On the exponential growth rates of lattice animals and interfaces

Agelos Georgakopoulos; Christoforos Panagiotis

REVIEW 3 major objections 4 minor 30 references

On the exponential growth rates of lattice animals and interfaces

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A universal inequality turns percolation thresholds into lower bounds on cluster growth rates.

desk verdict Genuinely new results on interface growth rates, but the advertised 'dictionary' inequality (2) is not proven as written—it depends on a false claim about exponential decay at pc. read the letter →

arxiv 1908.03426 v2 pith:64KDVVQS submitted 2019-08-09 math.PR math-phmath.COmath.MP

Agelos Georgakopoulos , Christoforos Panagiotis This is my paper · ORCID

classification math.PRmath-phmath.COmath.MP MSC 60K3505A1682B43

keywords percolationthresholdlatticeanimalsinterfacesexponentialgrowthratesBernoulliclustersizedistributiondualityCheegerconstant

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Lattice animals are the connected finite subgraphs that statistical physics counts; interfaces are a thin sub-species, the outermost layer separating a cluster from infinity. This paper proves that these two counting problems are quantitatively tied to Bernoulli percolation through one inequality: $b(G)\ge f(r(p_c(G)))$ with $f(r)=(1+r)^{1+r}/r^r$ and $r(p)=(1-p)/p$, where $b(G)$ is the exponential growth rate of interfaces. The matching Theorem 1.2 says that the interface-size distribution $P_p(|S_o|=n)$ fails to decay exponentially exactly when $b_{r(p)}(G)=f(r(p))$, giving a precise boundary between exponential and non-exponential decay. Since lattice animals are the coarsest case of the same interface construction, the same inequality recovers and refines the known animal-counting results, and for triangle-generated lattices it yields the duality $b_r=(b_{1/r})^r$. If correct, any improved upper bound on a percolation threshold automatically improves a lower bound on animal growth rates, and vice versa.

What carries the argument

The load-bearing object is $b_r(G)$, the exponential growth rate of interfaces of size $n$ with boundary size close to $rn$, together with the universal comparison function $f(r)=\frac{(1+r)^{1+r}}{r^r}$. Interfaces are defined from a chosen basis of the cycle space; at the coarse extreme they are all lattice animals, and at the fine extreme they are thin layers around clusters. The engine of the proof is the first-moment estimate $p(1-p)^r\le 1/b_r(G)$, obtained by counting occurring interfaces through the unique-interface theorem and bounding this count by a quasi-geodesic argument, together with a large-deviation lemma that most occurring interfaces have surface-to-volume ratio near $r(p)=(1-p)/p$. These two ingredients produce the threshold theorem, and the triangle-basis duality comes from a box-gluing construction that shows interfaces with fractal shapes do not dominate the count $b_r$.

What would settle it

Enumerate occurring interfaces in one percolation instance on the square lattice and look for two distinct outermost boundary layers that meet at a vertex; the paper's Lemma 3.2 forbids this, and finding one would falsify the uniqueness theorem on which the counting estimate rests. Alternatively, search numerically for an instance with more than a constant times $n$ occurring interfaces of size $n$ — the bound $N_n\le l n+1$ would fail, and the chain from counting to Theorem 1.2 would need to be re-examined.

Watch

Extended reading notes

Core claim

The central discovery is a two-way dictionary between percolation and enumeration. For every graph in the paper's class $\mathcal{S}$, the exponential growth rate $b_r(G)$ of interfaces with surface-to-volume ratio $r$ satisfies $b_{r(p)}(G)\le f(r(p))$ for every $p\in(0,1)$, and equality holds precisely when $P_p(|S_o|=n)$ does not decay exponentially. This equivalence is what converts an upper bound on the percolation threshold $p_c(G)$ into a lower bound on $b(G)$, hence on the lattice-animal growth rate $a(G)$, and conversely. The paper further proves that for any basis of the cycle space made of bounded cycles, interfaces are exponentially rarer than lattice animals, so the resulting inequality is strict: $a(G)>f(r(p_c(G)))$. It also establishes, for lattices whose cycle space has a triangle basis, the self-duality $b_r=(b_{1/r})^r$ of the growth-rate function, and, as a by-product, the continuity of the exponential decay rate of the cluster size distribution on $(0,1)$.

