REVIEW 3 major objections 4 minor 6 references
A unimodular random graph with large upper growth and no growth
T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read The paper constructs, for every $d\ge 3$, a unimodular random graph with maximal degree $d$, upper growth rate $d-1$, and lower growth rate $1$, hence with no ordinary growth rate.
desk verdict A sharp and likely correct counterexample to the AFH growth question, but the proof of the key pruning step is asserted rather than demonstrated. 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 central object is the Cartesian product $T\,\Box\,T$, where $T$ is the canopy tree, the infinite tree whose finite levels branch doubly toward the leaves. Each vertical fiber $T_v$ is cut by an unbounded edge set $J'_v$ into finite clusters, and each cluster is independently assigned one of two types: a path cluster, which is a long path through the cluster, or an exp cluster, which is a high-girth $d$-regular graph. The decisive mechanism is the turnable-edge inequality (3.1): a vertical edge $f^\uparrow(v)$ is turnable when $|t_\Box(u^+)|\le \epsilon(v_+)\log|\mathrm{comp}_{J_{v_+}}(u^+)|$, meaning the set of vertices that can reach the edge's upper endpoint by an upward path is small compared with the logarithm of the cluster size. At lucky vertices, turnable vertical edges are replaced by horizontal edges; the paper argues that every infinite path meets infinitely many lucky vertices, so the same path repeatedly absorbs the subexponential overhead of path clusters and the exponential growth of exp clusters. The cluster choices are local and independent, which preserves unimodularity.
What would settle it
Inspect a single ray in one vertical fiber after the thinning step of Section 3 and test the inequality $|t_\Box(u^+)|\le \epsilon(v_+)\log|\mathrm{comp}_{J_{v_+}}(u^+)|$ edge by edge; if along that ray only finitely many edges satisfy it, the proof's key ratio $L/r$ cannot be made to tend to $0$, so the claimed lower bound $d-1$ on the upper growth rate would not follow from this construction.
Extended reading notes
Core claim
The paper's central claim is Theorem 1: if $d\ge 3$, there exists a unimodular random graph $(U,o)$ with degree bounded above by $d$ whose upper growth rate is almost surely $d-1$ and whose lower growth rate is almost surely $1$. In words, balls of radius $n$ around the root grow like $(d-1)^n$ along selected radii and subexponentially along other radii, so $|B_n(o)|^{1/n}$ has no limit. The proof proceeds in stages: first a degree-$d+2$ construction with upper growth at least $d-1$, then a degree-$d$ construction with upper growth at least $(1-\epsilon)(d-1)$, then the maximal-upper-growth construction that makes the upper growth exactly $d-1$. A separate theorem gives, for every $d\ge 6$, a non-hyperfinite unimodular random graph with no growth rate and upper growth at least $d-4$.
Load-bearing premise
The load-bearing premise is that in every vertical fiber of the product graph one can thin the edge set so that infinitely many edges satisfy the turnable inequality (3.1); the paper supports this with 'Observe that ...' and 'using some local algorithm' rather than a proof of the algorithm or of the Borel-Cantelli step, and if the premise fails along some ray the ratio $L/r$ cannot be driven to $0$, so the upper growth rate claim collapses.
Editorial extensions
If this is right
- For every $d\ge 3$, the largest possible gap between upper and lower growth rate is realized inside degree bound $d$: upper growth $d-1$ and lower growth $1$.
- The growth dichotomy for unimodular random trees cannot be extended to arbitrary unimodular random graphs, even with a uniform degree bound.
- The main example is hyperfinite, so hyperfiniteness does not force the upper growth rate below $d-1$.
- For $d\ge 6$, a non-hyperfinite unimodular random graph with no growth rate and upper growth at least $d-4$ exists, giving a graph-side data point for the growth question on unimodular Riemannian surfaces.
- Every infinite path in a vertical fiber meets infinitely many lucky vertices, so the absence of a growth rate is produced by an infinite sequence of local switches rather than a single rare event.
Reading between the lines
- A natural next step would be to make the sketched local thinning rule explicit as a factor of iid (a measurable function of independent vertex labels); until then, the maximal-upper-growth result rests on an assertion the paper describes only informally.
- The turnable-edge inequality (3.1) has the shape of a cost-benefit comparison, which suggests a structural question the paper leaves open: whether any unimodular random graph with upper growth $d-1$ must contain an infinite family of similarly turnable edges.
