REVIEW 1 major objections 5 minor 17 references
Random Dehn function of groups
T0 review · 1 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict Solid extension of Sisto's random Dehn function model with a fixable but important indexing flaw in the definition of the loop word. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (1)
- [Definitions 2.9 and 2.10] 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.
minor comments (5)
- [Definition 2.6] 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.
- [Theorem 3.3] In the second bullet, 'quasi-homogenenous' is a typo for 'quasi-homogeneous'.
- [Proof of Theorem 3.4] 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.
- [Proof of Theorem 3.4] 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.
- [Proposition 2.11] 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.
Circularity Check
No circularity: the bound is derived from external deviation inequalities and the usual Dehn function; no fitted input is relabeled as a prediction.
full rationale
The derivation chain is: define the random Dehn function (Definition 2.10); import deviation inequalities from Mathieu--Sisto [MS20, Theorem 1.1], Goldsborough--Sisto [GS22, Theorem 1.4], and Goldsborough--Hagen--Petyt--Russell--Sisto [GHP+23, Theorem 3] (Theorem 3.3); use these to show that, with high probability, every random-walk sample point lies within O(log n) of a quasi-geodesic, so the filling can be assembled from O(n/log n) subloops of length O(log n), each with area at most delta_G(O(log n)); finally combine this with delta_G(n) <= n^d. No parameter is fitted to R_delta, and R_delta is not used as an input to its own bound. The cited deviation inequalities are prior theorems with independent proofs; the fact that the second author is a coauthor of [GS22] and [GHP+23] does not make the dependency circular, because the present paper does not derive the needed inequality from the claim it is proving. A separate correctness concern, not a circularity concern, is that the indexing alpha_i = alpha(x_{i+1} x_i^{-1}) in Definitions 2.9 and 2.10 appears to make the word W non-null-homotopic as printed, which would leave Fill and R_delta undefined; fixing this by reversing the increment to x_i^{-1} x_{i+1} does not change the logical structure of the proof.
Assumptions & free parameters
assumptions (3)
- domain assumption Deviation inequality for random walks and tame Markov chains on the listed groups (Theorem 3.3), from [MS20, Theorem 1.1], [GS22, Theorem 1.4], and [GHP+23, Theorem 3].
- domain assumption Tame Markov chains on non-amenable groups make linear progress with exponential decay (Definition 3.1 and Lemma 3.2), generalizing [Sis17, Lemma 4.5].
- domain assumption The groups in Theorems A-C satisfy the hypotheses needed for Theorem 3.3 and have at most polynomial Dehn function (e.g., hierarchically hyperbolic groups have quadratic Dehn function by [BHS19, Corollary 7.5]).
Cite this review
Pith. "Pith review of Random Dehn function of groups." pith.science (2026). https://pith.science/paper/BLOXVD4F
@misc{pith2026241112715,
author = {Pith},
title = {Pith review of: Random Dehn function of groups},
year = {2026},
howpublished = {\url{https://pith.science/paper/BLOXVD4F}},
note = {Machine review of arXiv:2411.12715}
}
read the original abstract
In this note, we study the notion of random Dehn function and compute an asymptotic upper bound for finitely presented acylindrically hyperbolic groups whose Dehn function is at most polynomial. By showing that in these cases, if the group is not hyperbolic, then the random Dehn function is strictly smaller than the usual Dehn function we confirm Gromov's intuition albeit in a different model. In fact, we show that in these cases the random Dehn function is at most quadratic.
Reference graph
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Reviewed August 12, 2026 · model on record in the stance chip above.
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