2.1 Abstract polymer models

An abstract polymer model [Reference Gruber and Kunz17] is given by a set $\mathcal{P}$ of polymers, a compatibility relation $\sim$ on polymers, and a weight $w_\gamma$ associated to each polymer $\gamma \in \mathcal{P}$ . In applications, polymers might be combinatorial objects such as vertex subsets or connected subgraphs of a graph. Let $\Omega$ denote the collection of all sets of pairwise compatible polymers, including the empty set (i.e. the set of independent sets in the graph on $\mathcal{P}$ with edges between incompatible polymers). Then the polymer model partition function is

\begin{equation*} \Xi = \sum _{\Lambda \in \Omega }\prod _{\gamma \in \Lambda }w_\gamma. \end{equation*}

The associated Gibbs measure $\nu$ on $\Omega$ is given by $\nu (\Lambda ) = \prod _{\gamma \in \Lambda }w_\gamma/ \Xi$ , in much the same way as the Gibbs measures associated to the partition function of the Potts and random cluster models. It may be helpful to observe that $\Xi$ is the multivariate independence polynomial of the graph on $\mathcal{P}$ with edges between incompatible polymers, evaluated at the polymer weights given by $w$ , and hence $\nu$ is the associated hard-core measure.

The cluster expansion of $\Xi$ is a formal power series for $\ln \Xi$ with terms given by clusters. A cluster $\Gamma$ is an ordered tuple of polymers, and to each $\Gamma$ we associate an incompatibility graph $H(\Gamma )$ . This graph has vertex set $\Gamma$ (i.e. a vertex for each polymer present in $\Gamma$ with multiplicity) and each pair of vertices representing incompatible polymers is an edge of $H(\Gamma )$ . By convention, a polymer is incompatible with itself. We write $\mathcal{C}$ for the set of all clusters. The formal power series corresponding to the cluster expansion is then

\begin{equation*} \ln \Xi = \sum _{\Gamma \in {\mathcal {C}}}\phi (\Gamma )\prod _{\gamma \in \Gamma }w_\gamma, \end{equation*}

where $\phi$ is the Ursell function

\begin{equation*} \phi (\Gamma ) = \sum _{\substack {F\subset E(H(\Gamma )) \\ (V(H(\Gamma )), F) \text { connected}}} (-1)^{|F|}. \end{equation*}

We remark that for any polymer $\gamma$ , the tuples $(\gamma ),(\gamma, \gamma ),\dotsc$ are valid clusters, and so the cluster expansion is an infinite sum. We use the following theorem to establish convergence and an important tail bound.

Theorem 6 (Kotecký–Preiss [Reference Kotecký and Preiss29]). If there exist functions $f\,{:}\, \mathcal{P} \to [0, \infty )$ and $g\,{:}\, \mathcal{P} \to [0, \infty )$ such that for all $\gamma \in \mathcal{P}$ ,

(5) \begin{equation} \sum _{\gamma^{\prime} \not \sim \gamma }w_{\gamma^{\prime}}e^{f(\gamma^{\prime}) + g(\gamma^{\prime})} \leq f(\gamma ), \end{equation}

then the cluster expansion converges absolutely. Moreover, if we let $g(\Gamma )=\sum _{\gamma \in \Gamma }g(\gamma )$ , then for every polymer $\gamma$ ,

(6) \begin{equation} \sum _{\substack{\Gamma \in \mathcal{C}\\ \Gamma \not \sim \gamma }}\left |\phi (\Gamma )\prod _{\gamma^{\prime} \in \Gamma }w_{\gamma^{\prime}}\right | e^{g(\Gamma )} \leq f(\gamma ), \end{equation}

where we write $\Gamma \not \sim \gamma$ to mean that the cluster $\Gamma$ contains a polymer that is incompatible with $\gamma$ .

2.2 Algorithms from convergent cluster expansions

Approximate counting and sampling algorithms for polymer models follow from Theorem 6 in a rather standard way due to [Reference Helmuth, Perkins and Regts23] and subsequent works. In this paper we consider two polymer models that we define for an $n$ -vertex, $d$ -regular graph $G$ , and then we consider approximating $\Xi$ or approximately sampling from $\nu$ with time complexities measured in terms of $n$ and a desired approximation error $\delta$ .

For a polymer model as above, let $g\,{:}\,\mathcal{P}\to [0,\infty )$ be as in Theorem 6, and extend $g$ to clusters $\Gamma \in{\mathcal{C}}$ by $g(\Gamma ) = \sum _{\gamma \in \Gamma }g(\gamma )$ . An FPTAS for $\Xi$ follows from a few steps that one must check can be done efficiently:

1. (1) Use (6) to show that there exists some $L= L(\epsilon )$ such that

   \begin{equation*} \Xi (L)\,{:\!=}\,\exp \left (\sum _{\substack {\Gamma \in {\mathcal {C}}\\ g(\Gamma ) \le L}}\phi (\Gamma )\prod _{\gamma \in \Gamma }w_\gamma \right ) \end{equation*}
   is a close enough relative approximation of $\Xi$ .

2. (2) List all clusters $\Gamma \in{\mathcal{C}}$ such that $g(\Gamma ) \le L$ .

3. (3) For each such cluster $\Gamma$ , compute $\phi (\Gamma )$ and $\prod _{\gamma \in \Gamma } w_\gamma$ .

4. (4) Compute $\Xi (L)$ directly from the above quantities.

The running time of this algorithm depends on $L$ , the definition of ‘close enough’, the time complexity of listing clusters and of computing polymer weights, $\phi$ and $g$ . It has been established [Reference Helmuth, Perkins and Regts23, Reference Jenssen, Keevash and Perkins24, Reference Borgs, Chayes, Helmuth, Perkins and Tetali2] that for the polymer models we consider, the above steps can be carried out sufficiently fast for the definition of an FPTAS.

Provided that the above algorithm yields an FPTAS for $\Xi$ , there is a generic polynomial-time approximate sampling algorithm for $\nu$ that follows from a self-reducibility property of the polymer model; see ([Reference Helmuth, Perkins and Regts23], Theorem 10). We omit the details and summarise these facts as the following theorem which applies when $\mathcal{P}$ is (a subset of) the connected subgraphs of some bounded-degree graph $G$ . For a subgraph $\gamma$ , we use $v_\gamma$ to denote the number of vertices in $\gamma$ . The important fact is that both the high-temperature and low-temperature polymer models we study satisfy the hypotheses of the result below.

3.1 Sketch of low-temperature argument

For the low-temperature regime, we decompose the partition function into colourings that have a majority colour, and the remaining colourings. Our polymer model can only represent colourings with a majority colour, so it is vital to prove that the remaining colourings have a small contribution to the partition function, which is the content of our Lemma 8. Working with fewer colours and smaller expansion makes this step more challenging than in [Reference Jenssen, Keevash and Perkins24]. The core of the proof involves obtaining upper bounds on the number of vertex-colorings of a graph that induce few monochromatic edges. In light of the heuristic (b), which relates colourings with few monochromatic edges to cuts, this is done by analysing a Karger-like random sequence of edge contractions and the probability that this yields a particular cut.

We then handle each majority colour with a polymer model where the polymers are connected, induced subgraphs of $G$ whose vertices do not receive the majority colour. The weight of a polymer is related to the $(q-1)$ -spin partition function on that subgraph. Since we are working with colourings with a majority colour, the ‘defects’ (vertices that don’t have the majority colour) occupy at most half the vertices, which is a global constraint on the polymers that we wish to avoid. We pass to a different polymer model which requires only that each polymer has at most half the vertices. This partition function is clearly an overestimate; however, in Lemma 9 we show that this overestimation is negligible.

