{"id":"b1f755f9-8fb2-4f21-a74c-9044c76c3704","arxiv_id":"2411.12715","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For finitely presented acylindrically hyperbolic groups with at most polynomial Dehn function, the random Dehn function is at most quadratic and strictly smaller than the classical Dehn function when the group is not hyperbolic.","lead":"A new invariant called the random Dehn function measures the expected area of loops made by random walks in a group. The paper proves that for a wide class of groups, this random version grows strictly slower than the usual worst-case Dehn function, confirming Gromov's intuition in a new model.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 2.9/2.10 indexes the loop word incorrectly: as printed, the word is not null-homotopic, so the random Dehn function in Theorem A is undefined until corrected.","rationale":"The reader already flagged the indexing problem in Definitions 2.9/2.10 and treated it as fixable; I agree it is fixable, but I regard it as the most load-bearing defect because it concerns the definition of the central quantity itself, not merely a step in the proof. The imported probabilistic input from [MS20]/[GS22] is a reasonable prior-work assumption; the reader's weakest_assumption about Theorem 3.3 is a genuine fragility but less immediate than the fact that, as printed, Fill_{P,α} is not defined on the word produced by the definition. Theorem A cannot be accepted as a definitive reference until the loop word is corrected to be genuinely null-homotopic. Because the error is local and the intended construction is clear, CONDITIONAL remains the appropriate verdict; no new evidence forces a full rejection.","tokens_in":10072,"tokens_out":15309,"duration_ms":144735,"concrete_test":"Take the free group F(a,b), presentation with generators {a,b}, set x0=1, x1=a, x2=ab, and use a genuine combing α (e.g., reduced normal form). Compute the reduced word given by Definition 2.9, including α0 as required for a full loop. If the reduced word is nonempty, the loop is not null-homotopic and Area is undefined; the definition must be altered. The same computation with α_i=α(x_i^{-1}x_{i+1}) gives the identity, confirming the fix.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 2.9 sets α_i = α(x_{i+1}x_i^{-1}) and defines W_{x1,...,xn} = α_1···α_{n-1}α_n^{-1} (Definition 2.10 repeats the same convention with α_{o,i}). Evaluating W in G gives (x_1x_0^{-1})(x_2x_1^{-1})···(x_nx_{n-1}^{-1})x_n^{-1}, not the telescoping product x_1·(x_1^{-1}x_2)···(x_{n-1}^{-1}x_n)·x_n^{-1}. The adjacent middle factors x_i^{-1}x_{i+1} are missing; what appears is x_{i+1}x_i^{-1} in the wrong order. For example, in F(a,b) with x0=1, x1=a, x2=ab, the word (with α0 included as required for a closed loop) is a(aba^{-1})(ab)^{-1} = a^2ba^{-1}b^{-1}a^{-1} ≠ 1. Since Definition 2.6 defines Area only for words representing 1 in G, Fill and hence Rδ are undefined exactly for the objects Theorem A quantifies over. The intended fix is plain — take α_i=α(x_i^{-1}x_{i+1}) or reverse the product — but the written statement and the proof's assertion that W is null-homotopic are false as printed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a random Dehn function for a finitely presented group: a random walk (or more general tame Markov chain) is combined with a quasi-geodesic combing to form a loop, and one takes the expected filling area. The main theorems, stated as Theorems A, B, and C, assert that for acylindrically hyperbolic groups with at most polynomial Dehn function, the random Dehn function is bounded above by (n/log n) δ_G(log n) ≼ n^2, and is strictly smaller than the usual Dehn function for non-hyperbolic groups. The proof uses known deviation inequalities for random walks and Markov chains on hyperbolic-like groups, together with a linear-progress lemma, to show that, away from exponentially unlikely events, sample paths stay close to a quasi-geodesic and can be filled by logarithmically short loops. The paper also proves a quasi-isometric invariance statement for tame Markov chains and combings.","tokens_in":10296,"tokens_out":10809,"duration_ms":96677,"significance":"If the indexing issue discussed below is repaired, the main result is a clean confirmation of Gromov's intuition in a random-walk model: generic loops are much cheaper to fill than worst-case loops. The paper is concise, relies on substantial external deviation inequalities rather than developing new