1.1. Lattice animals

A lattice animal is a connected subgraph $S$ of the hypercubic lattice ${\mathbb{Z}}^d$ or more generally of a vertex-transitive graph $G$ . The counts of lattice animals with prescribed parameters have been extensively studied by scholars in statistical mechanics as well as combinatorics and computer science [Reference Barequet, Barequet and Rote4, Reference Barequet and Shalah6, Reference Delyon11, Reference Gaunt and Peard15, Reference Hammond19, Reference Harris22, Reference Miranda and Slade36–Reference Peard and Gaunt38], both in ${\mathbb{Z}}^d$ and other lattices [Reference Barequet, Rote and Shalah5, Reference Barequet, Shalah and Zheng7, Reference Rands and Welsh39]. A lot of the motivation comes from the study of random configurations in ${\mathbb{Z}}^d$ , the central theme in many models of statistical mechanics. Our focus is Bernoulli (bond or site) percolation. Given a value of the percolation parameter $p\in [0,1]$ , we can express the probability $\theta (p)$ that the cluster $C_o$ of the origin $o$ is infinite as

(1) \begin{equation} 1-\theta (p) = \sum \nolimits_{A \text{ is a lattice animal containing } o}{\mathbb{P}}_p(C_o=A). \end{equation}

The probability ${\mathbb{P}}_p(C_o=A)$ is easily expressed as $p^{||A||} (1-p)^{|\partial A|}$ by the definition of bond percolation, where $||A||$ is the number of edges in $A$ , and the boundary $\partial A$ comprises the edges outside $A$ that are incident with $A$ . Thus, if we could enumerate the set of lattice animals $A_{n,m}$ with size $n$ and boundary size $m$ accurately enough, we could answer any question of percolation theory such as the continuity of $\theta (p)$ or the exact value of the critical threshold $p_c$ . In practice, however, $A_{n,m}$ is too difficult to enumerate, and one studies asymptotics as $n,m\to \infty$ . Such asymptotics, specifically the exponential growth rates, are still informative enough about the behaviour of percolation, and the present paper provides results in this vein.

1.2. Interfaces

Certain subfamilies of lattice animals, called interfaces, have also been extensively studied either as a tool in the study of statistical mechanics or for their own sake [Reference Cerf10, Reference Dobrushin, Kotecky and Shlosman12]. The term interface is commonly used to denote the common boundary of two components of a crystal or liquid that are in a different phase. The precise meaning of the term varies according to the model in question and the perspective of its study. When studying percolation on a planar lattice $L$ for example, the interface of the cluster $C_o$ of the origin can be thought of as the minimal cut separating $C_o$ from infinity, which forms a cycle in the dual $L^*$ . Peierls’ argument [Reference Grimmett18] is a famous application of the notion that uses an upper bound on the number of such cycles to deduce an upper bound on $p_c(L)$ .

In [Reference Georgakopoulos and Panagiotis17], we introduced a variant of the notion of interface – see Definitions 3.3 and 3.6 – and used it to prove the analyticity of the percolation density $\theta (p)$ for supercritical Bernoulli percolation on ${\mathbb{Z}}^d$ . Our proof relied on the fact that our interfaces satisfy a generalisation of (1). Therefore, as argued above for lattice animals, counting interfaces accurately enough would yield important conclusions in percolation theory. This observation is the main motivation of this paper, which studies the exponential growth rate $b(G)$ of interfaces of a ‘lattice’ $G$ . Our first result is the following inequality relating $b(G)$ to ${p}_c(G)$ :

(2) \begin{equation} {b(G) \geq f(r({p}_c(G)))}, \end{equation}

where $f(r)\,:\!= \frac{(1+r)^{1+r}}{r^r}$ and $r(p)\,:\!=\frac{1-p}{p}$ are universal functions. The class $\mathcal S$ of lattices $G$ we work with includes the standard cubic lattice in ${\mathbb{Z}}^d, d\geq 2$ , as well as all doubly periodic planar lattices, which include any regular tessellation of the Euclidean plane; see Section 3 for the definition of $\mathcal{S}$ .