All that remains is to verify the Kotecký–Preiss condition for this polymer model. There is, however, another challenge here: the aforementioned $(q-1)$ -spin partition function plays a significant role in this computation, and one needs to bound it in order to proceed. Fortunately, our understanding of extremal bounds on such functions has progressed recently, most notably due to Sah et al. [Reference Sah, Sawhney, Stoner and Zhao40]. We use their results to derive bounds on the partition functions of the subgraphs that make up our polymers. For triangle-free graphs the available bounds are stronger, which lets us work with weaker expansion in this case, and ultimately with the hypercube. The extremal results we derive from [Reference Sah, Sawhney, Stoner and Zhao40] are the content of Lemmas 11 and 12.

The final ingredient in verifying the Kotecký–Preiss condition is a bound on the number of connected subsets of vertices in $G$ with a given edge-boundary size. At first glance, this looks similar to a lemma used in [Reference Jenssen and Perkins25, Reference Jenssen, Perkins and Potukuchi26] that gives a bound on the number of connected subsets of vertices with a given vertex-boundary size (in bipartite graphs), which followed from mild adaptations of existing container methods [Reference Sapozhenko35, Reference Sapozhenko36].

While the case of edge boundaries does not seem to follow directly from these methods, we have developed Theorem 3 inspired by these previous works. At a high-level, container methods work by finding an ‘encoding’ or ‘certificate’ for each object of the type we wish to enumerate. If these certificates are small then the number of certificates, and thus objects, must be relatively small. In our case, we group sets of vertices $A$ with a fixed edge-boundary $B$ of size $b$ by identifying a ‘core’ subset $A_0\subset A$ of size $O(b/d)$ . We show these core sets must be connected in $G^7$ , and hence their number can be controlled. In this sense, each core set $A_0$ certifies a container which contains sets $A$ (of the type whose number we wish to bound) obtained by growing $A_0$ appropriately, see (19). To complete the bound on the number of sets $A$ of interest, we must bound for each core $A_0$ the number of possible $A$ , or equivalently edge boundaries $B$ , that can be associated with $A_0$ . We achieve this using Karger’s algorithm involving randomised edge contractions to count cuts.

3.2 Polymer model

In Theorem 1, we deal with the class of $d$ -regular $2$ -expanders and triangle-free $d$ -regular $1$ -expanders. Both these families of graphs satisfy

(7) \begin{equation} \text{every set}\,A \subset V\,\text{such that}\,2\leq |A| \leq |V|/2\,\text{satisfies}\,|\nabla (A)| \geq 2d-2. \end{equation}

In particular, this also holds for the hypercube $Q_d$ . Property (7) is immediate in $d$ -regular 2-expanders and follows from Mantel’s theorem [Reference Mantel33] in the triangle-free 1-expanding case. To see this, note that the expansion condition suffices for $|A|\ge 2d-2$ , and if $|A|\lt 2d-2$ then we use the bound

\begin{equation*} \nabla (A) = d|A| - 2|E(G[A])| \ge d|A| - \frac {|A|^2}{2}, \end{equation*}

which follows from Mantel’s theorem. For $2\le |A| \le 2d-2$ , this lower bound is at least $2d-2$ and property (7) follows.

The min-cut of a graph is the minimum number of edges that need to be deleted in order to disconnect the remaining graph. Any $d$ -regular graph $G$ that satisfies (7) also satisfies

(8) \begin{equation} \text{the min-cut of}\,G\,\text{has size at least}\,d. \end{equation}

In other words, the smallest set of edges that can be deleted to disconnect the graph is simply the set of all edges incident to any given vertex. Throughout this section we will use that $G$ satisfies (7), and therefore (8).

We now define a polymer model for the low-temperature regime, following [Reference Jenssen, Keevash and Perkins24]. Here we take $\epsilon \in (0,1)$ ; then, for some $d_0(\epsilon )$ and an absolute constant $c$ , we have $d\ge d_0$ , $q\ge d^c$ , and $\beta \ge (1\,{+}\,\epsilon )\beta _o(q,d)$ . We also assume that $q$ is large enough (by ensuring that $d_0$ and $c$ are large enough) that this implies $\beta \ge (2+\epsilon )\ln (q)/d$ .

Let a polymer $\gamma$ be a connected, induced subgraph of $G$ with at most $n/2$ vertices. For each polymer $\gamma$ , let $v_\gamma$ and $e_\gamma$ be the number of vertices and edges respectively. Let $\nabla _{\gamma }$ denote the number of boundary edges (i.e. edges of $G$ with one endpoint in the vertex set of $\gamma$ ), and let $\eta _{\gamma } \,{:\!=}\, \nabla _{\gamma }/v_\gamma$ , be a measure of the expansion of the vertex set of $\gamma$ in $G$ . The polymer weights $w_\gamma$ are given by

\begin{equation*} w_{\gamma } = e^{-\beta (\nabla _{\gamma } + e_{\gamma })} \cdot Z_{\gamma }(q-1, \beta ), \end{equation*}

where we interpret $\gamma$ as a graph and hence $Z_{\gamma }(q-1, \beta )$ is the Potts partition function of $\gamma$ with $q-1$ colours. We say that two polymers $\gamma$ and $\gamma^{\prime}$ are incompatible if $\mathrm{dist}_G(\gamma, \gamma^{\prime}) \leq 1$ . That is, $\gamma \not \sim \gamma^{\prime}$ if and only if they share a vertex or one contains a neighbour of the other. As before, we write $\Omega$ for the collection of pairwise compatible sets of polymers and $\Xi = \Xi _{G}(q,\beta )$ for the partition function of this polymer model.

For our low-temperature algorithm, there is not a precise correspondence between $Z_G(q,\beta )$ and $\Xi$ , and we must establish that approximating $\Xi$ does in fact yield an approximation of $Z_G(q,\beta )$ . The set $[q]^{V(G)}$ of $q$ -colorings of $V(G)$ admits a partition into $S_0, S_1, \dotsc, S_q$ where $S_0$ is the set of colourings such that each colour occupies at most $n/2$ vertices and for $j\in [q]$ , $S_j$ is the set of colourings in which $j$ is the majority colour (i.e. $S_j = \{\sigma \in [q]^{V(G)}\,{:}\, |\sigma ^{-1}(j)| \gt n/2\}$ ). We have the following lemma whose proof is postponed to Section 4.2.