probabilistic machinery, and the geometric reduction from probability to filling area is elegant. The quasi-isometric invariance of the random Dehn function for tame Markov chains is also a useful contribution. The result would be a meaningful addition to the literature on generic and average-case filling functions.","major_comments":[{"comment":"In the proof of Theorem 3.4, the step 'Let C1 be the constant from Theorem 3.3 associated to D, which we increase to make sure that n^d C1 n^{1-C1} → 0' is not justified by the quoted form of Theorem 3.3. The constant C1 appears both in the exponential e^{-l/C1} and in the chosen threshold l = C_1^2 log n, and simply increasing C1 does not obviously preserve the deviation inequality with the same l. The argument can be repaired by fixing the C1 supplied by the theorem and taking l = C^2 log n with C chosen so that C^2/C1 > d+1, but this must be written out because the complementary-event term is what makes the unconditional expectation small.","section":"Definitions 2.9 and 2.10"}],"minor_comments":[{"comment":"The area is said to be a 'positive integer' minimum over ℓ, but the empty word has area 0; 'non-negative integer' would be more accurate.","section":"Definition 2.6"},{"comment":"In the second bullet, 'quasi-homogenenous' is a typo for 'quasi-homogeneous'.","section":"Theorem 3.3"},{"comment":"The text writes P[B_n^c] ≤ C_3 n^{-k}, but Lemma 3.2 with r = d+1 gives C_3 n^{-(d+1)}; the exponent k is not defined and should be replaced by d+1.","section":"Proof of Theorem 3.4"},{"comment":"The loop P_{k_i} is bounded to have length at most 300 K C_1^2 C_3 D^2 log n, but two lines later its area is bounded by δ_G(300 K C_1 C_3 D^2 log n); the exponent on C_1 is inconsistent and should be corrected.","section":"Proof of Theorem 3.4"},{"comment":"The definition of β is written as β(x,y)=f(α(f^{-1}(x),f^{-1}(y))), which does not match the combing notation α:G→S^*; it should be stated as a combing β(h)=f(α(f^{-1}(h))), with the quasi-geodesic constants made explicit.","section":"Proposition 2.11"}],"recommendation":"major_revision","confidential_remarks":"The indexing problem in Definitions 2.9–2.10 is serious as written but appears to be a straightforward typo; the rest of the proof strategy is sound. The paper would benefit from stating the exact external deviation inequalities used, especially since the complementary-term estimate depends on the precise form of the constant. The second author is a co-author of some of the cited deviation-inequality papers, but those are separate published theorems and no circularity is apparent. With the indexing correction and a clarified constant choice, the paper should be publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main result: for finitely presented acylindrically hyperbolic groups with polynomial Dehn function, the random Dehn function (expected filling area of a random-walk loop closed by a quasi-geodesic combing) is O(n/log n * delta_G(log n)) ≼ n^2, so strictly below the classical Dehn function when the group is not hyperbolic. This confirms Gromov's intuition in Sisto's model and extends Sisto's relatively-hyperbolic result to a broader class. The proof is a clean combination of deviation inequalities (Mathieu-Sisto, Goldsborough-Sisto, Hagen-Petyt-Russell-Sisto) with a packing argument: split the walk into logarithmically spaced subwalks, use linear progress to force them to advance along a quasi-geodesic, fill the intervening loops with area delta_G(log n), and union-bound the exceptional events. The quasi-isometry invariance statement (Prop 2.11) is a nice addition.\n\nThe paper is honest about what it imports; the citation pattern is fine. The central argument is structurally sound.\n\nThe soft spots are real but fixable. Biggest: Definitions 2.9 and 2.10 do not define a null-homotopic word as printed. With alpha_i = alpha(x_{i+1} x_i^{-1}) and W = alpha_1 ... alpha_{n-1} alpha_n^{-1}, the product evaluates to x_n x_1^{-1} x_n^{-1}, not 1. You need alpha_i = alpha(x_i^{-1} x_{i+1}) and an initial segment from 1 to x_1. As printed, Area is undefined for the very words the theorem quantifies over. The local loop construction in the proof shows the authors knew the right picture, so this is a typo-class error, but a referee must require the correction.\n\nAlso, Subclaim 1 in Theorem 3.4's proof omits a factor D: the conclusion demands ell <= 10 C_1^2 D log n, while the case checks only give ell <= 10 C_1^2 log n. Minor constant slip; the argument can absorb it with slightly adjusted constants.