The fact that interfaces are a subspecies of lattice animals leads to the inequality $a(G)\geq b(G)$ . Inequality (2) allows us to translate 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 in both directions. We improve on the best known asymptotic lower and upper bounds on $a({\mathbb{Z}}^d)$ as $d\to \infty$ , answering a question of Barequet, Barequet and Rote [Reference Barequet, Barequet and Rote4]. We use percolation as a tool to obtain the latter, and conversely, we use the former to obtain lower bounds on $p_c({\mathbb{Z}}^d)$ .

1.3. Lattice animals vs. interfaces

Inequality (2) remains true if we replace $b(G)$ by the exponential growth rate $a(G)$ of lattice animals, and all ingredients for its proof are available in [Reference Hammond19]. In this paper, we provide a unified, and simpler, approach to proving such inequalities. Our definition of interface is parametrised by a choice of a basis $\mathcal{P}$ of the cycle space ${\mathcal{C}}(G)$ of the lattice $G$ . We elaborate on this notion in Sections 3 and 4. By varying the choice of $\mathcal{P}$ , we obtain a spectrum of notions of $\mathcal{P}$ interface, which are always relevant to percolation: we showed in [Reference Georgakopoulos and Panagiotis17, Theorem 10.4] (restated as Theorem 3.7 below) that every finite connected subgraph of $G$ , for example a percolation cluster, contains a unique $\mathcal{P}$ interface. One extreme of this spectrum is where $\mathcal{P}$ is just the set of all cycles of $G$ , in which case the $\mathcal{P}$ interfaces coincide with the lattice animals. The other extreme is where $\mathcal{P}$ is a minimal basis, in which case the $\mathcal{P}$ interface of a percolation cluster is a thin layer incident with its boundary to infinity. For example, when $G$ is the square lattice, and $\mathcal{P}$ comprises its squares of length $4$ , then the (unique) $\mathcal{P}$ interface contained in a finite connected subgraph $C$ consists of the vertices and edges at the boundary of the unbounded face $F$ of $C$ in the plane. Here, we think of $C$ as a plane graph inheriting its embedding into ${\mathbb{R}}^2$ from $G$ , and the unbounded face of $C$ is its unique face of infinite diameter. Some examples of such interfaces are depicted in Figure 2. The main message of this paper is that this second extreme of $\mathcal{P}$ interfaces provides finer information about percolation. In particular, we will prove (Theorem 10.1) that for every basis $\mathcal{P}$ comprising cycles of bounded length the following holds:

We remark that using this theorem and inequality (2), we obtain the strict inequality

\begin{equation*}{a(G) \gt f(r({p}_c(G)))}.\end{equation*}

1.4. Growth rates parametrised by ‘volume-to-surface ratio’

Before stating our other results, we need to introduce more terminology. Recall that when considering counts $A_{n,m}$ of lattice animals, it was important to parametrise them both by their size $n$ and their boundary size $m$ . Alternatively, instead of $m$ we could use the ‘volume-to-surface ratio’ $n/m$ . Similarly, to establish (2), we consider the exponential growth rate $b_r=b_r(G)$ of the number of interfaces of $G$ with size $n$ and volume-to-surface ratio approximating $r$ , as a function of $r\in{\mathbb{R}}_+$ . By ‘volume’ here we mean the number of edges contained in an interface $P$ , and by ‘surface’ we mean the cardinality of a set $\partial P$ of edges that are incident with $P$ and are ‘accessible’ from infinity. (See Definition 3.6 for details.) In the example of Figure 2, the edges in $\partial P$ are depicted by dashed lines.

We consider this function $b_r(G)$ to be of independent interest; in fact, most of this paper revolves around it. In particular, we prove that $b_r(G)$ is always continuous (Theorem 6.4) and log-concave (Theorem 6.3).

One of the best known results of percolation theory is the exponential decay, as $n\to \infty$ , of the cluster size distribution ${\mathbb{P}}_p(|C_o|= n)$ for $p$ in the subcritical interval $[0,p_c)$ [Reference Aizenman and Barsky1]. In the supercritical case $p\in (p_c,1)$ , this exponential decay holds for some, but not all, lattices and values of $p$ [Reference Aizenman, Delyon and Souillard2, Reference Hermon and Hutchcroft24].