Lemma 8. There exists an absolute constant $c$ such that the following holds. For every $\epsilon \in (0,1)$ there exists $d_0(\epsilon )$ such that for every $d\ge d_0$ , $q\ge d^c$ and $\beta \ge (1+\epsilon )\beta _o(q,d)$ , we have

\begin{equation*} \frac {\sum _{\sigma \in S_0}e^{\beta m(G,\sigma )}}{Z_{G}(q, \beta )} \leq q^{-\Omega \left (\frac {\epsilon n}{d}\right )}. \end{equation*}

Let us define the partition function-like quantity

\begin{equation*} \widetilde {\Xi } = \widetilde {\Xi }_G(q,\beta ) \,{:\!=}\, \sum _{\substack {\Lambda \in \Omega \\ \|\Lambda \| \lt n/2}}\prod _{\gamma \in \Lambda }w_{\gamma }, \end{equation*}

where $\|\Lambda \| = \sum _{\gamma \in \Lambda }v_{\gamma }$ is the size of a set $\Lambda$ of pairwise compatible polymers. We write $\widetilde \Omega = \{\Lambda \in \Omega\,{:}\, \|\Lambda \| \lt n/2\}$ and note that $\widetilde \Xi$ is not the true partition function of the polymer model because of the global constraint $\|\Lambda \| \lt n/2$ . In terms of the Potts partition function $Z_G(q,\beta )$ , the quantity $\widetilde{\Xi }$ faithfully represents the colourings of $G$ which have a majority colour because

\begin{equation*} \sum _{j\in [q]}\sum _{\sigma \in S_j}e^{\beta m(G,\sigma )} = e^{\beta dn/2} \sum _{j\in [q]}\sum _{\substack {\Lambda \in \Omega \\ \|\Lambda \| \lt n/2}} \prod _{\gamma \in \Lambda }w_\gamma = q e^{\beta dn/2}\widetilde {\Xi }. \end{equation*}

To see this, consider a colouring $\sigma \in [q]^{V}$ with majority colour $j$ and decompose $\sigma ^{-1}([q]\setminus \{j\})\,{\subset}\,V$ into sets such that their pairwise distance in $G$ is at least two. These sets induce subgraphs which form pairwise compatible polymers. Hence for any $j\in [q]$ the colourings with majority colour $j$ are in bijection with $\{\Lambda \in \Omega\,{:}\, \|\Lambda \| \lt n/2\}$ . The weight function and incompatibility criterion have been carefully defined to make the above calculation work on the level of partition function contribution as well.

Rewriting Lemma 8 in this terminology gives us that

\begin{equation*} \frac {q e^{\beta dn/2}}{Z_G(q,\beta )} \widetilde {\Xi } = 1 - q^{-\Omega \left (\frac {\epsilon n}{d}\right )}. \end{equation*}

Thus, approximating $\widetilde \Xi$ is a viable avenue for approximating $Z_G(q,\beta )$ . There is one final complication, however, related to the fact that $\widetilde \Xi$ is not precisely the partition function $\Xi$ of the low-temperature polymer model defined above. Since the standard algorithmic setup (Theorem 7) allows us to approximate $\Xi$ and approximately sample from the associated polymer measure $\nu$ , we must additionally show that restricting to $\Lambda$ such that $\|\Lambda \|\lt n/2$ does not harm the approximation too much. The proof of the low-temperature case of Theorem 1 therefore follows from the following two lemmas and the standard algorithmic setup of Theorem 7.

Lemma 9. For $\Xi$ and $\widetilde{\Xi }$ as defined above, we have

\begin{equation*} \widetilde {\Xi } \le \Xi \le \widetilde \Xi \left (1 + e^{-\Omega (n/d)}\right ). \end{equation*}

Note that the left-hand side of condition (5) decreases if $g$ decreases, and hence the above lemma also implies that Theorem 6 holds for any smaller choice of $g$ . We prove the above results, starting with Lemma 10, in the following subsections.

3.3 Proof of Lemma 10

We first give the proof in the case that $G$ is a 2-expander, and then show the modifications necessary for triangle-free 1-expanders.

We use $N(\gamma ) = N_G(\gamma ) = \{v\,{:}\,\exists u \in \gamma \text{ such that }\{u,v\} \in E\}$ for the neighbourhood of $\gamma$ in $G$ and $N_G[\gamma ] = N_G(\gamma ) \cup V(\gamma )$ for the closed neighbourhood of $\gamma$ . We upper bound the left-hand side of condition (5), here noting that the incompatibility relation on polymers yields

\begin{equation*} \sum _{\gamma^{\prime}\not \sim \gamma }w_{\gamma^{\prime}} e^{f(\gamma^{\prime})+g(\gamma^{\prime})} \le \sum _{u\in N_G[\gamma ]}\sum _{\gamma^{\prime} \ni u} w_{\gamma^{\prime}} e^{f(\gamma^{\prime}) + g(\gamma^{\prime})} = \sum _{u\in N_G[\gamma ]}\sum _{b\ge d}\sum _{\substack {\gamma^{\prime} \ni u \\ \text {s.t.} \nabla _{\gamma^{\prime}}=b}} w_{\gamma^{\prime}} e^{f(\gamma^{\prime}) + g(\gamma^{\prime})}, \end{equation*}

Since $|N_G[\gamma ]| \le v_\gamma + \nabla _\gamma$ , it suffices to show that for all $u\in V$ we have

(9) \begin{equation} \sum _{b\ge d}\sum _{\substack{\gamma^{\prime} \ni u \\ \text{s.t.} \nabla _{\gamma^{\prime}}=b}} w_{\gamma^{\prime}} e^{f(\gamma^{\prime}) + g(\gamma^{\prime})} \leq \min _{\gamma \in \mathcal{P}} \frac{f(\gamma )}{v_{\gamma } + \nabla _{\gamma }}. \end{equation}

We use the following lemma, whose proof is postponed to Section 5, to bound the weights $w_{\gamma^{\prime}}$ that appear in (9).

Lemma 11. For any polymer $\gamma \in \mathcal{P}$ we have

\begin{equation*} Z_{\gamma }(q-1, \beta ) \leq e^{\beta e_\gamma } q^{v_\gamma - \frac {2e_\gamma }{d}+\frac {e_\gamma }{\binom {d+1}{2}}} 2^{\frac {e_\gamma }{\binom {d+1}{2}}}. \end{equation*}

To bound the exponent of $q$ in Lemma 11, observe that $d v_\gamma = 2e_\gamma + \nabla _\gamma \ge 2e_\gamma$ by a double-counting argument, and recall that $\eta _{\gamma } = \nabla _{\gamma }/v_\gamma$ to obtain

\begin{equation*} v_\gamma - \frac {2e_\gamma }{d}+\frac {e_\gamma }{\binom {d+1}{2}} \leq \frac {\nabla _\gamma }{d}\left (1 + \frac {1}{\eta _\gamma }\right ). \end{equation*}

Plugging these facts and Lemma 11 into the definition of $w_\gamma$ gives

\begin{equation*} w_\gamma = e^{-\beta (e_\gamma +\nabla _\gamma )} Z_\gamma (q-1,\beta ) \leq e^{-\beta \nabla _\gamma } q^{\frac {\nabla _\gamma }{d}(1 + \frac {1}{\eta _\gamma })}2^{\frac {\nabla _\gamma }{d\eta _\gamma }}, \end{equation*}

and using $\beta \ge (2+\epsilon )\ln (q)/ d$ we have

\begin{equation*} e^{-\beta \nabla _{\gamma }} \leq e^{-(2+\epsilon )\nabla _\gamma \frac {\ln q}{d}} = q^{-(2+\epsilon )\frac {\nabla _\gamma }{d}}. \end{equation*}

We assume $q$ is large enough (by assuming $d_0$ and $c$ are large enough) so that $2^{\frac{\nabla _\gamma }{d\eta _\gamma }}\leq q^{\frac{\nabla _\gamma }{d} \cdot \frac{\epsilon }{4}}$ , and thus

\begin{equation*} w_\gamma \leq q^{-\frac {\nabla _\gamma }{d}\left (1+\frac {3\epsilon }{4} - \frac {1}{\eta _\gamma }\right )}. \end{equation*}

Using our choice of $f$ and $g$ , we also have

\begin{equation*} e^{f(\gamma )+g(\gamma )} = q^{\frac {\epsilon }{4d}(v_\gamma +\nabla _\gamma )}\cdot e^{v_\gamma/d} = q^{\frac {\nabla _\gamma }{d}\frac {\epsilon }{4}\left (1+\frac {1}{\eta _\gamma }\right )}\cdot e^{\frac {v_\gamma }{d}}, \end{equation*}

which combined with the bound $\eta _\gamma \ge 2$ gives

(10) \begin{equation} w_\gamma e^{f(\gamma )+g(\gamma )} \leq q^{-\frac{\nabla _\gamma }{d}\left (1+\frac{3\epsilon }{4} - \frac{1}{\eta _\gamma } - \frac{\epsilon }{4\eta _\gamma }-\frac{\epsilon }{4}\right )}e^{\frac{v_\gamma }{d}} \leq q^{-\frac{\nabla _\gamma }{d}\left (\frac{1}{2}+\frac{3\epsilon }{8}\right )}e^{\frac{v_\gamma }{d}} \leq q^{-\frac{\nabla _\gamma }{2d}}, \end{equation}

where the final inequality holds again when $q$ is large enough in terms of $\epsilon$ .

