{"id":"37e69463-86ea-4046-8221-9d45ea1736f5","arxiv_id":"2411.18465","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For every d at least 3, there exists a unimodular random graph of maximum degree d whose upper growth rate is d minus 1 and whose lower growth rate is 1, so it has no growth rate.","lead":"This paper constructs random infinite graphs with bounded degree where the number of vertices inside a ball grows at the fastest possible exponential rate along some radii and at a much slower rate along others, so the graph has no single growth rate. It answers an open question of Abert, Fraczyk and Hayes by showing their growth dichotomy for random trees cannot extend to general unimodular random graphs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3's central pruning step—producing unbounded J'_v with unbounded turnable subsets—is asserted but not proved; the upper-growth claim d−1 rests on it.","rationale":"The reader's conditional verdict correctly identifies the Section 3 sketch as the load-bearing weakness. I read the rest of the paper in good faith: Construction 1 appears coherent, and Construction 2 is plausible, with its random subset selection of J and the high-girth replacement being checkable in outline. However, Theorem 1, the strongest claim, depends on the existence of the pruned families J'_v with unbounded turnable subsets. This is asserted rather than proved, and the dependency of turnability at level v on the choice at level v+ makes the claimed 'local algorithm' nontrivial: one needs a simultaneous construction over all fibers that preserves unboundedness at every level while creating infinitely many turnable edges. The proof of the upper growth rate d−1 uses this precisely to force L/r tending to 0, so without a complete proof of the pruning step the central claim is not established. This is a rigor gap rather than a demonstrated contradiction, so conditional acceptance remains the appropriate disposition.","tokens_in":8823,"tokens_out":18069,"duration_ms":183481,"concrete_test":"Write out the claimed local algorithm on finite truncations: for each N let T^(N) be the set of vertices with ind(v) at most N plus their parents, and define J_v^N for all v in T^(N). Verify that the extension from N to N+1 produces, for every v with ind(v) at most N, at least one edge in J_v^{N+1} that is turnable with respect to J_{v+}^{N+1} and whose index is at least N. Concretely, for a fixed edge f in J_v at level l_m, compute the interval [ind(f), N(f)] that must be deleted from J_{v+}; check whether deleting all such intervals demanded by the two children of v+ still leaves J_{v+} with infinitely many edges and still permits an edge of J_{v+} to be turnable with respect to J_{v++}. If no uniform extension exists, Theorem 1 is not proved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The construction of U in Section 3 relies on the unproved assertion following (3.1): one can replace each J_v by an unbounded subset J'_v so that J^turn_v, the set of edges turnable with respect to J'_{v+}, is itself unbounded. Turnability is not a per-edge local deletion: pruning J_{v+} to create a gap around one edge f works for a single f, but doing this simultaneously for all v and all required edges must leave J'_{v+} unbounded and must also leave J'_{v+} with infinitely many turnable edges with respect to J'_{v++}. The text says only 'using some local algorithm' and 'by induction on ind(v)' without specifying the induction order; since turnability of level v depends on level v+ (higher index), the stated base case ind(v)=0 does not obviously start an induction upward. If the pruning cannot be carried out, lucky vertices need not occur infinitely often, and the key estimate L/r tending to 0 in the upper-growth proof fails; the maximal upper growth rate d−1 is then not established. The Borel-Cantelli step for the high-probability ball lower bounds in Construction 2 is also only sketched, but the turnable-edge construction is the more basic gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":9039,"tokens_out":12425,"duration_ms":110063,"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":[{"comment":"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.","section":"§3, Eq. (3.1) and the following paragraph"},{"comment":"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).","section":"§2, HGW_K and the Borel-Cantelli paragraph"},{"comment":"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.","section":"§2, construction of the edge set J"}],"minor_comments":[{"comment":"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'.","section":"§2, Construction 2, definition of new*(K)"},{"comment":"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.","section":"§3, paragraph after the definition of lucky vertices"},{"comment":"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.","section":"§3, notation for vertices of T□T"},{"comment":"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.","section":"§1, Lemma 3 and surrounding text"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a good question and the proposed construction is clever. However, the key pruning lemma in Section 3 is not proved, and the Borel-Cantelli step in Section 2 is also only sketched. These are not mere presentation issues: without a rigorous construction of the J'_v, the maximal upper growth rate d−1 in Theorem 1 is not established. I would be supportive of a major revision that supplies the missing construction and proofs. I see no reason to doubt the overall strategy, but the manuscript in its current form is not yet complete."