Load-bearing premise

The load-bearing assumption is geometric — every finite cluster separating the origin from infinity has exactly one outermost boundary layer, and at most a linear number of such layers of size $n$ can touch a fixed path from the origin — and if either half fails, the first-moment estimate $p(1-p)^r\le 1/b_r$ collapses with the whole dictionary.

Editorial extensions

If this is right

Any upper bound on the percolation threshold of a lattice in $\mathcal{S}$ automatically becomes a lower bound on the exponential growth rate of its lattice animals, and any such lower bound becomes a threshold upper bound.
For every interface basis made of bounded cycles, $b(G)<a(G)$, so the translated inequality is strict: $a(G)>f(r(p_c(G)))$.
The interface-size distribution $P_p(|S_o|=n)$ decays exponentially exactly when $b_{r(p)}(G)<f(r(p))$; in particular it also fails to decay at $p=1-p_c$ for the triangle-generated lattices.
On triangle-generated lattices, $b_r=(b_{1/r})^r$ for every $r>0$, so computing the branch $r<1$ determines the branch $r>1$ and vice versa.
The exponential decay rate of the cluster size distribution, $c(p)=\lim_n P_p(|C_o|=n)^{1/n}$, is a continuous function of $p$ on $(0,1)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension is to check the dictionary on all 1-ended vertex-transitive graphs; the paper's own proof suggests the large-deviation lemma is the part that would need to be rebuilt in that generality.
The interface constant $I(G)$ opens a route to threshold bounds on amenable graphs, where the classical Cheeger constant vanishes but $I(G)$ can remain positive; the paper proves the one-sided bound $p_c\le 1/(I(G)+1)$ and does not explore whether that bound can be sharp.
The continuity of the decay exponent $c(p)$ suggests that other percolation observables near criticality, such as finite-size scaling exponents, might inherit Lipschitz regularity; the paper proves continuity only for $c(p)$ and its interface analogue.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth your time if you work on percolation thresholds or enumeration of lattice animals, but treat inequality (2) with suspicion. The paper has three genuinely new pieces: the duality b_r = (b_{1/r})^r for triangulated lattices, the strict inequality b(G) < a(G) between interface and animal growth rates, and an interface Cheeger constant I(G) giving p_c <= 1/(I(G)+1), sometimes beating the Benjamini–Schramm bound. Those arguments look coherent and do not rely on the questionable part.

The soft spot is exactly the central formula (2), the promised translation between upper bounds on p_c and lower bounds on a(G). The only justification in the text is the caption of Figure 1, which attributes to Kesten–Zhang [23] the claim that P_p(|S_o|=n) fails to decay exponentially for p in [1-p_c, p_c]. That is false when p_c > 1/2: for p in (1-p_c, p_c) with p < p_c, Aizenman–Barsky gives exponential decay for the cluster size, and because the interface is a subgraph of the cluster (|S_o| <= |C_o|), the interface size distribution decays exponentially too. Kesten–Zhang's paper is about supercritical percolation and does not prove failure at p_c. No alternative proof of non-exponential decay at p_c is supplied. That failure is exactly what would turn Proposition 4.4/4.5 into equality at r(p_c) and yield (2). So as written the dictionary formula is unsupported. It may well be repairable—non-exponential decay at criticality is likely provable for the graphs in S via divergence of susceptibility—but that work is not here.

Also, the abstract overreaches: it says the paper improves the best known asymptotic bounds on a(Z^d), while Section 1.2 correctly defers those improvements to the follow-up [15]. That is a real mismatch between abstract and content.

What does hold up? The continuity of the decay exponent c(p), the log-concavity and continuity of b_r, the duality theorems, and the strict inequalities b < a and p_c < 1/(h+1). The proof of Lemma 5.5 is intricate; I did not find a clear hole, but it deserves close referee scrutiny. The dependence on the authors' own interface machinery from [14] is heavy but legitimate—Theorem 3.5 is restated from that work and is not obviously wrong.

Recommendation: send to a serious referee, but flag the status of (2) and the Kesten–Zhang citation explicitly. If the critical-point non-decay can be supplied, this is a strong paper; as it stands, the advertised dictionary formula is not proven.