We now use our container-type result. Applying Theorem 3 and (10) to bound the left-hand side of (9), we have (for $q$ large enough that the series converges)

\begin{equation*} \sum _{b\ge d}\sum _{\substack {\gamma^{\prime} \ni u \\ \text {s.t.} \nabla _{\gamma^{\prime}}=b}} w_{\gamma^{\prime}} e^{f(\gamma^{\prime}) + g(\gamma^{\prime})} \leq \sum _{b\ge d} d^{O\left ((1+1/\eta )b/d\right )} q^{-\frac {b}{2d}} \le \frac {d^{c'}/\sqrt q}{1-(d^{c'}/\sqrt q)^{1/d}},\end{equation*}

where the absolute constant $c'\ge 4$ is large enough that the term $O\left ((1+1/\eta )b/d\right )$ is at most $c'b/d$ for all $b\ge d$ and all $\eta \ge 1$ . Note that the right-hand side above is a decreasing function of $q$ . To satisfy (9) we want this to be at most

\begin{equation*} \min _{\gamma \in \mathcal {P}} \frac {f(\gamma )}{v_{\gamma } + \nabla _{\gamma }} = \frac {\epsilon \ln q}{4d} \min _{\gamma \in \mathcal {P}} \frac {v_\gamma }{v_\gamma +\nabla _\gamma } = \frac {\epsilon \ln q}{4d(d+1)}, \end{equation*}

which is an increasing function of $q$ . Thus, it suffices to take $q$ larger than some known value for the desired inequality holds. It is straightforward to check that $q=d^{3c'}$ suffices for all $d$ large enough.

3.4 The triangle-free case

If $G$ is triangle free then a stronger upper bound on partition functions of the form $Z_{\gamma }(q-1, \beta )$ holds.

Lemma 12. If $G$ is triangle free, then

\begin{equation*} Z_{\gamma }(q-1, \beta ) \leq e^{\beta e_\gamma } \cdot q^{v_\gamma - \frac {2e_\gamma }{d}+\frac {e_\gamma }{d^2}} \cdot 2^{\frac {e_\gamma }{d^2}}. \end{equation*}

Plugging this inequality in the computation in the previous section, we have

\begin{equation*} v_\gamma - \frac {2e_\gamma }{d} + \frac {e_\gamma }{d^2} \le \frac {\nabla _\gamma }{d}\left (1+\frac {1}{2\eta _\gamma }\right ),\end{equation*}

and hence only assuming $\eta \ge 1$ we have the bound

\begin{equation*} w_\gamma e^{f(\gamma )+g(\gamma )} \leq q^{-\frac {\nabla _\gamma }{d}\left (\frac {1}{2}+\frac {\epsilon }{4}\right )}e^{\frac {v_\gamma }{d}} \leq q^{-\frac {\nabla _\gamma }{2d}}. \end{equation*}

We may then apply Theorem 3 to reach the same conclusion as before.

3.5 Proof of Lemma 9

Let $\nu$ and $\widetilde{\nu }$ denote the probability distributions on $\Omega$ given by the partition functions $\Xi$ and $\widetilde{\Xi }$ respectively, meaning that $\nu (\Lambda ) = \prod _{\gamma \in \Lambda }w_\gamma/ \Xi$ for any $\Lambda \in \Omega$ and $\widetilde{\nu }(\Lambda )=0$ if $\|\Lambda \|\ge n/2$ but $\widetilde{\nu }(\Lambda ) = \prod _{\gamma \in \Lambda }w_\gamma/ \widetilde{\Xi }$ otherwise. We briefly prove a large deviation inequality on the size of the defects, similar to [Reference Jenssen and Perkins25, Reference Jenssen, Perkins and Potukuchi26]. Here, we carefully make use of the term $e^{v_\gamma/d}$ in the definition of $g(\gamma )$ and the monotonicity of condition (5) in $g$ . This immediately implies that Theorem 6 holds for an alternative polymer model on the same polymers but with weights

\begin{equation*} w^{\prime}_{\gamma }\,{:\!=}\, w_{\gamma } \cdot e^{\frac {v_{\gamma }}{d}}, \end{equation*}

because for the same choice $f(\gamma )=\frac{\epsilon v_\gamma \ln q}{4d}$ and now taking $g(\gamma ) = \frac{\epsilon \nabla _\gamma \ln q}{4d}$ , condition (5) is the same for both models.

Write

\begin{equation*} \Xi^{\prime}\,{:\!=}\, \sum _{\Lambda \in \Omega }\prod _{\gamma \in \Lambda }w^{\prime}_{\gamma } = \sum _{\Lambda \in \Omega }\prod _{\gamma \in \Lambda }w_{\gamma } e^{v_\gamma/d}, \end{equation*}

and let $\boldsymbol{\Lambda }$ denote a random set of pairwise compatible polymers drawn from $\nu$ . Observe that $\Xi^{\prime}$ naturally appears in an expression for $\mathbb{E} e^{\|\boldsymbol{\Lambda }\|/d}$ as follows:

\begin{equation*} \mathbb {E} e^{\|\boldsymbol{\Lambda }\|/d} = \sum _{\Lambda \in \Omega }\nu (\Lambda )e^{\|\Lambda \|/d} = \frac {1}{\Xi }\sum _{\Lambda \in \Omega }\prod _{\gamma \in \Lambda }\left (w_\gamma e^{v_\gamma/d}\right ) = \frac {\Xi^{\prime}}{\Xi }. \end{equation*}

Then Theorem 6 and the fact that $\Xi \ge 1$ gives us

\begin{equation*} \ln \mathbb {E} e^{\|\boldsymbol{\Lambda }\|/d} = \ln \frac {\Xi^{\prime}}{\Xi } \le \ln \Xi^{\prime} = \sum _{\Gamma \in \mathcal {C}}\phi (\Gamma )\prod _{\gamma \in \Gamma } w^{\prime}_\gamma, \end{equation*}

where the final equality is the cluster expansion for the abstract polymer model defined by $\Xi^{\prime}$ , and hence the sum is over clusters $\Gamma \in \mathcal{C}$ as defined in Section 2. We can crudely upper bound this cluster expansion of by summing over all polymers of the form $\gamma =(\{v\},\emptyset )$ induced by a vertex and applying the tail bound (6). This gives

