2. The even and odd polymer models

Let $\mathcal {E} \subset V(Q_d)$ be the set of even vertices of the hypercube, those whose coordinates sum to an even number, and let $\mathcal {O} \subset V(Q_d)$ be the odd vertices. Note that $Q_d$ is a bipartite graph with bipartition $(\mathcal {E}, \mathcal {O})$ and that $|\mathcal {E}| = | \mathcal {O} | =N:=2^{d-1}$ . A key insight of [Reference Jenssen and Perkins15] is that, for $\lambda$ not too small, the hard-core measure $\mu_{Q_d, \lambda}$ can be closely approximated by a random perturbation of a random subset of either $\mathcal {O}$ or $\mathcal {E}$ . The random perturbation takes the form of a polymer model with convergent cluster expansion, two notions that we introduce now.

For a set $S \subseteq \mathcal {O}$ (and analogously for $S \subseteq \mathcal {E}$ ), let $|S|$ denote the number of vertices of S, N(S) be the set of neighbours of S and $[S]= \{ v \in \mathcal {O} \,{:}\, N(v) \subseteq N(S) \}$ the bipartite closure of S. We call a set $S\subseteq \mathcal {O}$ an odd polymer if (i) the subgraph of $Q_d$ induced by the vertex set $S\cup N(S)$ is connected and (ii) $ |[ S]| \le N/2$ . We let $\mathcal {P}_\mathcal {O}$ denote the set of all odd polymers. The weight of an odd polymer S is

(7) \begin{align}w(S)=\frac{\lambda^{|S|}}{(1+\lambda)^{|N(S)|}}\, .\end{align}

We say that two odd polymers $S_1$ and $S_2$ are compatible and write $S_1\sim S_2$ , if the graph distance between $S_1, S_2$ is $Q_d$ is $>2$ . We let $\Omega_\mathcal {O}$ denote the set of all collections of mutually compatible odd polymers and define the following Gibbs measure on $\Omega_\mathcal {O}$ : for $\Gamma\in \Omega_\mathcal {O}$ ,

\begin{align*} \nu_\mathcal {O}(\Gamma)=\frac{\prod_{S\in \Gamma}w(S)}{\Xi_\mathcal {O}}\, \quad \text{where} \quad \Xi_{\mathcal {O}}= \sum_{\Gamma\in \Omega_{\mathcal {O}}} \prod_{S\in \Gamma}w(S) \end{align*}

is the odd polymer model partition function. Using $\nu_\mathcal {O}$ we define a measure $\mu_{\mathcal {O}, \lambda}$ on independent sets in $Q_d$ .

Let $Z_{\mathcal {O}}(\lambda) = (1+\lambda)^N \Xi_{\mathcal {O}}$ , the independence polynomial of $Q_d$ restricted to independent sets achievable in the odd polymer model; that is, those for which $\mu_{\mathcal {O}, \lambda}$ assigns positive probability.

We think of Step 1 in Definition 8 as a perturbation of the ‘ground state’ measure that simply selects a p-random subset of $\mathcal {E}$ with $p=\lambda/(1+\lambda)$ . The polymer configuration chosen in Step 1 will be typically small and so this process typically returns an independent set that is highly unbalanced with the majority of vertices even. We define even polymers, the even polymer model partition function $\Xi_\mathcal {E}$ , and measures $\nu_\mathcal {E}$ , $\mu_{\mathcal {E}, \lambda}$ analogously. It was shown in [Reference Jenssen and Perkins15] that for $\lambda>C\log d/d^{1/3}$ , the hard-core measure $\mu_{Q_d, \lambda}$ can be closely approximated by the mixture $\tfrac{1}{2}\mu_{\mathcal {O}, \lambda} + \tfrac{1}{2}\mu_{\mathcal {E}, \lambda}$ .

Theorem 9 ([Reference Jenssen and Perkins15]) For $\lambda\ge C \log d/d^{1/3}$ , we have

(8) \begin{align}\left| \log Z(\lambda) - \log \left[ 2 Z_{\mathcal {O}} (\lambda)\right ] \right | = O\left( \exp(\!-N/d^4) \right ) \,.\end{align}

Moreover, letting $\hat\mu_{\lambda}=\tfrac{1}{2}\mu_{\mathcal {O}, \lambda} + \tfrac{1}{2}\mu_{\mathcal {E}, \lambda}$ , we have

\begin{align*}\|\mu_{\lambda} - \hat\mu_{\lambda} \|_{TV} =O\left(\exp (\!-N/d^4)\right) \,.\end{align*}

Finally, with probability at least $1- O(\exp(\!-N/d^4))$ any defect vertices of I drawn from $\mu_{\mathcal {O}, \lambda} $ are on the odd side of the bipartition; that is, the defects are the polymers of the polymer configuration.

The lower bound on $\lambda$ in Theorem 9 is an artefact of Sapozhenko’s graph container method as implemented by Galvin [Reference Galvin10]. Theorem 9 quite possibly remains true for $\lambda=\tilde\Omega(1/d)$ though proving this would require significant new ideas.

The power of Theorem 9 stems from the fact that the even and odd polymer models admit convergent cluster expansions allowing for a detailed understanding of the measures $\mu_{\mathcal {O}, \lambda}, \mu_{\mathcal {E}, \lambda}$ . Let us now introduce the cluster expansion formally.

For a tuple $\Gamma$ of odd polymers, the incompatibility graph, $H(\Gamma)$ , is the graph with vertex set $\Gamma$ and an edge between any two incompatible polymers. An odd cluster $\Gamma$ is an ordered tuple of even polymers so that $H(\Gamma)$ is connected. The size of a cluster $\Gamma$ is $\|\Gamma \| = \sum_{S \in \Gamma} |S|$ . Let $\mathcal {C}$ be the set of all odd clusters. For a cluster $\Gamma$ we define

\begin{align*}w(\Gamma)&= \phi (H(\Gamma)) \prod_{S \in \Gamma} w(S) \,,\end{align*}

where $\phi(H)$ is the Ursell function of a graph H, defined by

(9) \begin{align}\phi(H) &= \frac{1}{|V(H)|!} \sum_{\substack{A \subseteq E(H)\\ \text{spanning, connected}}} (\!-1)^{|A|} \, .\end{align}

The cluster expansion is the formal infinite series

(10) \begin{align}\log \Xi_\mathcal {O}= \sum_{\Gamma\in \mathcal {C}} w(\Gamma)\, .\end{align}

We define the cluster expansion of $\log \Xi_\mathcal {E}$ analogously and note that by symmetry the expansions are identical.

In light of Theorem 9, throughout this section will we assume that $ \lambda\ge C \log d/d^{1/3}$ . We will also assume that $\lambda=O(1)$ as $d\to\infty$ . The following result from [Reference Jenssen and Perkins15] shows that for such $\lambda$ the cluster expansion converges, we have good tail bounds on the expansion, and the terms of the cluster expansion can be efficiently computed. We say that a polynomial is computable in time t, if its coefficients can be computed in time t.