Referee Report

3 major / 4 minor

Summary. The paper develops a percolation-based method for bounding the exponential growth rates of lattice animals and of a subclass called interfaces. It introduces growth rates b_r(G) of interfaces with a fixed surface-to-volume ratio r, proves the universal upper bound b_r(G) ≤ f(r) with f(r)=(1+r)^{1+r}/r^r, and proves an equivalence (Theorem 1.2) between exponential decay of the interface size distribution at parameter p and the strictness of the inequality b_{r(p)}<f(r(p)). It then derives duality relations b_r=(b_{1/r})^r for triangulated lattices, continuity and log-concavity of b_r, an interface analogue of the Cheeger constant giving new upper bounds on p_c, a strict inequality between interface and animal growth rates, and an appendix result on continuity of the cluster-size decay exponent. The abstract and introduction also claim improved asymptotic bounds on a(Z^d) as d→∞, but Section 1.2 states that those bounds are obtained in the follow-up paper [15].

Significance. If the main dictionary formula (2) were established, the paper would provide a genuinely new two-way bridge between percolation thresholds and enumerative growth constants, with the universal function f(r) playing a parameter-free role. The duality formula b_r=(b_{1/r})^r, the continuity and log-concavity results for b_r, and the strict comparison b(G)<a(G) are interesting and potentially influential. The paper is also commendably explicit about the dependence on imported results, especially the unique-interface theorem from [14]. However, the central inequality (2) is currently supported by a claimed interval of non-decay that is inconsistent with Aizenman–Barsky, so the main advertised conclusion is not established as written.

major comments (3)

[§4, Figure 1 caption; Theorem 1.2; Eq. (2)] The derivation of inequality (2), b(G) ≥ f(r(p_c(G))), rests on the claim, made only in the caption of Figure 1, that exponential decay of P_p(|S_o|=n) fails for all p in [1-p_c, p_c] and that this follows from Kesten–Zhang [23]. This interval claim is false when p_c>1/2. Indeed, by Theorem 3.5 the interface is a subgraph of the cluster, P⊂E(C), so |S_o|≤|C_o|. For p∈(1-p_c,p_c) with p<p_c, Aizenman–Barsky [1] gives exponential decay of P_p(|C_o|=n), hence P_p(|S_o|=n)≤P_p(|C_o|≥n) decays exponentially. By the 'only if' direction of Theorem 1.2 this forces b_{r(p)}(G)<f(r(p)) on that interval, contradicting the plateau claimed in Figure 1. Since the equality b_{r(p_c)}=f(r(p_c)) is needed for (2), and no alternative proof of non-decay at p=p_c is supplied in the paper, the central formula (2) is unsupported as written. The authors should either prove non-decay at p_c directly or explicitly restrict and re-derive all claims that depend on it.
[Abstract and §1.2] The abstract states that the paper improves the best known asymptotic bounds on a(Z^d) as d→∞, but Section 1.2 says these bounds are obtained in the follow-up paper [15] and the present paper only sets up the machinery. This is a mismatch between the advertised contribution and the actual content. The abstract and introduction should be rewritten so that the claims made for this paper are limited to the results proved here, with the asymptotic bounds clearly attributed to [15].
[§8, Theorem 8.1 vs §1.3, Theorem 1.1] Theorem 1.1 is stated as b(G)<a(G) for every G∈S, but the proof in Section 8 is carried out only for site-interfaces and site-animals, giving ˙b(G)<˙a(G) under additional assumptions (5) and (14). The manuscript does not explain how the site result implies the bond-interface statement b(G)<a(G), nor does it define clearly whether Theorem 1.1 is meant in the bond or site sense. This leaves a gap between the theorem as stated and the proof as written; the statement and proof should be reconciled, or the theorem should be reformulated for the site objects actually treated.

minor comments (4)

[Throughout] There are numerous typographical and grammatical slips, including 'in this vain' for 'in this vein', missing spaces such as 'LetSo', and inconsistent use of superscripts in the definitions of b°_r and b⊙_r. A careful proofreading pass is needed.
[§4, Definition 4.1] The notation b°_r and b⊙_r is introduced with a degree symbol and a circled dot, but the superscripts are dropped immediately afterward; the reader must infer which variant is meant. Please keep the notation explicit for at least the statements of Lemmas 4.2 and Proposition 4.6.
[§7, Theorem 7.1] The proof invokes the fact that θ is not analytic at p_c and refers to [14, Corollary 4.14] for analyticity of the inclusion-exclusion expansion, but it does not give a reference or argument for non-analyticity of θ at p_c. A citation or a short explanation would help.
[§2.4 and §3] The paper assumes without comment that quasi-transitive planar lattices are 2-connected after a vertex-deletion operation, and it states that this operation preserves p_c. This is plausible but should be justified or referenced, since the class S is used throughout.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation of b(G) >= f(r(pc(G))) uses first-moment counting, external critical non-decay, and a prior interface-uniqueness theorem; no fitted constant or target inequality is re-used as an input.