(11) \begin{align} \ln \Xi^{\prime} &\leq \sum _{v\in V(G)}\sum _{\substack{\Gamma \in \mathcal{C}\\ \Gamma \not \sim (\{v\},\emptyset )}} \left |\phi (\Gamma )\prod _{\gamma \in \Gamma }w^{\prime}_{\gamma }\right | \end{align}

(12) \begin{align} &\leq \sum _{v\in V(G)}\sum _{\substack{\Gamma \in \mathcal{C}\\ \Gamma \not \sim (\{v\},\emptyset )}} q^{-\epsilon/4}\left |\phi (\Gamma )\prod _{\gamma \in \Gamma }w^{\prime}_{\gamma }\right |e^{g(\Gamma )} \end{align}

(13) \begin{align} &\leq n \cdot \frac{\epsilon \ln q}{4q^{\epsilon/4}d}. \end{align}

Line (12) holds because every cluster $\Gamma \not \sim (\{v\},\emptyset )$ contains at least one polymer (which has at least one vertex) and hence using (8) we have

\begin{equation*} g(\Gamma ) = \frac {\epsilon \ln q}{4d}\sum _{\gamma \in \Gamma }\nabla _\gamma \ge \frac {\epsilon \ln q}{4}. \end{equation*}

To obtain line (13) we use (6) with $f((\{v\},\emptyset )) = \frac{\epsilon \ln q}{4d}$ .

Combining the above calculations, we have $\mathbb{E} e^{\|\boldsymbol{\Lambda }\|/d} \le e^{\frac{n}{4d}}$ , and hence Markov’s inequality gives

(14) \begin{equation} \mathbb{P}(\|\boldsymbol{\Lambda }\| \geq n/2) = \mathbb{P}(e^{\|\boldsymbol{\Lambda }\|/d} \ge e^{n/(2d)}) \leq \exp \left (\frac{\epsilon n \ln q}{4q^{\epsilon/4}d} - \frac{n}{2d}\right ) = \exp \left (-\frac{n}{4d}\right ), \end{equation}

which implies $\nu (\widetilde \Omega )\ge 1-e^{-n/(4d)}$ . Here we have used the fact that $q$ is large in terms of $\epsilon$ so $\frac{\epsilon \ln q}{4q^{\epsilon/4}} \leq \frac{1}{4}$ . We would like to point out that similar computations are required for the proofs of Theorems 4 and 5.

Since the probabilities for $\widetilde \nu$ on outcomes in $\widetilde \Omega$ are made by redistributing the probability mass $\nu (\Omega \setminus \widetilde \Omega )\le e^{-n/(4d)}$ onto outcomes in $\widetilde \Omega$ , we immediately have a bound on the total variation distance

\begin{equation*} \|\widetilde {\nu } - \nu \|_{TV} \leq e^{-n/(4d)}. \end{equation*}

The conclusion follows because by the definitions of $\widetilde \nu$ and total variation distance we have

\begin{equation*} 0 \le \widetilde \nu (\widetilde \Omega )-\nu (\widetilde \Omega ) = 1-\nu (\widetilde \Omega ) \le \|\widetilde {\nu } - \nu \|_{TV}, \end{equation*}

and hence dividing by $\nu (\widetilde \Omega )$ gives

\begin{equation*} 0 \le \frac {1}{\nu (\widetilde \Omega )}-1 = \frac {\Xi }{\widetilde \Xi }-1 \le \frac {\|\widetilde {\nu } - \nu \|_{TV}}{\nu (\widetilde \Omega )} \le \frac {e^{-n/(4d)}}{1-e^{-n/(4d)}}. \end{equation*}

4.1 Connected sets with small edge boundaries: Proof of Theorem 3

The main combinatorial result underlying our container-like lemma is the following standard adaptation of Karger’s algorithm to count $\alpha$ -min-cuts.

We also require a well-known result giving an upper bound to the number of connected induced subgraphs containing a fixed vertex.

Proposition 14 ([Reference Knuth28], Vol. 3 p. 396, Ex.11). The number of rooted labelled trees of maximum degree $\Delta$ on $n \geq 1$ vertices is

\begin{equation*} \frac {\binom {\Delta n}{n}}{(\Delta -1)n + 1} \leq (e\Delta )^{n-1}, \end{equation*}

and hence, for a graph of maximum degree $\Delta$ and a fixed vertex $v$ , the number of connected induced subgraphs on $n$ vertices containing $v$ is also at most this number.

With this, we now prove Theorem 3.

Proof of Theorem 3. We first introduce some notation. Recall $V$ and $E$ are the sets of vertices and edges of $G$ respectively. For $u, v \in V$ , we use $\mathrm{dist}_G(u, v)$ to denote the distance in $G$ from $u$ to $v$ , that is, the number of edges on a shortest path between $u$ and $v$ . For every $k \in \mathbb{N}$ , we define the $k$ th neighbourhood of $u$ in $G$ as $N_{G}^k(u) = \{v \in V\,{:}\, 1 \leq \mathrm{dist}_G(u, v) \leq k\}$ . Then $N_G(u) = N_G^1(u)$ . We may also define the $k$ th neighbourhood of a set of vertices $A \subseteq V$ as $N_G^k(A) = \cup _{u \in A}N_G^k(u)$ . The $k$ th power of the graph $G$ is the graph $G^k$ on vertex set $V$ where $\{u,v\} \in E(G^k)$ if and only if $1 \leq \mathrm{dist}_G(u, v) \leq k$ . Observe that $N_{G^k}(u) = N_G^k(u)$ .

Let $I$ be the vertex-edge incidence graph of $G$ , which is a bipartite graph with vertex set $V \sqcup E$ and edges $\left \{(u,e) \in V\times E\mid u \in e\right \}$ . Let $J$ be the bipartite graph with vertex set $V \sqcup E$ and edges given by $\left \{(u,e)\in V\times E\mid \exists v\in N_G(u) \text{ s.t. } v\in e\right \}$ . That is, $(u,e)$ forms an edge of $J$ if and only if $e$ is an edge incident in $G$ to a neighbour of $u$ (this includes all edges incident to $u$ itself). Note that $I$ is a subgraph of $J$ .

Observe that for every $u\in V$ , the degree of $u$ in $J$ is at least $\binom{d+1}{2} \ge d^2/2$ and at most $d^2$ . To see the lower bound, consider that each of the $d$ neighbours of $u$ is itself incident to $d$ edges, and the number of distinct edges incident to $N_G(u)$ is minimised when $N_G(u)$ forms a clique on $d+1$ vertices. The upper bound holds because the maximum number of edges of $G$ incident to $N_G(u)$ is $d^2$ (which occurs if and only if $u$ is contained in no cycles of length 3 or 4).

Now consider a set of vertices $A \subseteq V$ of size at most $n/2$ such that $G[A]$ (the subgraph of $G$ induced by $A$ ) is connected and $x \in A$ . Let $B=\nabla (A)$ and $|B|=|\nabla (A)|=b$ .