Theorem 10 For fixed $k \ge 1$ ,

\begin{align*} \sum_{\substack{\Gamma \in \mathcal{C} \\ \|\Gamma\| \geq k}}|w(\Gamma)|= O \left ( \frac{2^d d^{2(k-1)}}{(1+ \lambda)^{dk} }\right )\end{align*}

and

(11) \begin{align} \sum_{\substack{\Gamma \in \mathcal{C} \\ \|\Gamma\|= k}} w(\Gamma) = N\cdot R_k(\lambda,d) (1+\lambda)^{-kd}\end{align}

where $R_k$ is a polynomial in d and $\lambda$ of degree at most 2k in d and of degree at most $3k^2$ in $\lambda$ . Moreover $R_k$ is computable in time $e^{O(k \log k)}$ . In particular,

\begin{align*}\log \Xi_\mathcal {O}&= N \sum_{j=1}^k R_j(\lambda,d) (1+\lambda)^{-jd} + O \left ( \frac{2^d d^{2k}}{(1+ \lambda)^{d(k+1)} }\right ) \, .\end{align*}

We note that Theorem 3 follows in [Reference Jenssen and Perkins15] from Theorems 9 and 10.

It will also be useful to record the following slightly strengthened tail bound on the cluster expansion. The following lemma is essentially contained in [Reference Jenssen and Perkins15], though it does not appear explicitly and so we provide the details.

Lemma 11 For $t, \ell\geq 1$ fixed,

\begin{align*} \sum_{\substack{\Gamma \in \mathcal{C} \\ \|\Gamma\| \geq t}}|w(\Gamma)| \|\Gamma\|^\ell = O \left ( \frac{2^d d^{2(t-1)}}{(1+ \lambda)^{dt} }\right )\, .\end{align*}

Proof. In [Reference Jenssen and Perkins15] (see Lemma 15) it is shown that

\begin{align*}\sum_{\Gamma \in \mathcal {C}} |w(\Gamma)| e^{\gamma(d,\|\Gamma\|)} & \le 2^{d-1} d^{-3/2} \,,\end{align*}

where

\begin{align*}\gamma(d,k) &= \begin{cases}\log(1+\lambda) (dk - 3 k^2)- 7 k\log d \text{ if } k \le \dfrac{d}{10} \\ \\[-7pt] \dfrac{d \log(1+\lambda) k}{20} \text{ if } \dfrac{d}{10} < k \le d^4\\ \\[-7pt] \dfrac{k}{d^{3/2}} \text{ if } k > d^4 \,.\end{cases}\end{align*}

Since $e^{\gamma(d,k)/2}\geq k^\ell$ for all k and d sufficiently large, it follows that

\begin{align*}\sum_{\Gamma \in \mathcal {C}} |w(\Gamma)|\|\Gamma\|^\ell e^{\gamma(d,\|\Gamma\|)/2} & =O( 2^{d} d^{-3/2}) \, .\end{align*}

Keeping only terms in the above inequality corresponding to clusters of size at least k we have

(12) \begin{align} \sum_{\substack{\Gamma \in \mathcal{C} \\ \|\Gamma\| \geq k}} |w(\Gamma)|\|\Gamma\|^\ell & \le O(2^{d} d^{-3/2}e^{-\gamma(d,k)/2}) \,.\end{align}

With $t\geq0$ fixed we have by Theorem 10 and (12) that

\begin{align*}\sum_{\substack{\Gamma \in \mathcal{C} \\ \|\Gamma\| \geq t}} |w(\Gamma)|\|\Gamma\|^\ell &= \sum_{\substack{\Gamma \in \mathcal{C} \\ t\leq\|\Gamma\| < 3t}} |w(\Gamma)|\|\Gamma\|^\ell + \sum_{\substack{\Gamma \in \mathcal{C} \\ \|\Gamma\| \geq 3t}} |w(\Gamma)|\|\Gamma\|^\ell \\[3pt] &= O \left ( \frac{2^d d^{2(t-1)}}{(1+ \lambda)^{dt} }\right ) + O\left(\frac{2^{d}d^{11t}}{(1+\lambda)^{3dt/2}}\right)\\[3pt] &= O \left ( \frac{2^d d^{2(t-1)}}{(1+ \lambda)^{dt} }\right ) \, .\end{align*}

Let $\boldsymbol\Gamma$ be a collection of compatible polymers sampled according to $\nu_{\mathcal {O}}$ (the polymer measure at Step 1 of Definition 8, the definition of $\mu_{\mathcal {O}, \lambda}$ ). We will use the above lemma to show that $\|\boldsymbol\Gamma\|$ and $|N(\boldsymbol\Gamma)|$ obey a central limit theorem. Formally, we say a sequence of random variables $(X_d)$ obeys a central limit theorem if $(X_d - \mathbb{E} X_d)/\sqrt{\text{var}(X_d)}$ converges to N(0,1) in distribution as $d\to\infty$ . To prove this central limit theorem we will make use of a connection between the cluster expansion and cumulant generating functions.

Recall the cumulant generating function of a random variable X, is $h_t(X) = \log \mathbb{E} e^{tX}$ . The $\ell$ th cumulant of X is defined by taking derivatives of $h_t(X)$ and evaluating at 0:

\begin{align*}\kappa_\ell(X) &= \frac{ \partial^\ell h_t(X)}{\partial t^\ell} \Bigg|_{t=0} \,.\end{align*}

In particular, $\kappa_1(X)=\mathbb{E} (X)$ and $\kappa_2(X)=\text{var}(X)$ . The cumulants of $|N(\mathbf\Gamma)|$ can be expressed in terms of the cluster expansion as follows. Consider the odd polymer model with modified weights $w_t(S)=w(S)e^{t|N(S)|}$ for $t>0$ and let $\Xi_t$ denote the corresponding partition function. We then have

\begin{align*}h_t(|N(\boldsymbol\Gamma)|)= \log \Xi_t - \log \Xi_{\mathcal {O}}\, .\end{align*}

Applying the cluster expansion to $\log \Xi_t$ , taking derivatives, and evaluating at $t=0$ shows that

(13) \begin{align}\kappa_\ell(|N(\boldsymbol\Gamma)|)= \sum_{\Gamma\in\mathcal {C}}w(\Gamma)|N( \Gamma)|^{\ell}\, .\end{align}

Similarly $\kappa_\ell(\|\boldsymbol\Gamma\|)= \sum_{\Gamma\in\mathcal {C}}w(\Gamma)\|\Gamma\|^{\ell}$ .