full rationale

The derivation chain is not circular. Proposition 4.4 bounds E_p(N_n) by counting occurring interfaces and using the disjointness bound N_n <= l n + 1 (Eq. 9); this is a first-moment counting argument, not a re-use of the target inequality. Theorem 1.2 / Proposition 4.6 is a large-deviation equivalence: equality b_{r(p)}(G) = f(r(p)) is obtained by comparing the exponential growth of interface counts at volume-to-surface ratio r(p) with the weighted probability p^n(1-p)^{|∂P|}; the non-decay conclusion is derived, not assumed. Inequality (2) is then obtained by combining Theorem 1.2 with the critical non-decay assertion in the Figure 1 caption, attributed to Kesten & Zhang [23]. Whether [23] supports the claimed p-interval is a correctness concern (the caption's interval appears to conflict with Aizenman-Barsky exponential decay for subcritical p), but that is not circularity because the cited result is external to this paper's fitted or derived quantities. The only in-house dependency is Theorem 3.5, the unique-interface theorem imported from the authors' earlier work [14]; it is load-bearing (it justifies P ⊂ E(C), the disjointness behind (9), and the definition of S_o), but it is a prior mathematical theorem with stated assumptions that do not include the paper's target inequality, and it is not a fitted or constructed surrogate for b(G) >= f(r(pc(G))). Under the independence rule, that citation is real evidence rather than a circular input. No fitted parameters are renamed as predictions, no uniqueness theorem is used to force the target formula by definition, and no equation is shown to reduce to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results rest on the interface theory imported from [14] and on geometric properties of the class S. No complex constants are fitted; the universal function f(r) is derived. The paper is self-contained except for prior theorems that are cited as background.

assumptions (4)

domain assumption Unique interface theorem (Theorem 3.5, restated from [14, Theorem 10.4]): every finite cluster separating the origin from infinity contains exactly one occurring interface.
This is the backbone of all interface counting arguments; it is cited from the authors' prior companion paper and not proved in the current manuscript.
domain assumption Quasi-geodesic intersection bound (Eq. (9)): for graphs in S, the number N_n of occurring interfaces of size n satisfies N_n <= l n + 1 for a fixed constant l.
Used in Proposition 4.4 and Lemma 4.7 to obtain first-moment bounds; it relies on the geometric properties of the lattice class S.
standard math Hardy-Ramanujan partition estimate (Theorem 2.1).
Used in Lemma 4.2 to show that the number of component-size partitions of a multi-interface is subexponential.
standard math Aizenman-Barsky exponential decay for p < pc.
Invoked in the introduction and Figure 1 to place the subcritical branch of br below f(r); it is not part of the paper's new derivations.

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Pith. "Pith review of On the exponential growth rates of lattice animals and interfaces." pith.science (2026). https://pith.science/paper/64KDVVQS

@misc{pith2026190803426,
  author       = {Pith},
  title        = {Pith review of: On the exponential growth rates of lattice animals and interfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/64KDVVQS}},
  note         = {Machine review of arXiv:1908.03426}
}

abstract

We introduce a formula for translating any upper bound on the percolation threshold of a lattice \g into a lower bound on the exponential growth rate of lattice animals $a(G)$ and vice-versa. We exploit this to improve on the best known asymptotic bounds on $a(\mathbb{Z}^d)$ as $d\to \infty$. Our formula remains valid if instead of lattice animals we enumerate certain sub-species called interfaces. Enumerating interfaces leads to functional duality formulas that are tightly connected to percolation and are not valid for lattice animals, as well as to strict inequalities for the percolation threshold. Incidentally, we prove that the rate of the exponential decay of the cluster size distribution of Bernoulli percolation is a continuous function of $p\in (0,1)$.

Figures

Figures reproduced from arXiv: 1908.03426 by the authors.