We then have $N_I(A)= B\sqcup W$ where $W=E(G[A])$ . Note that by summing degrees, $d|A|=b + 2|W|$ , and by the expansion assumption (3) we have $b \ge \eta |A|$ . Hence,

(15) \begin{equation} |W|\le \frac{b}{2}\left (\frac{d}{\eta }-1\right ) \le \frac{bd}{2\eta }. \end{equation}

Observe that $I$ is a subgraph of $J$ , so $N_I(A) \subseteq N_J(A)$ . Let $B' = N_J(A) \setminus N_I(A)$ . If $e \in B'$ , then $e$ must contain the external endpoint of some boundary edge of $A$ . There are at most $b$ such external endpoints, and each is incident to at most $d-1$ such non-boundary edges. Thus,

(16) \begin{equation} |B'| \leq b(d-1). \end{equation}

Let $A_0 \subseteq A$ be a maximal subset of vertices with pairwise-disjoint neighbourhoods in $J$ . We will first determine how many choices there are for $A_0$ . We claim that

(17) \begin{equation} |A_0| \le \frac{2}{d^2}\left (|B'| +|B| +|W|\right ) \le \frac{b}{d}\left (2+\frac{1}{\eta }\right ). \end{equation}

The first inequality holds because the above definitions give $|N_J(A)| = |B'| + |B| + |W|$ , and each vertex in $A$ has degree at least $d^2/2$ in $J$ . The second inequality follows from (15) and (16).

The maximality of $A_0$ gives us that

(18) \begin{equation} \text{for any}\,u \in A \setminus A_0,\,\text{there is a}\,v \in A_0\,\text{such that}\,N_{J}(u) \cap N_{J}(v) \neq \emptyset. \end{equation}

As a result, we have,

(19) \begin{equation} A_0 \cup N^2_{J}(A_0) \supseteq A. \end{equation}

Moreover, we make the following claim (where we recall $x \in A$ is some specified vertex).

Claim 15. The set of vertices $A_0 \cup \{x\}$ is connected in $G^7$ .

Proof. We start by proving that $A_0$ is connected in $G^7$ . Since $A$ is connected in $G$ , and $A_0\subseteq A$ , for any pair of distinct vertices $u,v\in A_0$ , there must be a path $P$ from $u$ to $v$ along edges of $G$ using only vertices in $A$ . Let the vertices of $P$ be $w_0 = u, w_1, \dots, w_k = v$ .

Observe that every vertex in $P$ is distance at most three in $G$ from $A_0$ . This follows from (18); indeed, either $w_i \in A_0$ or there is some $u_i \in A_0$ and $e \in G$ such that $e \in N_J(w_i) \cap N_J(u_i)$ . This means $e$ contains a vertex in $N_G(w_i)$ and a vertex in $N_G(u_i)$ . There are three possibilities for $e$ : either $e = u_iv_i$ , $e$ is incident to exactly one of $u_i$ or $w_i$ , or $e$ is incident to neither $u_i$ nor $w_i$ . These cases give $\mathrm{dist}_G(w_i, u_i)$ as 1, 2, and 3, respectively. See Fig. 1 for an illustration.

Figure 1. The three different cases for an edge $e$ to be in $N_J(u_i) \cap N_J(w_i)$ .

We can now construct a walk $P'$ along edges of $G$ from $u$ to $v$ that visits a vertex in $A_0$ at most every seven steps. Let $Q_i$ be a path from $w_i$ to $A_0$ such that $|Q_i| \leq 3$ , and let $Q_i^{-1}$ be the reverse path from $A_0$ to $w_i$ . If $w_i \in A_0$ , set $Q_i$ and $Q_i^{-1}$ to be empty. Then the edges of $P'$ are

\begin{equation*}P' = (uw_1, Q_1, Q_1^{-1}, w_1w_2, Q_2, Q_2^{-1}, \dots, w_{k-2}w_{k-1}, Q_{k-1}, Q_{k-1}^{-1}, w_{k-1}w_k)\end{equation*}

See Fig. 2 for an example. The walk $P'$ visits vertices of $A_0$ at most every seven steps, so there is a walk in $G^7$ from $u$ to $v$ using only vertices in $A_0$ . As $u$ and $v$ were arbitrary, it follows that $A_0$ is connected in $G^7$ .

Figure 2. An example of part of the path $P'$ in red arrows.

This completes the proof of the claim in the case $x \in A_0$ . If $x \notin A_0$ , then it is distance at most three in $G$ from the nearest vertex in $A_0$ and hence is adjacent to a vertex of $A_0$ in $G^7$ . Thus, $A_0\cup \{x\}$ is connected in $G^7$ as required.

As a result, we can specify $A_0$ by specifying a tree of degree at most $d^7$ rooted at $v$ (since $G^7$ has maximum degree $d(d-1)^6\le d^7$ ). Using Proposition 14 and the bound on $|A_0|$ given by (17), there are at most $(ed^7)^{\frac{b}{d}\left (2+\frac{1}{\eta }\right )} = d^{O((1+1/\eta )b/d)}$ possibilities for $A_0$ .

We now want to count the number of choices for $B$ . Let $E_0$ be the edges of $G$ in $N^3_J(A_0)$ , meaning the edges which have an endpoint at distance at most 5 from $A_0$ , and let $G'$ be the graph $G$ with every edge in $E \setminus E_0$ contracted. By (19) we have that

\begin{equation*} B \subseteq N_{I}(A) \subseteq N_{J}(A) \subseteq N_{J}^3(A_0), \end{equation*}

so $B \subset E(G')$ . We also have $v \in V(G')$ if and only if $\mathrm{dist}_G(u, v) \leq 5$ for some $u \in A_0$ . Since $|N_G^{k-1}(A_0)| \leq (d+1)|N_G^k(A_0)|$ for $k \geq 1$ , this implies $|V(G')| \leq (d+1)^4|A_0| = O(bd^3)$ .

By (8), the min-cut size of $G$ is at least $d$ (and in fact exactly $d$ since $G$ is $d$ -regular). Contracting edges does not decrease the min-cut size of a graph, so the min-cut size of $G'$ is also at least $d$ .

Thus, the number of possible boundaries $B$ is upper bounded by the number of cuts of size at most $b$ in $G'$ , which by Theorem 13 is at most

\begin{equation*} \binom {O(bd^3)}{\leq 2b/d} \cdot 2^{2b/d} = d^{O(b/d)}. \end{equation*}

We are done, as we can uniquely determine $A$ from its boundary.

4.2 Colourings with small colour classes: Proof of Lemma 8

Let us now turn to Lemma 8. For a colouring $\sigma$ , let us use $\textrm{nm}\!(\sigma ) = \textrm{nm}\!(G,\sigma )$ to denote the number of non-monochromatic edges in $\sigma$ ; observe that $m(G,\sigma ) = |E(G)| - \textrm{nm}\!(\sigma )$ . Recall that $S_0$ is the set of colourings in which each colour occupies at most $n/2$ vertices. The following lemma is the main point of this section.

Lemma 16. Let $G$ be as in Theorem 1. For $k \geq d$ , let $\mathcal{L}_k$ denote the number of $q$ -colorings from $S$ which give exactly $k$ non-monochromatic edges. Then for $\delta \gt \frac{12}{\ln d}$ ,

\begin{equation*} \mathcal {L}_k \leq n^4 \cdot q^{\frac {(2+\delta )k}{d}-\Omega \left (\frac {\delta k}{d}\right )}. \end{equation*}

We would like to briefly comment on parameter $\delta$ . If one is interested in a combinatorial bound, one could simply plug in the ‘best possible’ value (i.e., $\delta = 12/\ln d$ ). We allow for a choice in the parameter $\delta$ solely to make the subsequent calculations slightly easier.

Before using this to prove Lemma 8, we will need the following.

Proof. Let $A_i=\sigma ^{-1}(i)$ be the set of vertices which get colour $i$ under $\sigma$ , which by the fact that $\sigma \in S_0$ satisfies $|A_i|\le n/2$ . The non-monochromatic edges of $G$ under $\sigma$ are precisely those that appear as $\nabla (A_i)$ for two distinct choices of $i\in [q]$ . This gives

\begin{equation*} 2\operatorname {nm}(\sigma ) = \sum _{i \in [q]}|\nabla (A_i)| \ge \eta \sum _{i\in [q]}|A_i| = \eta n. \end{equation*}

We are now ready to prove Lemma 8.