- The non-hyperfinite example is made by taking a Cartesian product with a $3$-regular tree, so the same product construction might convert other no-growth examples into non-hyperfinite ones while preserving the absence of a growth rate; the paper only demonstrates this for its own construction.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper constructs unimodular random rooted graphs with bounded degree d ≥ 3 whose upper growth rate is a.s. d−1 and whose lower growth rate is a.s. 1, so that no growth rate exists. This directly answers in the negative an extension question of Abért, Fraczyk and Hayes, who proved a growth dichotomy for unimodular random trees. The construction proceeds in stages: first a graph Γ_d on the canopy tree with degree d+2 and upper growth at least d−1 (Construction 1); then a degree-d graph U_{d,ε} with upper growth arbitrarily close to d−1 (Construction 2); and finally a graph U built on the Cartesian product T□T of two canopy trees, with degree at most d and upper growth exactly d−1 (Theorem 1). A non-hyperfinite example with no growth rate is then obtained by taking a Cartesian product with a 3-regular tree (Theorem 2).
Significance. If the construction is fully valid, Theorem 1 is a strong and surprising counterexample: it shows that the tree-growth dichotomy fails in the most extreme possible way for general unimodular graphs, since the upper growth rate is maximal (d−1) while the growth rate still fails to exist. The paper's strategy is elegant and uses appropriate tools: the canopy tree as a unimodular random graph, high-girth regular graphs from Linial-Simkin, the mass transport principle, and the martingale convergence theorem for Galton-Watson processes. The hyperfinite examples underline the point that cycles can be 'useless' for growth even in an amenable, hyperfinite setting. The non-hyperfinite example is also of independent interest in connection with Abért's conjecture on unimodular surfaces. However, as it stands, a central technical step in the proof of Theorem 1 is only sketched, and the Borel-Cantelli arguments in Construction 2 also need to be made explicit. The main result is therefore not yet fully established, but the approach is promising and the gaps appear local rather than fatal.
major comments (3)
- [§3, Eq. (3.1) and the following paragraph] The central step of the paper is the passage from the arbitrary unbounded edge sets J_v of Construction 2 to smaller unbounded edge sets J'_v with unbounded turnable subsets. The text says 'Observe that ...' and 'using some local algorithm' and then defines J'_v 'by induction on ind(v)'. This is not a proof. Turnability (3.1) of an edge f in T_v is defined through J'_{v+}, and ind(v+)=ind(v)+1; hence the given base case ind(v)=0 does not start an induction unless the sets for larger indices are already defined, and no ordering of the induction is supplied. More importantly, the simultaneous constraints on all fibers are nontrivial: making f turnable by deleting a middle range of indices from J_{v+} can interfere with unboundedness of J'_{v+} or with turnability of edges in T_{v+}. The upper-growth proof for U depends on the existence of infinitely many lucky vertices along the relevant paths---in particular on L/r→0 via (3.1)---so without a rigorous construction of J'_v the maximal upper growth rate d−1 is not established. The authors need to provide an explicit recursive or algorithmic construction and prove that J'_{v+} remains unbounded and J^{turn}_v is unbounded for every v.
- [§2, HGW_K and the Borel-Cantelli paragraph] The claim that |B_r(v, HGW_K)| stochastically dominates the first r generations of a two-type Galton-Watson process and that this yields growth ((1−ε*)(d−1))^r with high probability is only sketched. The subsequent statement that 'a Borel-Cantelli argument works' to produce infinitely many good vertices along the relevant paths needs a precise tail bound, with ε* depending on ε in a controlled way, and an explanation of how independence across the random clusters and the random internal choices yields the almost-sure limsup statement. This is load-bearing for Construction 2's upper growth rate, and it is inherited by the Section 3 construction, which uses W*(J_v).
- [§2, construction of the edge set J] In the construction of J, the assertions that J is unbounded a.s., satisfies property (1) (at most one J-edge incident to a vertex), and satisfies property (2) for every cluster are compressed into 'It is easy to see...' and 'we may use a Borel-Cantelli argument'. A complete proof is needed because property (2) controls |out(K)|, which is used both for the lower-growth estimate and for the choice of ext(K), and property (1) is part of the definition of W*(J). In particular, the distribution of the number of selected representatives per equivalence class and the resulting bound |out(K)| ≤ ε/2 |V(K)| should be written out explicitly.
minor comments (4)
- [§2, Construction 2, definition of new*(K)] The phrase 'define new*(K) as HK if the type of K is exp' appears to be a typo: the preceding sentences define HGW_K by deleting edges incident to out(K)∪ext(K), and the following estimates concern HGW_K. It should say 'define new*(K) as HGW_K if the type of K is exp'.