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know that this paper is one of the genuinely important preprints in this area in the last couple of years: if the main construction goes through, it gives the sharpest possible negative answer to the Abert-Fraczyk-Hayes growth dichotomy question, with a unimodular random graph of degree d, upper growth d-1, and lower growth 1. The non-hyperfinite example in Section 4 is a nice bonus. The core ideas—splicing high-girth regular graphs into the canopy tree at chosen scales, then using a Cartesian product with the canopy tree to hide cycles—are clever and mostly well executed.\n\nConstruction 1 is solid. Lemma 3 is the right tool, and the growth estimates are checkable. Construction 2 adds the ext(K) set and a Galton-Watson domination argument; that is plausible, but the Borel-Cantelli step is only sketched. The real issue is Section 3. Theorem 1 rests on the claim that one can choose unbounded subsets J'_v of the J_v so that each fiber has infinitely many turnable edges. The text says 'using some local algorithm' and 'by induction on ind(v)', but no algorithm or induction is actually given. The stress-test note is right that turnability of f in T_v depends on J'_{v+}, so defining J'_v by induction starting at ind(v)=0 seems backwards: to make a leaf-level edge turnable you need to prune the parent's fiber, which is not yet defined. There may be a way to do it by working from high indices downward, but the infinite canopy tree has no top, so you would need a different argument. This is a real gap, not a minor exposition issue. The paper itself calls Section 3 a sketch, which is honest, but the main theorem depends on it.\n\nThat said, nothing here looks wrong enough to suggest the result is false. The gap is plausibly fillable. The paper deserves a serious referee; it should not be desk-rejected. But the referee should demand a complete proof of Section 3 before publication. If you work on unimodular random graphs or growth of random graphs, this is worth reading. I would not cite Theorem 1 in its current form, but I would keep an eye on the revision.\n\nRecommendation: send to peer review with the expectation of major revision.","headline":"A sharp and likely correct counterexample to the AFH growth question, but the proof of the key pruning step is asserted rather than demonstrated.","tokens_in":9619,"tokens_out":5398,"would_cite":false,"duration_ms":44890,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C80","60J80","60G42"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["unimodular random graph","growth rate","upper growth rate","lower growth rate","canopy tree","Cartesian product","hyperfinite"],"falsifier":"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.","tokens_in":8583,"feed_emoji":"📈","tokens_out":13398,"duration_ms":108272,"temperature":0.7,"pith_summary":"Unimodular random graphs are random rooted graphs satisfying the mass-transport principle, the probabilistic analogue of vertex-transitivity; they arise as limits of finite graphs. The paper constructs, for every degree bound $d\\ge 3$, a unimodular random graph whose upper growth rate is the maximum possible value $d-1$ while its lower growth rate is $1$, so the limit that would define an ordinary growth rate does not exist. This refutes the natural extension of a growth dichotomy already proved for unimodular random trees, and it does so with the largest possible upper growth. The same construction is adapted to give a non-hyperfinite unimodular random graph with no growth rate, providing a graph-theoretic data point for the question of whether unimodular Riemannian surfaces of bounded negative curvature always have growth.","feed_headline":"Degree-d random graphs can have maximal growth, no growth rate","feed_subtitle":"Upper growth d-1 with lower growth 1 means no limiting growth rate, killing the tree-only extension.","key_machinery":"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.","core_discovery":"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 $d-4$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Proved the growth dichotomy for unimodular random rooted trees whose extension to general graphs is the question answered negatively.","marker":"[AFH]"},{"why":"Supplied the earlier counterexample of a random tree with no growth rate, the baseline the new construction builds on.","marker":"[T]"},{"why":"Gives the high-girth $d$-regular graphs used to build the exp clusters with near-maximal ball growth.","marker":"[LS]"},{"why":"Provides the Galton-Watson martingale convergence theorem used to estimate ball growth inside exp clusters.","marker":"[LP16]"},{"why":"Gives the definition of unimodular random graphs and the mass-transport principle used to verify unimodularity of the constructions.","marker":"[AL]"}],"fun_headline_variants":["Unimodular graph: max upper growth, no growth rate","Upper growth d-1, lower one: no growth rate for these graphs","Ball growth oscillates: upper d-1, lower 1, no limit rate","Graph has top growth but no growth rate, new construction"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Unimodular graph: max upper growth, no growth rate","Upper growth d-1, lower one: no growth rate for these graphs","Ball growth oscillates: upper d-1, lower 1, no limit rate","Graph has top growth but no growth rate, new construction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0009,"raw_usage":{"total_tokens":3838,"prompt_tokens":871,"completion_tokens":2967,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":2888}},"tokens_in":487,"tokens_out":2967,"duration_ms":19805,"temperature":1.0,"reasoning_tokens":2888,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:12:08.251025+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}