Proof of Lemma 8. We start off with the easy lower bound $Z_{G}(q,\beta ) \geq q \cdot e^{\beta |E(G)|}$ obtained by considering only the colourings that give every vertex the same colour. So for $\beta \geq (2+\epsilon )\frac{\ln q}{d}$ , we have

\begin{align*} \frac{\sum \limits _{\sigma \in S_0}e^{\beta \left (|E(G)| - \textrm{nm}\!(\sigma )\right )}}{Z_{G}(q,\beta )} &\le \frac{\sum \limits _{k \geq \eta n/2}\sum \limits _{\substack{\sigma \in S_0 \\ \textrm{nm}\!(\sigma ) = k}}e^{\beta (|E(G)| - k)}}{q \cdot e^{\beta |E(G)|}} \\[5pt] & \leq \frac{1}{q}\sum _{k \geq \eta n/2}\sum _{\substack{\sigma \in S_0 \\ \textrm{nm}\!(\sigma ) = k}}q^{-(2+\epsilon )\frac{k}{d}} \\[5pt] & \leq \frac{n^4}{q}\sum _{k \geq \eta n/2}q^{-\Omega \left (\frac{\epsilon k}{d}\right )} \end{align*}

where the first inequality uses Lemma 17 and the last inequality uses Lemma 16 with $\delta = \epsilon$ (note that we already assume that $d \geq \exp \left (\Omega (1/\epsilon )\right )$ ). If $d \geq \sqrt{n}$ , then $\frac{n^4}{q} \lt 1$ . Otherwise, $n^4 = q^{o\left (\frac{\epsilon k}{d}\right )}$ . In either case,

\begin{equation*} \frac {n^4}{q}\sum _{k \geq \eta n/2}q^{-\Omega \left (\frac {\epsilon k}{d}\right )} \leq q^{-\Omega \left (\frac {\epsilon n}{d}\right )}. \end{equation*}

One ingredient in the proof of Lemma 16 is the following, which requires the same techniques as in the proof of Theorem 13.

Proof. For part 1: Fix any $q$ -coloring $\sigma$ with exactly $\ell$ non-monochromatic edges and run the following algorithm on $G$ .

1. I. While there are more than $\frac{2\ell }{d}$ vertices, choose a uniformly random edge and contract it. Delete self-loops, if any, after each contraction.

2. II. Colour every vertex uniformly and independently from $[q]$ . Output the final graph $G'$ and colouring $\sigma '$ .

Observe that $\sigma '$ naturally corresponds to a $q$ -coloring $\sigma$ in the original graph defined by $\sigma (v) = \sigma '(v')$ where $v'$ is the vertex in $G'$ into which $v$ was contracted.

Recall that $G$ has a min-cut of size $d$ by (8), and it does not decrease with the contraction operation during step I. So at any point in this step, if the current graph has $m$ vertices, then it has at least $\frac{md}{2}$ edges.

Moreover, each contraction reduces the number of vertices by $1$ , and all deleted self-loops must be monochromatic. So the probability that the $\ell$ non-monochromatic edges remain uncontracted in $G'$ is at least

\begin{equation*} \left (1-\frac {2\ell }{nd}\right )\left (1 - \frac {2\ell }{(n-1)d}\right )\cdots \frac {d}{2\ell } = \binom {n}{2\ell/d}^{-1}. \end{equation*}

During step II, the probability that every contracted vertex gets the correct colour is $q^{-\frac{2\ell }{d}}$ . Therefore, the probability that $\sigma$ was recovered by this procedure is at least $\binom{n}{2\ell/d}^{-1}q^{-\frac{2\ell }{d}}$ . Since this holds for any $\sigma$ , there are at most $\binom{n}{2\ell/d}q^{\frac{2\ell }{d}}$ many $q$ -colorings with $\ell$ non-monochromatic edges.

For part 2: Fix any $q$ -coloring $\sigma$ with at least $\ell$ non-monochromatic edges where each monochromatic component has at least $2$ vertices, and consider the following algorithm.

1. I. While there are more than $\frac{2\ell }{d}$ vertices, choose a uniformly random edge and contract it. Delete self-loops, if any, after each contraction.

2. II. While a vertex in this contracted graph has degree $d$ , contract it with one of its neighbours uniformly at random, deleting self-loops at each stage.

3. III. While there are more than $\frac{2\ell }{2d - 2}$ vertices, choose a uniformly random edge and contract it. Delete self-loops, if any, after each contraction.

4. IV. Colour every vertex uniformly and independently at random from $[q]$ .

Again, the final colouring naturally corresponds to a $q$ -coloring in $G$ . Step I is analysed in an identical manner as $(1)$ . The probability that the $\ell$ non-monochromatic edges remain uncontracted at the end of this step is at least $\binom{n}{2\ell/d}^{-1}$ .

Since every monochromatic component in $\sigma$ has at least two vertices, every uncontracted vertex at the end of step I is incident to at least one monochromatic edge. So during step II, the probability that the $\ell$ non-monochromatic edges remain uncontracted is at most $d^{-\frac{2\ell }{d}}$ .

After the end of step II, (7) guarantees that the min-cut is at least $2d-2$ , and this does not decrease with the contraction operation in step III. So, using a calculation similar to that in the analysis of step I, the probability that the $\ell$ non-monochromatic edges remain uncontracted at the end of this step is at least $2^{-\frac{2\ell }{d}}$ .

During step IV, the probability that every contracted vertex gets the correct colour is $q^{-\frac{2\ell }{2d-2}}$ . Thus, $\sigma$ is recovered with probability at least $\binom{n}{2\ell/d}^{-1}q^{-\frac{2\ell }{(2d-2)}}(2d)^{-\frac{2\ell }{d}}$ . Since this holds for any $\sigma$ , we have that there are at most $\binom{n}{2\ell/d}q^{\frac{2\ell }{(2d-2)}}(2d)^{\frac{2\ell }{d}}$ many $q$ -colorings with $\ell$ non-monochromatic edges and each monochromatic component having size at least $2$ .

One can check by hand that our bound is not tight in several cases, such as when $s = 1$ and $\ell = d$ . Indeed, the original algorithm by Karger for min-cuts gives a reasonable tight lower bound on the number of near-minimum cuts when $G$ is a cycle – a case that is not included in the class of graphs under consideration for Theorem 1. However, recent work obtaining a more fine-grained understanding of the Karger process may be adaptable to our setting of $q$ -colorings; we refer the interested reader to [Reference Gupta, Harris, Lee and Li15].