Letting $S_o \subseteq C_o$ denote the interface of $C_o$ , we can analogously ask for which $p\in (0,1)$ we have exponential decay of the probability ${\mathbb{P}}_p(|S_o|= n)$ . We prove that this is uniquely determined by the value $b_{r(p)}$ , where $r(p)\,:\!=\frac{1-p}{p}$ is a bijection between the parameter spaces of edge density $p$ and volume-to-surface ratio $r$ (we consider bond percolation in this section, but we will also discuss site percolation later on). More concretely, we observe that, firstly, $b_{r(p)}(G) \leq f(r(p))$ holds for every lattice $G$ and every $p\in (0,1)$ , where $f(r)$ is the aforementioned universal function (Proposition 4.4), and secondly, ${\mathbb{P}}_p(|S_o|= n)$ decays exponentially in $n$ for exactly those values of $p$ for which this inequality is strict:

Incidentally, we prove in the Appendix that the rate of the exponential decay of ${\mathbb{P}}_p(|C_o|= n)$ , defined as $c(p)\,:\!= \lim _n \left ({\mathbb{P}}_p(|C_o|= n)\right )^{1/n}$ (see [Reference Bandyopadhyay, Steif and Timár3, Reference Grimmett18] for a proof that the limit exists for every $p\in (0,1)$ ), is a continuous function of $p$ (Theorem 12.1). Our proof boils down to elementary calculations not involving our notion of interface. Another contribution of this paper is the subexponential decay at $1-p_c$ for triangulated lattices in ${\mathbb{R}}^d$ (Corollary 5.7).

We expect our results to hold for more general vertex-transitive 1-ended graphs, but decided to restrict our attention to $\mathcal{S}$ to avoid technicalities that would add little to the understanding of the matter.

It is interesting that Theorem 1.2 holds for every choice of basis $\mathcal{P}$ with respect to which our interfaces are defined. In particular, since lattice animals are a special case of $\mathcal{P}$ interfaces as mentioned above, we can replace the interface size distribution ${\mathbb{P}}_p( |S_o|=n )$ by the cluster size distribution ${\mathbb{P}}_p( |C_o|=n )$ and $b_{r(p)}(G)$ by its analogue $a_{r(p)}(G)$ counting lattice animals. This form of Theorem 1.2 was proved by Hammond [Reference Hammond19] building on a result of Delyon [Reference Delyon11].

Our proof of Theorem 1.2 is based on a large deviation principle for interfaces (Lemma 4.7) that may be of independent interest. Roughly speaking, the latter result says that for any $p\in (0,1)$ , most occurring interfaces have a volume-to-surface ratio close to the value $r(p)$ .

We mentioned above that the most refined extreme of such results is obtained when $\mathcal{P}$ is a minimal basis. This is even more so for ‘triangulated’ lattices, that is lattices having a basis $\mathcal{P}$ consisting of triangles. For such a lattice, we show that the following holds.

In other words, the values of $b_r$ for $r\lt 1$ determine those for $r\gt 1$ (Theorems 5.1 and 5.2). This is the technically most involved result of this paper. It shows that considering interfaces rather than lattice animals yields a differently behaved function $b_r$ , namely one with no intersection with $f(r)$ not only for $r\gt r(p_c)$ but also for $r\lt 1/r(p_c)$ . Most of our knowledge about $b_r$ is summarised in Figure 1.

Figure 1.