Proof. We show that $|N(\boldsymbol\Gamma)|$ obeys a central limit theorem and the proof for $\|\boldsymbol\Gamma\|$ is identical. Let $Z=(|N(\boldsymbol\Gamma)| - \mathbb{E} |N(\boldsymbol\Gamma)|)/\sqrt{\text{var}(|N(\boldsymbol\Gamma)|)}$ . To show that Z converges to N(0,1) in distribution, it suffices to show that the cumulants of Z converge to the cumulants of a standard normal i.e. it suffices to show that $\kappa_\ell(Z)\to 0$ for $\ell \geq 3$ . Now by (13) and Lemma 11,

\begin{align*}\kappa_\ell(|N(\boldsymbol\Gamma)|)= \sum_{\Gamma\in \mathcal {C}}w(\Gamma)|N(\Gamma)|^{\ell}\leq d^\ell \sum_{\Gamma\in \mathcal {C}}w(\Gamma)\|\Gamma\|^{\ell}= O \left ( \frac{2^d d^\ell }{(1+ \lambda)^{d} }\right )\, .\end{align*}

On the other hand, by (13) and Lemma 11 again we have

\begin{align*}\text{var}(|N(\boldsymbol\Gamma)|)= \sum_{\Gamma\in \mathcal {C}}w(\Gamma)|N(\Gamma)|^{2}\geq \sum_{\Gamma\in \mathcal {C}}w(\Gamma)\|\Gamma\|^2= 2^{d-1}\frac{\lambda}{(1+\lambda)^d}+ O \left ( \frac{2^d d^2}{(1+ \lambda)^{2d} }\right )\, .\end{align*}

It follows that if $\ell\geq 3$ then

\begin{align*}\kappa_\ell(Z)= \text{var}(|N(\boldsymbol\Gamma)|)^{-\ell/2} \kappa_\ell(|N(\boldsymbol\Gamma)|)\to 0\end{align*}

as desired.

Next we will use the cluster expansion to give bounds on $\mathbb{E}_{\mathcal {O},\lambda} ( | \mathbf I |)$ , the expected size of an independent set sampled according to $\mu_{\mathcal {O}, \lambda}$ . We begin by recording a useful expression for $\mathbb{E}_{\mathcal {O},\lambda} ( | \mathbf I |)$ in terms of the cluster expansion.

Lemma 13

\begin{align*}\mathbb{E}_{\mathcal {O},\lambda} ( | \mathbf I |)&= \frac{\lambda}{1+ \lambda} N + \sum_{\Gamma\in\mathcal {C}} w(\Gamma)\left(\|\Gamma\|- \frac{\lambda}{1+\lambda}|N(\Gamma)|\right).\end{align*}

Proof. Note that for an independent set I such that $\mu_{\mathcal {O},\lambda}(I)>0$ , we have

\begin{align*}\mu_{\mathcal {O},\lambda}(I)= \frac{\lambda^{|I|}}{Z_\mathcal {O}(\lambda)} = \frac{\lambda^{|I|}}{(1+\lambda)^N \Xi_\mathcal {O}}\, .\end{align*}

It follows that

\begin{align*}\mathbb{E}_{\mathcal {O},\lambda} ( | \mathbf I |)=\sum_I \frac{|I|\lambda^{|I|}}{(1+\lambda)^N \Xi_\mathcal {O}}=\lambda\frac{d}{d\lambda} \log\left((1+\lambda)^N \Xi_\mathcal {O} \right)= \frac{\lambda}{1+\lambda}N + \lambda(\log \Xi_\mathcal {O})'\, .\end{align*}

We expand $\log \Xi_\mathcal {O}$ via the cluster expansion as in (10) (which converges absolutely by Theorem 10). Recalling that $w(\Gamma)= \phi(H(\Gamma))\lambda^{\|\Gamma\|}(1+\lambda)^{-|N(\Gamma)|}$ for a cluster $\Gamma$ , we have

\begin{align*}\mathbb{E}_{\mathcal {O},\lambda} ( | \mathbf I |)&=\frac{\lambda}{1+ \lambda} N + \sum_{\Gamma\in\mathcal {C}} \phi(H(\Gamma)) \frac{ \lambda^{|\Gamma|} ( (1+\lambda)\|\Gamma\| - \lambda|N(\Gamma)| ) }{ (1+\lambda)^{|N(\Gamma)|+1} } \\&= \frac{\lambda}{1+ \lambda} N + \sum_{\Gamma\in\mathcal {C}} w(\Gamma)\left(\|\Gamma\|- \frac{\lambda}{1+\lambda}|N(\Gamma)|\right)\, .\\[-45pt]\end{align*}

Corollary 14 For fixed $k\geq 0$ ,

\begin{align*}\mathbb{E}_{\mathcal {O},\lambda} ( | \mathbf I |) &= \frac{\lambda}{1+ \lambda} N + \lambda N \sum_{j=1}^k \frac{ \frac{\partial}{\partial \lambda}R_j(\lambda,d) }{ (1+\lambda)^{jd} } - \frac{ jd R_j(\lambda,d) }{ (1+\lambda)^{jd+1} } + O\left( \frac{2^d d^{2k+1}}{ (1+\lambda)^{d(k+1)}} \right)\, .\end{align*}

Where the $R_j(\lambda, d)$ are as in Theorem 10.

Proof. By Lemmas 11 and 13, noting that $|N(\Gamma)|\leq d\|\Gamma\|$ for any cluster $\Gamma$ , we have

\begin{align*}\mathbb{E}_{\mathcal {O},\lambda} ( | \mathbf I |)= \frac{\lambda}{1+ \lambda} N + \sum_{\substack{\Gamma \in \mathcal{C} \\ \|\Gamma\| \leq k}} w(\Gamma)\left(\|\Gamma\|- \frac{\lambda}{1+\lambda}|N(\Gamma)|\right) + O \left ( \frac{2^d d^{2k+1}}{(1+ \lambda)^{d(k+1)} }\right )\, .\end{align*}

The result follows by recalling the definition of $R_j(\lambda, d)$ at (11) .

3. Independent sets of a given size

Theorem 6 will follow from several lemmas. The first says that almost all independent sets of size $m = \lfloor \beta N \rfloor$ are accounted for by exactly one of the two polymer distributions. The second says that if we find $\lambda_\beta$ so that the expected size of the independent set drawn from $\mu_{\mathcal {O}, \lambda_\beta}$ (as in Definition 8) is close to m then we have an asymptotic formula for the number of independent sets of size m in terms of $\lambda_\beta$ and $Z_{\mathcal {O}}(\lambda_{\beta})$ . The third lemma gives an efficiently computable formula for a suitable such $\lambda_\beta$ . We will then prove Theorem 5 by analysing expansions of $\log Z_{\mathcal {O}}(\lambda_{\beta})$ and $\log \lambda_\beta$ in powers of $(1-\beta)^d$ .

Let $i_m(\mathcal {O}) $ be the number of independent sets I of size m in $Q_d$ that are achievable in the odd polymer model (i.e. $\mu_{\mathcal {O},\lambda}(I)>0$ ).

Lemma 15 For any $\beta >0$ ,

\begin{align*} i_{\lfloor \beta N \rfloor} (Q_d) = (2+o(1)) i_{\lfloor \beta N \rfloor} (\mathcal {O}) \end{align*}

as $d \to \infty$ .