\n\nVerdict: worth a serious referee. The result is solid and the errors are correctable. Not a paradigm shift, but a useful confirmation of Gromov's intuition in a new model. I'd take it after minor revisions. I wouldn't cite it in my own work in the next year, but specialists should know it.","headline":"Solid extension of Sisto's random Dehn function model with a fixable but important indexing flaw in the definition of the loop word.","tokens_in":10873,"tokens_out":9891,"would_cite":false,"duration_ms":79736,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20P05","20F65","20F69","20F67"],"pacs":[],"model":"deepseek-v4-flash","headline":"For acylindrically hyperbolic groups with polynomial Dehn function, the expected filling area of random-walk loops is at most quadratic and strictly below the worst case when the group is non-hyperbolic.","keywords":["random Dehn function","Dehn function","filling invariants","acylindrically hyperbolic groups","random walks","tame Markov chains","deviation inequalities","quasi-geodesic combing"],"falsifier":"Exhibit a finitely presented acylindrically hyperbolic group with quadratic Dehn function and a quasi-geodesic combing for which the expected filling area of a length-$n$ random-walk loop grows faster than $C n \\log n$ for every constant $C$, or for which the deviation probability in Theorem 3.3 fails at the scale $l = C\\log n$; either would disprove the paper's quadratic bound.","tokens_in":9825,"feed_emoji":"","tokens_out":14353,"duration_ms":116211,"temperature":0.7,"pith_summary":"The paper studies the random Dehn function of a finitely presented group: the expected area needed to fill the closed loop obtained by taking the first $n$ steps of a random walk and closing it with a quasi-geodesic path from the endpoint back to the start. Its main theorem says that if the group is acylindrically hyperbolic and its usual worst-case Dehn function is at most a polynomial of degree $d$, then this expected filling area is at most $(n/\\log n)\\,\\delta_G(\\log n)$, and in particular at most quadratic. For non-hyperbolic groups with quadratic or higher polynomial Dehn function, the bound is strictly smaller than the worst-case Dehn function, so typical loops are genuinely easier to fill than the hardest loops. This confirms the long-standing intuition that average-case filling in such groups should be faster than worst-case filling.","feed_headline":"Random-walk loops need only quadratic filling area","feed_subtitle":"In many non-hyperbolic groups, typical loops fill with area n log n, below worst-case n squared.","key_machinery":"The argument is carried by a deviation inequality (Theorem 3.3) imported from prior work on random walks and Markov chains: for each $D>0$ there is a constant $C_1$ such that the probability that any of the first $n$ random-walk positions lies at distance at least $l$ from every $(D,D)$-quasi-geodesic from the starting point to the final position is at most $C_1 n e^{-l/C_1}$. A companion lemma (Lemma 3.2) says that sub-walks of logarithmic length make linear progress in the Cayley graph. Together they partition the time interval into about $n/\\log n$ blocks of length roughly $\\log n$, over which the random walk tracks the closing quasi-geodesic; each block contributes a loop of length $O(\\log n)$, whose filling area is at most $\\delta_G(O(\\log n))$, giving the total bound.","core_discovery":"On its own terms, the paper establishes that for every finitely presented acylindrically hyperbolic group whose Dehn function $\\delta_G$ is bounded above by a polynomial of degree $d \\ge 1$, and for every quasi-geodesic combing $\\alpha$, the random Dehn function satisfies $R_\\delta(G,(Z_n),\\alpha)(n) \\preceq \\frac{n}{\\log n}\\,\\delta_G(\\log n) \\preceq n^2$. Consequently, when the group is hyperbolic the random Dehn function is at most linear; when the group is non-hyperbolic and $\\delta_G$ is quadratic it is at most $n\\log n$; and when the polynomial degree is larger than 2 it is strictly smaller than $\\delta_G$. The same quadratic bound is obtained for tame Markov chains on relatively hyperbolic groups, acylindrically hyperbolic 3-manifold groups, and well-behaved hierarchically hyperbolic groups such as mapping class groups and extra-large Artin groups.","pith_inferences":["The paper leaves implicit that the proof template should apply to any stochastic