An approximate, conjectural, visualisation of $b_r(G)$ when $G$ is a lattice in ${\mathbb{R}}^d, d\geq 3$ . The graph of $b_r(G)$ (depicted in bold ink) lies below the graph of $f(r)\,:\!= \frac{(1+r)^{1+r}}{r^r}$ (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 $b_r= f(r)$ for $r$ in the interval $(r(1-p_c), r(p_c)]$ , where $r(p)\,:\!= \frac{1-p}{p}$ , follows by combining a theorem of Kesten & Zhang [Reference Kesten and Zhang29], saying that exponential decay of ${\mathbb{P}}_p(|S_o|= n)$ fails in that interval, with our Theorem 1.2. That $b_r\lt f(r)$ for $r\gt r(p_c)$ follows from the well-known exponential decay of ${\mathbb{P}}_p(|C_o|= n)$ for $p\lt p_c$ [Reference Aizenman and Barsky1]. We also know that $b_r$ is continuous and log-concave. The continuity of $b_r$ , combined with Theorem 1.2 again, implies failure of the exponential decay at $p=1-p_c$ (Corollary 5.7), which was not obtained in [Reference Kesten and Zhang29]. If the cycle space of $G$ is generated by its triangles, then Theorem 1.3 determines the subcritical branch $r\gt r(p_c)$ given the branch $r\lt r(1-p_c)$ and vice versa. For the planar triangular lattice, the picture degenerates as $p_c=1-p_c=1/2$ , and so $b_r= f(r)$ for $r=r(1/2)=1$ only. Note that $b_r(G)$ is an invariant of $G$ defined without reference to any random experiment. The connection to percolation is established by Theorem 1.2 via the above transformation $r(p)$ . Since $r(p)$ is monotone decreasing in $p$ , the right-hand side of Figure 1 corresponds to the subcritical percolation regime, and the left-hand side to the supercritical. Using the transformation $r\to \frac 1{r}$ (from volume-to-surface into surface-to-volume ratio), we could reverse the picture to have the ‘subcritical’ interval on the left. For ‘triangulated’ lattices, the picture would look exactly the same due to Theorem 1.3, only the positions of $r(p_c)$ and $r(1-p_c)$ would be interchanged.

Amusingly, our universal function $f(r)$ also satisfies the equation of Theorem 1.3, that is $f(r)= f(1/r)^r$ .

1.5. Bounds on lattice animals

Obtaining reasonable bounds on the count $a_n({\mathbb{Z}}^d)$ of $d$ -dimensional lattice animals of size $n$ containing the origin is very difficult to come by even in 2 dimensions, and so the mainstream focuses on their exponential growth rates $a({\mathbb{Z}}^d)\,:\!=\lim _{n\to \infty }{a_n({\mathbb{Z}}^d)}^{1/n}$ . These have important interactions with statistical mechanics models such as percolation theory, the present paper being an instance of this interaction. Some precise asymptotic expansions for $a({\mathbb{Z}}^d)$ and its site counterpart $\dot{a}({\mathbb{Z}}^d)$ were reported in the physics literature [Reference Gaunt and Peard15, Reference Harris22, Reference Peard and Gaunt38] but without any rigorous bounds on the error terms. Miranda and Slade [Reference Miranda and Slade36, Reference Miranda and Slade37] determined the first three terms of the $1/d$ expansion of $a({\mathbb{Z}}^d)$ rigorously.

Much less is known about its site variant $\dot{a}({\mathbb{Z}}^d)$ . Barequet, Barequet, and Rote in [Reference Barequet, Barequet and Rote4] proved that $\dot{a}({\mathbb{Z}}^d)=2de-o(d)$ . Peard and Gaunt had previously made involved, but nonrigorous, calculations that yield $\dot{a}({\mathbb{Z}}^d)=2de-3e+O(1/d)$ [Reference Peard and Gaunt38, (2.22)] and [Reference Barequet, Barequet and Rote4] expressed the belief that this is correct. Our first result is that this prediction is indeed right as a lower bound (Theorem 7.5). We deduce this from a recent bound of Heydenreich and Matzke [Reference Heydenreich and Matzke25] on the site percolation threshold ${\dot{p}_c}({\mathbb{Z}}^d)$ , obtained using an involved technique called lace expansion. (The dot in ${\dot{p}_c}({\mathbb{Z}}^d),\dot{a}({\mathbb{Z}}^d)$ etc. means that we are considering site percolation, or lattice site animals; most of our results have a bond and a site version.) To do so, we exploit the aforementioned formula (2).

