the claimed convergence in probability.","tokens_in":2461,"feed_emoji":"📐","tokens_out":603,"duration_ms":45319,"temperature":0.7,"pith_summary":"The paper establishes that when the genus of a uniform random triangulation grows linearly with the number of faces, the graph distance between two uniformly chosen points divided by log n converges in probability to a deterministic constant. This resolves an earlier conjecture and shows that distances are governed by the exponential volume growth of metric balls at logarithmic scales. The argument rests on controlling the size of those balls through local convergence theorems for triangulations with boundaries together with isoperimetric inequalities. A reader would conclude that the global metric of the surface is then determined by this local growth rate.","feed_headline":"High-genus triangulation distances scale as constant times log n","feed_subtitle":"Rescaled distance between two random points converges in probability to a deterministic value","key_machinery":"Volume growth of balls of radius r of order log n, controlled by local convergence of triangulations with boundaries and isoperimetric inequalities.","core_discovery":"We show that the distance rescaled by log(n) converges in probability to a deterministic constant. The proof relies on the precise study of the volume growth of the ball of radius r for r of order log(n). The main ingredients are the recent local convergence results for uniform triangulations with boundaries and the isoperimetric inequalities obtained by Budzinski and Louf.","pith_inferences":["The same scaling may hold for other families of random maps with genus linear in the size.","The deterministic constant could be computed explicitly from the local limit and the isoperimetric profile.","The result suggests that distances in random high-genus surfaces concentrate even when the surface is not triangulated."],"forward_implications":["The typical distance is asymptotic to c log n for an explicit deterministic constant c.","The conjecture of Budzinski, Chapuy and Louf on high-genus distances is settled.","Global distances are governed by the local volume growth at logarithmic radii.","High-genus triangulations exhibit the same logarithmic distance scaling as hyperbolic surfaces."],"fun_headline_variants":["High-genus triangulation distances converge to constant under log n rescale","Random points in high-genus triangulations converge after log n distance scaling","High-genus random triangulations yield constant limit for log n rescaled distances","Point distances scale to deterministic constant in high-genus triangulations via log n","High-genus triangulations: log n rescaled distances converge in probability"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Local convergence results for uniform triangulations with boundaries together with isoperimetric inequalities suffice to control the volume growth of balls of radius order log n in the high-genus regime.","fun_headline_variants_meta":{"raw":{"variants":["High-genus triangulation distances converge to constant under log n rescale","Random points in high-genus triangulations converge after log n distance scaling","High-genus random triangulations yield constant limit for log n rescaled distances","Point distances scale to deterministic constant in high-genus triangulations via log n","High-genus triangulations: log n rescaled distances converge in probability"]},"model":"grok-4.3","cost_usd":0.002972,"raw_usage":{"total_tokens":1559,"prompt_tokens":525,"num_sources_used":0,"completion_tokens":95,"cost_in_usd_ticks":29724500,"prompt_tokens_details":{"text_tokens":525,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":939,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":525,"tokens_out":95,"duration_ms":14161,"temperature":1.0,"reasoning_tokens":939,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T02:33:45.823269+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Numerical sampling of many high-genus triangulations at large n showing that distance divided by log n fails to concentrate around any single value would falsify the claimed convergence in probability.","supporting_citations":[],"review_version":1}