Lemma 16 Fix $\beta>0$ . Suppose $\lambda= \lambda(\beta, d)$ is such that

(14) \begin{equation}\left | \mathbb{E}_{\mathcal {O},\lambda} | \mathbf I | -\lfloor \beta N \rfloor \right| = o( N^{1/2}) \,.\end{equation}

Then

\begin{align*} i_{\lfloor \beta N \rfloor}(\mathcal {O}) = (1+o(1)) \frac{ (1+\lambda) Z_{\mathcal {O}}(\lambda) }{ \lambda^{\lfloor \beta N \rfloor} \sqrt{2 \pi N \lambda} } \, .\end{align*}

Lemma 17 There exists a sequence of rational functions $B_j(d,\beta)$ , $j\in \mathbb N$ , such that $B_j$ can be computed in time $e^{O( j \log j)}$ and the following holds. Fix $t \ge 1$ and let $r=\lceil t/2 \rceil-1$ . Suppose that $m = \lfloor \beta N \rfloor$ with $\beta > 1- 2^{-1/t} $ , then if

(15) \begin{equation}\lambda_{\beta} = \frac{\beta}{1-\beta} + \sum_{j=1}^r B_j(\beta, d) (1-\beta)^{jd}\, ,\end{equation}

then

\begin{align*} \left | \mathbb{E}_{\mathcal {O},\lambda_{\beta}} | \mathbf I | -m \right| = o( N^{1/2}) \,. \end{align*}

We prove Lemma 16 first, for which we need the following basic binomial local central limit result.

Lemma 18 Fix $p \in (0,1)$ and suppose $X \sim \text{Bin}(n,p)$ . Suppose $n \to \infty$ and $k=o(\sqrt{n})$ , then

\begin{align*} \mathbb{P}(X= np +k) = \frac{1+o(1)}{\sqrt{ 2\pi n p(1-p)}} \,.\end{align*}

Proof of Lemma 16. Let $m = \lfloor \beta N\rfloor$ . For any $\lambda>0$ , we have

\begin{align*}i_m(\mathcal {O}) &= \frac{ Z_{\mathcal {O}}(\lambda)}{\lambda^m} \mathbb{P}_{\mathcal {O},\lambda} [ | \mathbf I| = m] \,,\end{align*}

where the probability is with respect to the measure $\mu_{\mathcal {O}, \lambda}$ . We therefore need to show that if $\lambda$ satisfies (14), then

(16) \begin{equation} \mathbb{P}_{\mathcal {O},\lambda} [ | \mathbf I| = m] = (1+o(1)) \frac{1+ \lambda}{ \sqrt{2 \pi N \lambda} } \,.\end{equation}

By considering first the collection of polymers $\boldsymbol\Gamma$ chosen at Step 1 in the definition of $\mu_{\mathcal {O}, \lambda}$ (Definition 8), and then the probability that the correct number of additional vertices are chosen at Step 2, we see that

(17) \begin{align} \mathbb{P}_{\mathcal {O},\lambda} [ | \mathbf I| = m] =& \sum_{\Gamma\in\Omega_{\mathcal {O}}}\mathbb{P}\left[\boldsymbol\Gamma = \Gamma \right] \cdot \mathbb{P} \left [ \text{Bin}\left(N-|N( \Gamma)|, \frac{\lambda}{1+\lambda}\right) = m - \|\Gamma\| \right]\, ,\end{align}

where we recall that $\Omega_\mathcal {O}$ denotes the set of all collections of mutually compatible odd polymers. By the large deviation bound [Reference Jenssen and Perkins15, Lemma 16] we have

\begin{align*}\mathbb{P}\left[|N(\boldsymbol\Gamma)|\geq \frac{2N}{d}\right]=O( \exp(\!-N/d^4))\, ,\end{align*}

and so we can condition on the event $|N(\boldsymbol\Gamma)|\leq \frac{2N}{d}$ throughout (17) and only change the resulting probability by an additive factor of $O( \exp(\!-N/d^4))=o(N^{-1/2})$ . Under this conditioning, the binomial probabilities in (17) are uniformly bounded by $O(1/\sqrt{N})$ . To establish (16), it therefore suffices to show that with high probability in the choice of $\boldsymbol\Gamma$ , the binomial probabilities in (17) are in fact equal to $(1+o(1)) \frac{1+ \lambda}{ \sqrt{2 \pi N \lambda} }$ . By Lemma 18, it suffices to show that with high probability in the choice of $\boldsymbol\Gamma$ we have

(18) \begin{align}\frac{\lambda}{1+\lambda}\left(N-|N( \Gamma)|\right)=m-\|\Gamma\| + o(N^{1/2})\, .\end{align}

Now, by our assumption on $\lambda$ , (13) and Lemma 13,

\begin{align*}m= \mathbb{E}_{\mathcal {O},\lambda} | \mathbf I | + o(N^{1/2}) = \mathbb{E} \|\boldsymbol\Gamma\|+ \frac{\lambda}{1+\lambda}\left(N-\mathbb{E} |N(\boldsymbol\Gamma)|\right) + o(N^{1/2})\, .\end{align*}

It follows that to show (18) holds whp with respect to $\boldsymbol\Gamma$ , it suffices to show that

(19) \begin{align}\mathbb{P}\left[\|\boldsymbol\Gamma\|= \mathbb{E} \|\boldsymbol\Gamma\|+o(N^{1/2})\right]=1+o(1)\, ,\end{align}

and similarly for $|N(\boldsymbol\Gamma)|$ . This is an immediate consequence of Lemma 12 and the fact that

(20) \begin{align}\mathbb{E}\|\mathbf\Gamma\|\leq\mathbb{E}|N(\boldsymbol\Gamma)|=\sum_{\Gamma\in\mathcal {C}}w(\Gamma)|N(\Gamma)|=o(N)\, ,\end{align}

where we have used (13) and Lemma 11.

Next we prove Lemma 15.

Proof of Lemma 15. Let $\mathcal I_m$ denote the set of all independent sets of size m in $Q_d$ . Then by Theorem 9 (and the symmetry between even and odd) we have

\begin{align*}\left|\mu_{\lambda}(\mathcal I_m) - \hat\mu_{\lambda}(\mathcal I_m) \right|=\left| \frac{i_m(Q_d)\lambda^m}{Z_{Q_d}(\lambda)} - \frac{i_m(\mathcal {O})\lambda^m}{ Z_{\mathcal {O}}(\lambda) } \right| =O\left(\exp (\!-N/d^4)\right)\, .\end{align*}

By Theorem 9 again it follows that

\begin{align*}\left| i_m(Q_d)- (2+o(1))i_m(\mathcal {O})\right| =O\left(\exp (\!-N/d^4)\right)\frac{ Z_{\mathcal {O}}(\lambda)}{\lambda^m}\, .\end{align*}

It therefore suffices to show that there is a choice of $\lambda$ such that $\frac{i_m(\mathcal {O})\lambda^m}{ Z_{\mathcal {O}}(\lambda)}$ is much larger than $\exp (\!-N/d^4)$ . This follows from Lemma 16 by choosing $\lambda$ satisfying (14).

Next we prove Lemma 17. In the following, if P(x, y) is a polynomial in x, y, we write ${\textrm{deg}}_x(P)$ for the degree of P in x.