- [§3, paragraph after the definition of lucky vertices] The symbol λ(v) is used without definition in 'conditioned on λ(v) ∈ J^turn_v'; it presumably means the vertical edge f↑(v), but this should be stated.
- [§3, notation for vertices of T□T] The notation 'For (v,u)=v' overloads the symbol v, since v is already used for the base vertex of the fiber T_v and for an arbitrary vertex of T□T. Using a different letter, such as x=(v,u), would improve readability.
- [§1, Lemma 3 and surrounding text] There are minor grammatical issues, e.g., 'any two path Q1, Q2' should be 'any two paths Q1, Q2'. Also, the inequality 'leaf(K) ≤ 2/3 |V(K)|' is stated without justification; it is straightforward from the canopy structure but should be given for completeness.
Circularity Check
No circularity; the construction is self-contained even though Section 3 contains a sketched technical pruning step that is a proof gap rather than a circular step.
full rationale
The derivation chain is self-contained and does not reduce to its inputs by construction. Construction 1 uses the canopy tree, independent Bernoulli cluster types, and high-girth regular graphs from Linial--Simkin; the lower-growth estimate is a direct ball count with Lemma 3 and the cluster-size bounds, while the upper-growth estimate uses girth-regular balls. Construction 2 replaces I by a random unbounded J, and the Galton-Watson domination argument uses the martingale convergence theorem and a Borel-Cantelli step. Section 3's turnable-edge pruning is asserted via 'using some local algorithm' and is under-proved, but it is not circular: inequality (3.1) is a new sufficient condition used to control L/r, not a restatement of the desired upper growth rate d-1, and the lucky-vertex probability is derived from independent ext(K) choices rather than fitted to the target growth rate. The self-citation [T] is used only as prior context for the no-growth phenomenon and is not load-bearing; no external uniqueness theorem is imported from same-author work, and no prediction is fitted to data. The main correctness risk is the missing proof of the local pruning algorithm, which is a gap in the construction, not a circularity.
Assumptions & free parameters
free parameters (2)
- Cluster level sequence l_i =
any sequence with l_{i+1} - l_i >= 2^{2 l_i + 2} log_2(d-1)
- Epsilon in Construction 2 =
any value in (0, 1/(d(d-1)^2))
assumptions (5)
- standard math For fixed d, for every n there exists a d-regular graph on n vertices with girth at least (1 - o(1)) log_{d-1} n.
- standard math The canopy tree with root at generation k with probability 2^{-k} is a unimodular random graph.
- domain assumption A random graph obtained from a unimodular random graph by local finite rewiring is unimodular.
- standard math In a supercritical Galton-Watson process with mean mu, the normalized generation size Z_n / mu^n converges almost surely to a positive limit.
- domain assumption The Cartesian product of two unimodular random graphs with the product root measure is unimodular.
Cite this review
Pith. "Pith review of A unimodular random graph with large upper growth and no growth." pith.science (2026). https://pith.science/paper/UUAUNZ4I
@misc{pith2026241118465,
author = {Pith},
title = {Pith review of: A unimodular random graph with large upper growth and no growth},
year = {2026},
howpublished = {\url{https://pith.science/paper/UUAUNZ4I}},
note = {Machine review of arXiv:2411.18465}
}
abstract
We construct a unimodular random rooted graph with maximal degree $d\geq 3$ and upper growth rate $d-1$, which does not have a growth rate. Ab\'ert, Fraczyk and Hayes showed that for a unimodular random tree, if the upper growth rate is at least $\sqrt{d-1}$, then the growth rate exists, and asked with some scepticism if this may hold for more general graphs. Our construction shows that the answer is negative. We also provide a non-hyperfinite example of a unimodular random graph with no growth rate. This may be of interest in light of a conjecture of Ab\'ert that unimodular Riemannian surfaces of bounded negative curvature always have growth.
Reference graph
Works this paper leans on
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[1]
Growth dichotomy for unimodular random rooted trees
M. Abért, M. Fraczyk, B. Hayes. Growth dichotomy for unimodular random rooted trees. (2023) To appear in the Annals of Probability. https://arxiv.org/pdf/2312.04611
work page Pith review arXiv 2023
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[4]
R. Lyons and Y. Peres. Probability on trees and networks . Cambridge University Press, 2016. Available at http://mypage.iu.edu/ rdlyons
work page 2016
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[6]
Á. Timár. A stationary random graph of no growth rate Annales de l'I.H.P. Probabilités et statistiques 50.4 (2014), 1161-1164
work page 2014
Reviewed August 12, 2026 · model on record in the stance chip above.
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