process with exponential deviation from quasi-geodesics; testing non-backtracking or lazy random walks in the same groups would check whether the $n/\\log n$ factor is universal.","Because the bound is independent of the particular combing up to constants, it suggests the random Dehn function records the geometry of typical geodesics rather than worst-case words, and could distinguish groups with the same worst-case Dehn function.","An analogous block argument may give upper bounds for expected filling areas of random $k$-cycles in higher-dimensional spaces, provided a deviation inequality controls the distance of the process from a quasi-convex model.","The strict gap for non-hyperbolic groups suggests the random Dehn function is a quasi-isometry invariant that could measure how far a group is from being hyperbolic in a way the worst-case Dehn function does not."],"forward_implications":["If the group is hyperbolic, the random Dehn function is at most linear, matching the linear worst-case bound.","If the group is non-hyperbolic with quadratic Dehn function, the expected filling area is at most $n\\log n$, asymptotically smaller than the worst-case $n^2$.","If the polynomial degree is $d>2$, the random Dehn function is strictly smaller than the usual Dehn function, so the random-walk model cannot see the higher-degree worst-case behavior.","The same $n\\log n$ upper bound applies to tame Markov chains on relatively hyperbolic groups, acylindrically hyperbolic 3-manifold groups, and well-behaved hierarchically hyperbolic groups, not just to simple random walks.","In the non-amenable setting, the random Dehn function is a quasi-isometry invariant up to the usual equivalence of asymptotic growth functions."],"supporting_citations":[{"why":"Supplies the deviation inequality for simple random walks on acylindrically hyperbolic groups used as Theorem 3.3.","marker":"[MS20, Theorem 1.1]"},{"why":"Supplies the deviation inequality for tame Markov chains on hyperbolic-like groups used as Theorem 3.3.","marker":"[GS22, Theorem 1.4]"},{"why":"Extends the deviation inequality to tame quasi-homogeneous Markov chains on well-behaved hierarchically hyperbolic groups.","marker":"[GHP+23, Theorem 3]"},{"why":"Is the linear-progress lemma for sub-walks that Lemma 3.2 generalises to processes with exponential decay.","marker":"[Sis17, Lemma 4.5]"},{"why":"Shows that simple random walks are tame quasi-homogeneous Markov chains, connecting the random-walk theorem to the Markov-chain framework.","marker":"[GS22, Lemma 2.8]"},{"why":"Gives the at-most-quadratic Dehn function for the 3-manifold groups covered by Theorem B.","marker":"[ECH+92, Theorem 12.4.7]"},{"why":"Gives the at-most-quadratic Dehn function for hierarchically hyperbolic groups covered by Theorem C.","marker":"[BHS19, Corollary 7.5]"}],"fun_headline_variants":["Random loops fill quadratically, not worse","Typical loops need only quadratic filling area","Random Dehn function beats worst-case bound","Random filling area capped at quadratic for many groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on an imported probability estimate asserting that a random walk stays exponentially close, at every time step, to some straight path from its starting point to its current position, with a constant large enough to absorb the polynomial degree of the Dehn function.","fun_headline_variants_meta":{"raw":{"variants":["Random loops fill quadratically, not worse","Typical loops need only quadratic filling area","Random Dehn function beats worst-case bound","Random filling area capped at quadratic for many groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000157,"raw_usage":{"total_tokens":1152,"prompt_tokens":806,"completion_tokens":346,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":422,"completion_tokens_details":{"reasoning_tokens":289}},"tokens_in":422,"tokens_out":346,"duration_ms":4218,"temperature":1.0,"reasoning_tokens":289,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:14:24.200770+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a finitely presented acylindrically hyperbolic group with quadratic Dehn function and a quasi-geodesic combing for which the expected filling area of a length-$n$ random-walk loop grows faster than $C n \\log n$ for every constant $C$, or for which the deviation probability in Theorem 3.3 fails at the scale $l = C\\log n$; either would disprove the paper's quadratic bound.","supporting_citations":[],"review_version":1}