Proof of Lemma 17. Let $r=\lceil t/2 \rceil - 1$ and set

\begin{align*}\lambda=\lambda_{\beta} = \frac{\beta}{1-\beta} + \sum_{j=1}^r B_j(\beta, d) (1-\beta)^{jd}\, ,\end{align*}

where the functions $B_j$ are rational polynomials in $\beta, d$ of constant degree (independent of d) to be determined later. Let $X:=\sum_{j=1}^r B_j(\beta, d) (1-\beta)^{jd+1}$ and note that $X=o(1)$ . It will be useful to note that for $k=O(d)$ ,

(21) \begin{align}(1+\lambda)^{-k}=\frac{(1-\beta)^{k}}{(1+X)^{k}}= (1-\beta)^{k}\sum_{i=0}^{r}\binom{-k}{i}X^i + O\left(X^{r+1}\right) \, .\end{align}

In particular, since $\beta > 1-2^{-1/t}$ ,

(22) \begin{align}(1+\lambda)^{-d(r+1)}=(1+o(1))(1-\beta)^{d(r+1)}=O\left(e^{-cd} \cdot 2^{-d(r+1)/t}\right)=O\left(e^{-cd} N^{-1/2}\right)\, ,\end{align}

for some constant $c>0$ .

By Corollary 14 and Theorem 10,

(23) \begin{align}\mathbb{E}_{\mathcal {O},\lambda} ( | \mathbf I |)&= N \frac{\lambda}{1+ \lambda} + \lambda N \sum_{j=1}^r \frac{ \frac{\partial}{\partial \lambda}R_j(\lambda,d) }{ (1+\lambda)^{jd} } - \frac{ jd R_j(\lambda,d) }{ (1+\lambda)^{jd+1} } + O\left(N \frac{d^{2r+1}}{ (1+\lambda)^{d(r+1)}} \right)\nonumber \\&=\frac{\lambda}{1+ \lambda} N + N \sum_{j=1}^r F_j(\lambda, d)(1+\lambda)^{-jd-1} + o\left(N^{1/2} \right)\, , \end{align}

where the $F_j$ are polynomials in $\lambda, d$ with ${\textrm{deg}}_d(F_j)\leq 2j+1$ and ${\textrm{deg}}_\lambda(F_j)\leq 3j^2$ . Our goal is to show that there exists an appropriate choice of $B_1, \ldots, B_r$ that makes this final expression (23) equal to $m+o\left(N^{1/2} \right)$ .

Since ${\textrm{deg}}_\lambda(F_j)\leq 3j^2$ , and $\lambda=(\beta+X)/(1-\beta)$ we may write

\begin{align*}F_j(\lambda,d)=(1-\beta)^{-c_j}G_j(\beta, d, X)\end{align*}

for some non-negative integer $c_j\leq 3j^2$ and $G_j$ a polynomial in $\beta, d, X$ such that ${\textrm{deg}}_d(G_j)\leq 2j+1$ . It follows by (21) that

(24) \begin{align}\nonumber &\frac{\lambda}{1+\lambda} + \sum_{j=1}^r F_j(\lambda, d) (1+\lambda)^{-jd-1}=(\beta+X)\sum_{i=0}^{r}(\!-X)^i \\&\quad + \sum_{j=1}^rG_j(\beta, d, X) (1-\beta)^{jd+1-c_j}\sum_{i=0}^{r}\binom{-jd-1}{i}X^i + O(d^{3r}X^{r+1})\, .\end{align}

We now recall that $X=\sum_{j=1}^r B_j (1-\beta)^{jd+1}$ and we expand this final expression as a polynomial in $(1-\beta)^d$ . This yields

(25) \begin{align}\frac{\lambda}{1+\lambda} + \sum_{j=1}^r F_j(\lambda, d) (1+\lambda)^{-jd-1}&=\beta + \sum_{j=1}^{r} Q_j(\beta, d, B_1,\ldots, B_r)\cdot (1-\beta)^{jd} + O(d^{3r}X^{r+1})\end{align}

where $Q_j=Q_j(\beta, d, B_1,\ldots, B_r)$ is a rational function $\beta, d, B_1, \ldots, B_r$ with denominator $(1-\beta)^{b_j}$ for some $b_j\leq 3j^2$ and with ${\textrm{deg}}_d(Q_j)\leq 2j+1$ . Moreover, by examining the expansion (24), we see that $Q_j$ is linear in $B_j$ where the coefficient of $B_j$ is $(1-\beta)^2$ (in particular the coefficient is non-zero). It follows inductively that there is a choice of $B_1, \ldots, B_r$ such that $Q_1=\ldots=Q_r=0$ where $B_j$ is a rational function of $\beta, d$ of constant degree (depending on j but not d). With this choice of $B_1, \ldots, B_r$ it follows from (23) and (25) that

\begin{align*}\mathbb{E}_{\mathcal {O},\lambda} ( | \mathbf I |) = \beta N+ O(Nd^{3r}X^{r+1})=\beta N+o(N^{1/2})\, ,\end{align*}

where for the last bound we used that $d^{3r}X^{r+1}=d^{O_r(1)}(1-\beta)^{d(r+1)}=o(N^{-1/2})$ by (22).

Finally we note that the above argument gives an algorithm for computing the $B_j$ . Since the $R_j$ , and so also the $F_j$ and $G_j$ , can be computed in $e^{O(j \log j)}$ time, we see that the $Q_j$ can be computed in $e^{O(j \log j)}$ time. The $B_j$ can then be computed by solving j successive linear equations.

To give a concrete example of the algorithm above in action, we pause for a moment to calculate the rational function $B_1$ . Using the definition of $R_1$ at (11), we see that $R_1=\lambda$ . In the notation of the proof of Theorem 17, it follows that $F_1=\lambda+(1-d)\lambda^2$ . Noting that $\lambda=(\beta+X)/(1-\beta)$ we have

\begin{align*}F_1=(1-\beta)^{-2}\left[(1-\beta)(\beta+X) + (1-d)(\beta+X)^2 \right]\end{align*}

and so $G_1= (1-\beta)(\beta+X) + (1-d)(\beta+X)^2$ and $c_1=2$ . Recalling that $X:=\sum_{j=1}^r B_j (1-\beta)^{jd+1}$ and examining the coefficient of $(1-\beta)^d$ in (24), we see that

\begin{align*}Q_1=B_1(1-\beta)^2 + \frac{\beta(1-\beta)+(1-d)\beta^2}{1-\beta}\, .\end{align*}

Solving $Q_1=0$ yields

(26) \begin{align}B_1=\frac{(d\beta-1)\beta}{(1-\beta)^3}\, .\\ \nonumber\\[-40pt] \nonumber\end{align}

3.2 Proof of Theorem 5

We now prove Theorem 5. Given the formula (5) from Theorem 6, we need to extract the binomial coefficient $\binom{N}{ \lfloor \beta N \rfloor}$ and expand the logarithm of what remains.

Lemma 19 Fix $\beta \in (0,1)$ . With $\lambda_0 = \frac{\beta}{1 - \beta}$ ,

\begin{align*}\binom{N}{ \lfloor \beta N \rfloor} &= (1+o(1)) \frac{ (1+\lambda_0)^N} { \lambda_0^{ \lfloor \beta N \rfloor} \sqrt{2\pi N \beta (1-\beta)} }\end{align*}

as $N \to \infty$ .