**Figure 1.** An approximate, conjectural, visualisation of br(G) when G is a lattice in R d , d ≥ 3. The graph of br(G) (depicted in bold ink) lies below the graph of f(r) := (1+r) 1+r rr (depicted in blue, if colour is shown). The fact that f(r) plots almost like a straight line can be seen by rewriting it as (1+r)(1+1/r) r . The fact that br = f(r) for r in the interval (r(1−pc), r(pc)], where r(p) := 1−p p , follows by combin… view at source ↗

**Figure 2.** An example of a multi-interface M, comprising two nested interfaces P1, P2. We depict M with bold lines, and ∂M := ∂P1 ∪ ∂P2 with dashed lines (green, if colour is shown). The edges not participating in M are depicted in plain lines (blue, if colour is shown). In the case where G ∈ S is a graph the cycle space of which is generated by its triangles, (8) site-interfaces admit an equivalent definition that is more st… view at source ↗

**Figure 3.** The grid Bn and the interfaces when d = 2 in the proof of Proposition 5.3. 4)2d 2 possible shapes for interfaces of size n. On the other hand, there are exponentially many interfaces in In,r,, so we can choose n large enough to ensure that there is a non-empty set K ⊆ In,r, of cardinality at least N := cn,r,/p(n) consisting of interfaces P with |P| = n and (r −)n ≤ |∂P| ≤ (r +)n that have the same shape. We now… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The interface Q in the proof of Proposition 5.3. For d > 2, since Q is a connected graph, there is an interface associated to it. We will verify that Q coincides with its interface. Let B denote the minimal edge cut of Q. Consider an interface Ki and the smallest box B…

**Figure 5.** Figure 5: If the vertex incident to the two dashed lines is attached to the site-interface, the vertices of which are depicted with big disks, then the new graph is not a siteinterface any more. Lemma 5.4. Let G be a graph satisfying (3) for some c > 0, d ≥ 2. Assume that C(G) …

**Figure 6.** Figure 6: The situation in the proof of Lemma 5.4. We can now conclude that P v∈∂V P |Qv| ≤ |∂P| ≤ ∆|P|, where ∆ is the maximal degree of G. It follows that the number of v ∈ ∂ V P such that |Qv| ≥ |P| 3/4 is at most ∆|P| 1/4 . By the isoperimetric inequality (3) there is c > 0 …

**Figure 7.** Figure 7: The vertices of Q are depicted with big disks, and the vertices of ∂Q are depicted with smaller disks. The edges spanned by P and C are depicted in solid lines, while the edges of Π are depicted in dashed lines. It is not hard to see that there is a cycle C at bounded …

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [23]

Kesten and Y

H. Kesten and Y. Zhang. The probability of a large ﬁnite cluster in super- critical Bernoulli percolation. Annals of Probability, 18(2):537–555, 1990

work page 1990

[2] [15]

Georgakopoulos and C

A. Georgakopoulos and C. Panagiotis. On the exponential growth rates of lattice animals and interfaces II: new asymptotic bounds. Preprint 2021

work page 2021

[3] [14]

Georgakopoulos and C

A. Georgakopoulos and C. Panagiotis. Analyticity results in Bernoulli Per- colation. To appear in Memoirs of the AMS

work page

[4] [1]

Aizenman and D

M. Aizenman and D. J. Barsky. Sharpness of the phase transition in per- colation models. Communications in Mathematical Physics , 108:489–526, 1987

work page 1987

[5] [2]

Aizenman, F

M. Aizenman, F. Delyon, and B. Souillard. Lower bounds on the cluster size distribution. Journal of Statistical Physics , 23(3):267–280, 1980

work page 1980

[6] [3]

Bandyopadhyay, J

A. Bandyopadhyay, J. Steif, and ´A. Tim´ ar. On the cluster size distribution for percolation on some general graphs. Revista Matem´ atica Iberoameri- cana, 26(2):529–550, 2010

work page 2010

[7] [4]

Barequet, R

G. Barequet, R. Barequet, and G. Rote. Formulae and growth rates of high-dimensional polycubes. Combinatorica, 30(3):257–275, 2010

work page 2010

[8] [5]

Barequet, G

G. Barequet, G. Rote, and M. Shalah. An improved upper bound on the growth constant of polyiamonds. Acta Mathematica Universitatis Comeni- anae, 88(3):429–436, 2019. 36

work page 2019