The aforementioned upper bound of [Reference Barequet, Barequet and Rote4] was improved to $\dot{a}({\mathbb{Z}}^d)\leq 2de-2e+1/(2d-2)$ in simultaneous work by Barequet and Shalah [Reference Barequet and Shalah6]. We improve this further asymptotically to $\dot{a}({\mathbb{Z}}^d)\leq 2de-5e/2+O(1/\log\! (d))$ (Theorem 8.1). For this, we use direct combinatorial arguments that do not involve percolation. We can then plug this bound into (the site version of) (2) to obtain the lower bound ${\dot{p}_c}({\mathbb{Z}}^d)\geq \frac{1}{2d}+\frac{2}{(2d)^2}-O(1/d^2\log\! (d))$ (Theorem 8.4). This bound was improved by Heydenreich and Matzke [Reference Heydenreich and Matzke25] shortly after the first draft of our work appeared, see (17).

In this paper, we used percolation as a tool to bound $\dot{a}({\mathbb{Z}}^d)$ from above. Another method was introduced by Eden [Reference Eden13] using more direct counting arguments. This method was enhanced by Klarner and Rivest [Reference Klarner and Rivest31] in the case of ${\mathbb{Z}}^2$ , who obtained that $\dot{a}({\mathbb{Z}}^2)\leq 4.6496$ , and more recently by Barequet and Shalah [Reference Barequet and Shalah6], who obtained the asymptotic inequality $\dot{a}({\mathbb{Z}}^d)\leq 2de-2e+1/(2d-2)$ . In dimension $3$ , the same paper proves $\dot{a}({\mathbb{Z}}^3)\lt 9.3835$ . Plugging this into (2), we deduce $\dot{p}_c({\mathbb{Z}}^3)\gt 0.2522$ , which is the best rigorous lower bound known.

1.6. Improvements for non-amenable graphs

A well-known theorem of Benjamini & Schramm [Reference Benjamini and Schramm8] states that $p_c(G)\leq \frac{1}{h(G)+1}$ , where $h(G)$ denotes the Cheeger constant. We show that this inequality is in fact strict, that is

\begin{equation*} p_c(G)\lt \frac{1}{h(G)+1},\end{equation*}

when $G$ is $1$ -ended, has bounded degrees, and its cycle space admits a basis consisting of cycles of bounded length (Theorem 10.2). Moreover, in Section 9 we define a variant $I(G)$ of the Cheeger constant by considering interfaces rather than arbitrary finite subgraphs of $G$ . We obtain the strengthening $p_c(G)\leq \frac{1}{I(G)+1}$ of the aforementioned theorem (Theorem 9.2), which again has a site and a bond version. We remark that, unlike $h(G)$ , our $I(G)$ can be positive even for amenable graphs. When $G$ is the planar square lattice for example, it is not hard to see that $I(G)=1/2$ in the bond case, which yields the Peierls bound $p_c\leq 2/3$ . Moreover, one can have $I(G)\gt h(G)$ even in the non-amenable case: this turns out to be the case for regular triangulations and quadrangulations of the hyperbolic plane as proved in a companion paper [Reference Haslegrave and Panagiotis23].

1.7. Structure

This paper is structured as follows. Section 2 contains some standard definitions and results that will be used in several places later on. Section 3 recalls the notion of an interface and some of its properties obtained in [Reference Georgakopoulos and Panagiotis17]. Section 4 introduces $b_r$ and presents the proof of the inequality $b_r\leq f(r)$ and Theorem 1.2. Sections 5 and 6 contain a proof of Theorem 1.3 and the fact that $b_r$ is well-behaved, namely continuous and log-concave. In Sections 7 and 8, we obtain the bounds on lattice animals and the lower bound on $\dot{p}_c({\mathbb{Z}}^3)$ . The last two sections are devoted to the new bounds on $p_c$ for non-amenable graphs and to the proof of Theorem 1.1. The Appendix contains a proof that the rate of the exponential decay of the cluster size distribution of Bernoulli percolation is a continuous function of $p\in (0,1)$ . An analogous result is proved for interfaces as well.