The proof follows from Stirling’s formula.

Proof of Theorem 5. Given Lemma 19, Theorems 3 and 6, we are left to compute coefficients $P_j = P_j(\beta, d)$ , $j \ge 1$ , so that for $t \ge 1 $ and $\beta > 1- 2^{-1/t}$ we have

(27) \begin{align} \log \left( \frac{1+\lambda_\beta}{1+\lambda_0} \right) - \beta \log \frac{\lambda_\beta}{\lambda_0} + \sum_{j=1}^{t-1} R_j(d,\lambda_\beta) (1+\lambda_\beta)^{-dj} &= \sum_{j=1}^{t-1} P_j \cdot (1-\beta)^{jd} +o(N^{-1})\end{align}

where $\lambda_\beta$ is given by (4). We proceed by expanding each term on the left-hand side of (27) as a power series in $(1-\beta)^d$ . As in the proof of Lemma 17 we set $X:=\sum_{j=1}^r B_j(\beta, d) (1-\beta)^{jd+1}$ and note that $X^t=o(N^{-1})$ since $\beta > 1- 2^{-1/t}$ . It follows by Taylor expansion that

(28) \begin{align}\nonumber \log \left( \frac{1+\lambda_\beta}{1+\lambda_0} \right) - \beta \log \frac{\lambda_\beta}{\lambda_0} &= \log(1+X)-\beta \log(1+X/\beta)\\ &= \sum_{i=1}^{t-1}\frac{(\!-1)^{1+i}}{i}(1-\beta^{1-i})X^i + o(N^{-1})\, .\end{align}

We now turn to the sum on the left-hand side of (27). Since $R_j$ is a polynomial in $\lambda_\beta, d$ such that ${\textrm{deg}}_{\lambda_\beta}(R_j)\leq 3j^2$ and ${\textrm{deg}}_d(R_j)\leq 2j$ and the fact that $\lambda_\beta=(\beta+X)/(1-\beta)$ , we may write

\begin{align*}R_j(\lambda_\beta,d)=(1-\beta)^{-c_j}S_j(\beta, d, X)\end{align*}

for some non-negative integer $c_j\leq 3j^2$ and $S_j$ a polynomial in $\beta, d, X$ such that ${\textrm{deg}}_d(S_j)\leq 2j$ . It follows by (21) that

(29) \begin{align}\sum_{j=1}^t R_j(\lambda_\beta, d) (1+\lambda_\beta)^{-dj}&=\sum_{j=1}^{t-1}S_j(\beta, d, X) (1-\beta)^{jd-c_j}\sum_{i=0}^{t-1}\binom{-jd}{i}X^i + O(d^{2t}X^t)\, .\end{align}

We note that $O(d^{2t}X^t)=O(d^{3t}(1-\beta)^{td})=o(N^{-1})$ . To compute the $P_j$ we simply sum (28) and (29), expand the powers of X and collect the coefficients of $(1-\beta)^d, \ldots, (1-\beta)^{(t-1)d}$ . Finally we note that since $R_j, S_j$ and $B_j$ can each be computed in time $e^{O(j\log j)}$ , $P_j$ can be computed in time $e^{O(j\log j)}$ .

3.2.1 Computation of $P_1, P_2$

To illustrate the algorithm for computing the $P_j$ in Theorem 5, we use it to compute $P_1$ and $P_2$ . First we note that in [Reference Jenssen and Perkins15] it was shown that

\begin{align*}R_1=\lambda\text{ and } R_2= \frac{(2\lambda^3+\lambda^4)d(d-1)-2\lambda}{4}\, .\end{align*}

It follows that (using the notation of the proof of Theorem 5)

\begin{align*}S_1=\beta+X \text{ and }S_2= \frac{1}{2} (\beta - 1)^3 (X+\beta) - \frac{1}{4} d(d - 1) (\beta - X - 2) (X+\beta)^3\end{align*}

and $c_1=1$ , $c_2=4$ . Now, to compute $P_1, P_2$ we compute the coefficients of $(1-\beta)^d, (1-\beta)^{2d}$ in the sum of (28) and (29). The coefficient of $(1-\beta)^d$ in (28) is 0 and in (29) it is $\beta/(1-\beta)$ and so

\begin{align*}P_1=\frac{\beta}{1-\beta}\, .\end{align*}

The coefficient of $(1-\beta)^{2d}$ in (28) is

\begin{align*}\frac{1}{2\beta}B_1^2(1-\beta)^3\, .\end{align*}

The coefficient of $(1-\beta)^{2d}$ in (29) is

\begin{align*}(1-d\beta)B_1+(1-\beta)^{-4}\left(\frac{1}{2} (\beta - 1)^3 \beta - \frac{1}{4} d(d - 1) (\beta - 2) \beta^3 \right)\, .\end{align*}

Recalling (26), the formula for $B_1$ , and summing the above two expressions yields

\begin{align*}P_2= \frac{2(\beta - 1)^3 \beta - d(d - 1) (\beta - 2) \beta^3}{4(1-\beta)^4}-\frac{\beta(1-d\beta)^2}{2(1-\beta)^3}\, .\end{align*}

4. Local central limit theorems for polymer models

In the odd polymer model, let T be a defect type and $X_T$ be the random variable counting the number of defects of type T in a sample from $\mu_{\mathcal {O}, \lambda}$ . Recall that $m_T = \mathbb{E} X_T$ and let $\sigma^2_T = \text{var}(X_T)$ . Moreover, let $n_T$ denote the number of polymers of type T and let $w_T$ denote the weight w(S) (defined at (7)) of a representative polymer S of type T.

Throughout this section we assume that $\lambda\ge C\log d /d^{1/3}$ as in Theorem 9 and probabilities and expectations are with respect to the odd polymer model. The main result of this section is a multivariate local central limit theorem for the number of polymers of different types, extending the multivariate central limit theorem of [Reference Jenssen and Perkins15, Theorem 6].

Theorem 20 Let $\mathcal {T}_1$ and $\mathcal {T}_2$ be two fixed sets of defect types so that for each $T \in \mathcal {T}_1$ , $m_T \to \rho_T$ for some constant $\rho_T>0$ , and for each $T \in \mathcal {T}_2$ , $m_T \to \infty$ as $d \to \infty$ . Let $ \{ k_T \}_{T \in \mathcal T_1}$ be a collection of non-negative integers and let $ \{ k_T \}_{T \in \mathcal T_2}$ be such that $k_T= \lfloor m_T +s_T \rfloor$ where $| s_T| = O(\sqrt{ m_T})$ for all $T \in \mathcal T_2$ . Then

\begin{align*}\mathbb{P} \left( \bigcap_{T \in \mathcal {T}_1\cup \mathcal {T}_2} X_T=k_T\right )&= (1+o(1)) \prod_{T \in \mathcal {T}_1} \frac{\rho_T^{k_T} e^{-\rho_T}}{(k_T)!}\prod_{T \in \mathcal T_2} \frac{e^{ - \frac{ s_T^2}{2 m_T}}}{\sqrt{2 \pi m_T}} \,.\end{align*}

The probability in the theorem statement is with respect to the odd polymer model, but the statement also holds for defects of an independent set drawn from the hard-core model on $Q_d$ via Theorem 9.

Before we proceed it will be useful to recall a result from [Reference Jenssen and Perkins15] on the cumulants of the random variables $X_T$ . Recall that for a random variable X we use $\kappa_k(X)$ to denote the kth cumulant of X. The following result appears as Lemma 20 in [Reference Jenssen and Perkins15].

Lemma 21 For a defect type T, let $Y_T(\Gamma)$ denote the number of polymers of type T in the cluster $\Gamma$ . Then for any fixed $k \ge 1$ ,

(30) \begin{equation} \kappa_k(X_T)= \sum_{\Gamma\in \mathcal {C}} w(\Gamma) Y_T(\Gamma)^k = (1+o(1)) n_T w_T \,,\end{equation}

and

(31) \begin{equation} \sum_{\substack{\Gamma \in \mathcal{C} \\ \|\Gamma\| > |T|}} \left | w(\Gamma) Y_T(\Gamma)^k \right| = o( n_T w_T) \,.\end{equation}

Given a random vector $X=(X_1, \ldots, X_q)\in\mathbb{R}^q$ its characteristic function is

\begin{align*}\varphi_X(t)=\mathbb{E} e^{i\langle X, t \rangle}\,\end{align*}

for $t \in \mathbb{R}^q$ .

Lemma 22 Fix $q\in\mathbb N$ and a list $T_1, \ldots, T_q$ of defect types. Let $X=(X_{T_1}, \ldots, X_{T_q})$ . There exists $c>0$ such that

\begin{align*} |\varphi_{X}(t) | \le \exp\left\{- c \sum_{i=1}^q t_i^2 n_{T_i}w_{T_i}\right\} \, , \end{align*}

for all $t\in[\!-\pi,\pi]^q$ .

Proof. Given a cluster $\Gamma\in \mathcal {C}$ , let $Y(\Gamma)=(Y_{T_1}(\Gamma),\ldots, Y_{T_q}(\Gamma))$ . Using the cluster expansion we write

\begin{align*}\log \mathbb{E} e^{i\langle t, X\rangle} &= \sum_{\Gamma\in \mathcal {C}} w(\Gamma) e^{i \langle t, Y(\Gamma)\rangle} - \sum_{\Gamma\in\mathcal {C}} w(\Gamma) \\[3pt] &= \sum_{\Gamma\in \mathcal {C}} w(\Gamma) (e^{i \langle t, Y(\Gamma)\rangle}-1) \,.\end{align*}

Let

\begin{align*}\mathcal{M}_j=\left\{\Gamma\in \mathcal {C}: j=\max\{i: Y_{T_i}(\Gamma)>0\}, \sum_{i=1}^q Y_{T_i}(\Gamma)>1 \right\}\, .\end{align*}

Then,

\begin{align*}\textrm {Re} \log \mathbb{E} e^{i\langle t, X\rangle} &= \sum_{\Gamma\in \mathcal {C}} w(\Gamma) (\cos( \langle t, Y(\Gamma)\rangle)-1) \\[2pt] &=\sum_{i=1}^q n_{T_i} w_{T_i} (\cos (t_i) -1) +\sum_{j=1}^q\sum_{\Gamma\in \mathcal{M}_j} w(\Gamma) (\cos( \langle t, Y(\Gamma)\rangle)-1) \\[2pt] &\le-\frac{1}{5}\sum_{i=1}^q t_i^2 n_{T_i} w_{T_i} +\sum_{j=1}^q\sum_{\Gamma\in \mathcal{M}_j} |w(\Gamma)| \langle t, Y(\Gamma)\rangle^2 \\[2pt] &\le-\frac{1}{5}\sum_{i=1}^q t_i^2 n_{T_i} w_{T_i} +\sum_{j=1}^q\sum_{\Gamma\in \mathcal{M}_j} |w(\Gamma)|j\sum_{i=1}^jt_i^2Y_{T_i}(\Gamma)^2 \\[2pt] &\le -\frac{1}{5}\sum_{i=1}^q t_i^2 n_{T_i} w_{T_i} +\sum_{i=1}^qt_i^2o(n_{T_i}w_{T_i}) \\\end{align*}

where for the first inequality we used that $-t^2\leq\cos(t)-1\leq -t^2/5$ for $t\in[\!-\pi,\pi]$ . For the next inequality we used Cauchy–Schwarz, and for the final inequality we used Lemma 21.

Proof of Theorem 20. Let $\mathcal {T}_1=\{T_1,\ldots, T_p\}$ , $\mathcal {T}_2=\{T_{p+1},\ldots, T_q\}$ and $\mathcal {T}=\mathcal {T}_1\cup\mathcal {T}_2$ . Let $X_i=X_{T_i}$ , $m_i=m_{T_i}$ , $\sigma_i=\sigma_{T_i}$ , $k_i=k_{T_i}$ for $i\in [q]$ . Let $X=(X_1,\ldots, X_q)$ and let

\begin{align*}\tilde X=\left(X_1,\ldots, X_p, \frac{X_{p+1}-m_{p+1}}{\sigma_{p+1}}, \ldots, \frac{X_{q}-m_{q}}{\sigma_{q}}\right)\, .\end{align*}

By Fourier inversion,

\begin{align*}\mathbb{P}\left( \bigcap_{T \in \mathcal T} X_T=k_T\right) &= \frac{1}{(2 \pi)^q} \int_{[\!-\pi,\pi]^q} \varphi_X(t) \cdot e^{-i\langle t, k \rangle } \, dt\, .\end{align*}

Making the substitution $t_i=x_i$ for $i\in[p]$ and $t_i=x_i/\sigma_i$ for $i>p$ , we have

(32) \begin{align}\mathbb{P}\left( \bigcap_{T \in \mathcal T} X_T=k_T\right) = \frac{1}{(2 \pi)^q} \prod_{i>p}\sigma_i^{-1} \int_{B_1} \int_{B_2} \varphi_{\tilde X}(x)g(x)dx_q\ldots dx_1\end{align}

where $B_1=[\!-\pi,\pi]^p$ , $B_2=[\!-\pi\sigma_{p+1}, \pi\sigma_{p+1}]\times\ldots\times [\!-\pi\sigma_{q}, \pi\sigma_{q}]$ and

\begin{align*}g(x)=\exp\left\{i\sum_{j>p}x_j\frac{m_j-k_j}{\sigma_j}-i\sum_{j\leq p}x_jk_j\right\}\, .\end{align*}

Let $Y=(Y_1, \ldots, Y_q)$ where $Y_{i}\sim {\textrm{Po}}(\rho_i)$ for $i\in [p]$ , $Y_{i} \sim N(0,1)$ for $i>p$ , and $Y_1,\ldots,Y_q$ are jointly independent. Then by Fourier inversion, we have the identity:

\begin{align*}\frac{1}{(2 \pi)^q} \int_{B_1} \int_{\mathbb R^{q-p}} \varphi_{Y}(x)g(x)dx_q\ldots dx_1 &= \prod_{T \in \mathcal {T}_1} \frac{\rho_T^{k_T} e^{-\rho_T}}{(k_T)!}\prod_{T \in \mathcal T_2} \frac{e^{- \frac{(k_T-m_T)^2}{2\sigma_T^2}}}{\sqrt{2\pi}}\, . \\\end{align*}

By (32) (noting that $m_i=(1+o(1))\sigma^2_i$ by Lemma 21), it therefore suffices to show that

\begin{align*}\int_{B_1} \int_{B_2} \varphi_{\tilde X}(x)g(x)dx_q\ldots dx_1 = \int_{B_1} \int_{\mathbb R^{q-p}} \varphi_{Y}(x)g(x)dx_q\ldots dx_1 + o(1)\, .\end{align*}

Since, $m_i\to\infty$ for $i>p$ , Lemma 21 implies that $\sigma_i\to\infty$ for $i>p$ also. It follows that

\begin{align*} \int_{B_1} \int_{\mathbb{R}^{q-p}\backslash B_2}\varphi_{Y}(x)g(x)dx_q\ldots dx_1 =o(1)\,\end{align*}

and so it suffices to show that

\begin{align*} \int_{B_1} \int_{B_2} |\varphi_{\tilde X}(x)-\varphi_Y(x)|dx_q\ldots dx_1 =o(1)\, .\end{align*}

In [Reference Jenssen and Perkins15, Theorem 6] it was shown that $\tilde X$ converges to Y in distribution and so $\varphi_{\tilde X}\to \varphi_Y$ pointwise. It therefore suffices, by dominated convergence, to show that $|\varphi_{\tilde X}(x)-\varphi_Y(x)|$ is bounded by an integrable function. By Lemmas 21 and 22,

\begin{align*}|\varphi_{\tilde X}(x)|= |\varphi_{X}(x_1,\ldots,x_p, x_{p+1}/\sigma_{p+1},\ldots, x_q/\sigma_q)|\leq e^{-\Theta(\sum_{i=1}^q x_i^2)}\, .\end{align*}

We have a similar bound for $|\varphi_Y(x)|$ and so we are done.

5. Independent sets of a given size and structure

Here we prove Theorem 7. Recall that for a set of defect types $\mathcal {T}$ and a vector of integers $ \mathbf x=( x_T )_{T \in \mathcal T}$ , we let $ i_{m, \mathbf x}(Q_d) $ denote the number of independent sets in $Q_d$ of size m with exactly $ x_T $ defects of type T for each $T \in \mathcal T$ .

We use the following identity, an easy extension of (3). For any $\lambda>0$ ,

(33) \begin{equation}i_{m, \mathbf x} (Q_d) = \frac{Z(\lambda)}{\lambda^m} \mathbb{P}_\lambda\left[ |\mathbf I | =m, (X_T)_{T \in \mathcal {T}} = \mathbf x \right ] \,,\end{equation}

where $X_T$ be the random variable counting the number of defects of type T in a sample from $\mu_{\lambda}$ . Theorem 7 follows immediately from (33), Theorem 6 and the following lemma.

Lemma 23 Fix $\lambda>0$ and let $m = m(d)$ be such that that $|\mathbb{E}_{\lambda} | \mathbf I| -m | = o(N^{1/2})$ . Let $\mathcal {T}_1, \mathcal {T}_2$ be the sets of defect types such that $m_T \to \rho_T$ for some fixed $\rho_T > 0$ as $d \to \infty$ for all $T\in \mathcal {T}_1$ and $m_T\to\infty$ for all $T\in \mathcal {T}_2$ . Let $ ( k_T )_{T \in \mathcal T_1}$ be a vector of fixed non-negative integers and let $ ( k_T )_{T \in \mathcal T_2}$ be such that $k_T= \lfloor m_T +s_T \rfloor$ where $| s_T| = O(\sqrt{ m_T})$ for all $T \in \mathcal T_2$ . Let $\mathbf x= (k_T)_{T\in \mathcal {T}_1\cup\mathcal {T}_2}$ . Then

\begin{align*}\mathbb{P}_{\lambda} \left [ |\mathbf I | =m, (X_T)_{T \in \mathcal {T}_1 \cup \mathcal {T}_2} = \mathbf x \right ] &= (1+o(1)) \mathbb{P}_{\lambda} [ |\mathbf I | =m] \prod_{T \in \mathcal {T}_1} \frac{\rho_T^{k_T} e^{-\rho_T}}{(k_T)!} \prod_{T \in \mathcal T_{2}} \frac{e^{ - \frac{ s_T^2}{2 m_T}}}{\sqrt{2 \pi m_T}} \,.\end{align*}

Proof. Using Theorems 9 and 20, it is enough to show that

\begin{align*}\mathbb{P}_{\mathcal{O},\lambda} \left[ | \mathbf I |= m \Big | (X_T)_{T \in \mathcal{T}_1 \cup \mathcal{T}_2} = \mathbf x \right ] & = (1+o(1) ) \mathbb{P}_{\mathcal{O},\lambda} \left[ | \mathbf I |= m \right ] \, ,\end{align*}

where the above probabilities are with respect to $\mu_{\mathcal {O}, \lambda}$ and $X_T$ is now the random variable counting the number of defects of type T in a sample from $\mu_{\mathcal {O}, \lambda}$ . Let $\boldsymbol\Gamma$ be the random collection of compatible polymers chosen at Step 1 in the definition of $\mu_{\mathcal {O}, \lambda}$ (Definition 8) and let $\Gamma$ be a fixed collection of compatible polymers such that

(34) \begin{equation}\|\Gamma\| = \mathbf{E}[\|\boldsymbol{\Gamma}\|] + o(N^{1/2})\end{equation}

and

(35) \begin{equation}|N(\Gamma)| = \mathbf{E}[|N(\mathbf\Gamma)|] + o(N^{1/2}).\end{equation}

Then the proof of Lemma 16 gives us that

\begin{align*}\mathbb{P}_{\mathcal{O},\lambda} \left[ | \mathbf I |= m \Big | \boldsymbol{\Gamma} = \Gamma \right ] = (1+o(1) ) \mathbb{P}_{\mathcal{O},\lambda} \left[ | \mathbf I |= m \right ] \,\end{align*}

and so in particular, if $\Gamma$ is also consistent with $\mathbf x$ (i.e. $\Gamma$ has precisely $k_T$ polymers of type T for all $T\in \mathcal {T}_1 \cup \mathcal {T}_2$ ),

\begin{align*}\mathbb{P}_{\mathcal{O},\lambda} \left[ | \mathbf I |= m \Big | (X_T)_{T \in \mathcal{T}_1 \cup \mathcal{T}_2} = \mathbf x ,\ \boldsymbol{\Gamma} = \Gamma \right ] & = (1+o(1) ) \mathbb{P}_{\mathcal{O},\lambda} \left[ | \mathbf I |= m \right ] \,.\end{align*}

Finally, Lemma 12 gives us that (34) and (35) both hold with probability $1 - o(1)$ , completing the proof.