We are now ready to prove Lemma 16. Recall that $\mathcal{L}_k$ is the number of $q$ -colorings from $S_0$ which induce exactly $k$ non-monochromatic edges, and we want to show that

\begin{equation*}\mathcal {L}_k \leq n^4 \cdot q^{\frac {(2+\delta )k}{d}-\Omega \left (\frac {\delta k}{d}\right )}. \end{equation*}

Proof of Lemma 16. Case 1: $k \leq nd^{1 - \frac{\delta }{4}}$ . For a subset $A \subset V$ , let $E(A)$ denote the set of edges induced by $A$ . Consider a partition of the vertices of $G$ into $L$ monochromatic components. Let $T$ be the set of components containing exactly one vertex. Let $N$ be the set of non-monochromatic edges. We then have the partition

\begin{equation*} N = E(T) \sqcup \nabla (T) \sqcup \left (N \cap E\left (\overline {T}\right )\right ). \end{equation*}

Let $e_T \,{:\!=}\, |E(T)|$ , $b_T \,{:\!=}\, |\nabla (T)|$ , and $\ell \,{:\!=}\, \left |N\cap E (\overline{T})\right | = k - e_T - b_T$ . We have that $|T| \leq \frac{2k}{d} \leq nd^{-\delta } \lt n/3$ and each monochromatic component has size at most $n/2$ . So, one can choose some $S \subseteq V$ such that $n/3 \leq |S|\leq 2n/3$ by greedily including all vertices from the smallest remaining monochromatic component. In particular, $S \supseteq T$ . The expansion condition (3) then implies that $|\nabla (S)| \geq n/3$ . However, $\nabla (S) \subseteq \nabla (T) \cup (N \cap E(\overline{T}))$ , and so $|\nabla (S)| \leq b_T + (k - e_T - b_T) = k - e_T = \ell + b_T$ . As a result, we have

(20) \begin{equation} b_T + \ell = \Omega (n). \end{equation}

A $q$ -coloring counted by $\mathcal{L}_k$ can be chosen by (I) first choosing the cut $\left (T, \overline{T}\right )$ , (II) giving each element on the $T$ -side of the cut a colour, and (III) choosing the monochromatic components in $V \setminus T$ so that each component has at least two vertices.

Theorem 13 gives us that (I) can be done in at most $\binom{n}{2b_T/d} \cdot 2^{2b_T/d}$ ways. We also have that (II) can be done in at most $q^{|T|}$ ways. Part 2 of Lemma 18 gives us that (III) can be done in at most $\binom{n}{2\ell/d}\cdot q^{\frac{2\ell }{2d-2}}\cdot (2d)^{\frac{2\ell }{d}}$ ways. So, the number of $q$ -colorings of $V$ with the above parameters is at most

\begin{align*} Q & \,{:\!=}\,\binom{n}{2b_T/d} \cdot 2^{2b_T/d} \cdot q^{|T|} \cdot \binom{n}{2\ell/d}\cdot q^{\frac{2\ell }{2d-2}} \cdot (2d)^{\frac{2\ell }{d}} \\[5pt] & = \binom{n}{2b_T/d}\cdot \binom{n}{2\ell/d}\cdot q^{\frac{\ell + 2e_T + b_T}{d}} \cdot q^{\frac{\ell }{d^2}} \cdot (2d)^{\frac{2\ell }{d}}, \\[5pt] \end{align*}

where we have used $d|T| = 2e_T + b_T$ . So we have

\begin{align*} Q \cdot q^{-\frac{2k}{d}} & = \binom{n}{2\ell/d}\binom{n}{2b_T/d}\cdot q^{\frac{-b_T - \ell }{d}} \cdot q^{\frac{\ell }{d^2}}\cdot (2d)^{\frac{2\ell }{d}} \\[5pt] & \leq \binom{n}{2(\ell + b_T)/d} \cdot 2^{\frac{4(b_T + \ell )}{d}} q^{\frac{-b_T - \ell }{d}} \cdot q^{\frac{\ell }{d^2}} \cdot (2d)^{\frac{2\ell }{d}} \\[5pt] & \leq d^{O\left (\frac{\ell + b_T}{d}\right )}q^{-\frac{\ell + b_T}{d}} \cdot 2^{\frac{4(b_T + \ell )}{d}}q^{\frac{\ell }{d^2}} \\[5pt] & \leq q^{-\Omega \left (\frac{n}{d}\right )}, \end{align*}

where the first inequality used the fact that for $a+b \leq x$ , we have $\binom{x}{a}\binom{x}{b} \leq \binom{x}{a+b}4^{a+b}$ , and the next two inequalities use (20). So summing over all values for $\ell$ and $b_T$ gives us that for $k \leq nd^{1 - \frac{\delta }{4}}$ ,

\begin{align*} \mathcal{L}_k \leq n^4 \cdot Q \leq n^4 \cdot q^{\frac{2k}{d} - \Omega \left (\frac{n}{d}\right )}. \end{align*}

Case 2: $k \geq nd^{1 - \frac{\delta }{4}}$ . This case is relatively straightforward. Using part 1 of Lemma 18, we have

\begin{equation*} \mathcal {L}_k \leq \binom {n}{2k/d}q^{\frac {2k}{d}} \leq (ed)^{\frac {2\delta k}{3d}} \cdot q^{\frac {2 k}{d}} = q^{-\frac {(2 + \delta )k}{d} - \Omega \left (\frac {\delta k}{d}\right )}. \end{equation*}

5.1 General graphs and cliques

Theorem 19 (Sah et al. [Reference Sah, Sawhney, Stoner and Zhao40], Theorem 1.14 restated). Let $G$ be a graph, $q\ge 2$ be an integer and $\beta \ge 0$ . Then

\begin{equation*} Z_G(q,\beta ) \le \prod _{v\in V(G)} Z_{K_{d_v+1}}(q,\beta )^{\frac {1}{d_v+1}}, \end{equation*}

where $d_v$ is the degree of a vertex $v\in V(G)$ .

We briefly remark that their result was much more general; [Reference Sah, Sawhney, Stoner and Zhao40] shows that in fact every ferromagnetic model (using a definition introduced in [Reference Galanis, Štefankovič and Vigoda18] as a generalisation of the ferromagnetic Potts model) is ‘clique-maximizing’, which translates to the above statement on the level of partition functions.

Corollary 20. Let $G$ be a graph of maximum degree $\Delta$ with $n$ vertices and $m$ edges. Then for any integer $q\ge 2$ and $\beta \ge 0$ ,

\begin{equation*} Z_G(q,\beta ) \le q^{n-\frac {2m}{\Delta }} Z_{K_{\Delta +1}}(q,\beta )^{\frac {2m}{\Delta (\Delta +1)}}. \end{equation*}

5.2 Triangle-free graphs and bicliques

The upper bound in Theorem 19 can be improved in the case that $G$ is triangle-free.

Theorem 21 (Sah et al. [Reference Sah, Sawhney, Stoner and Zhao40], Theorem 1.9 and the subsequent remark). Let $G$ be a triangle-free graph with no isolated vertices, $q\ge 2$ be an integer and $\beta \ge 0$ . Then

\begin{equation*} Z_G(q,\beta ) \le \prod _{uv\in E(G)} Z_{K_{d_u,d_v}}(q,\beta )^{\frac {1}{d_u d_v}}, \end{equation*}

where $d_u$ is the degree of a vertex $u\in V(G)$ .

Corollary 22. Let $G$ be a triangle-free graph of maximum degree $\Delta$ with $n$ vertices and $m$ edges. Then for any integer $q\ge 2$ and $\beta \ge 0$ ,

\begin{equation*} Z_G(q,\beta ) \le q^{n-\frac {2m}{\Delta }} Z_{K_{\Delta,\Delta }}(q,\beta )^{\frac {m}{\Delta ^2}}. \end{equation*}

Proof. Let $G$ have $t$ isolated vertices. Then by Theorem 21 we have

\begin{equation*} Z_G(q,\beta ) \le q^t \prod _{uv\in E(G)} Z_{K_{d_u,d_v}}(q,\beta )^{\frac {1}{d_u d_v}}. \end{equation*}

The desired result is now a consequence of the fact that for $1\le a\le c$ and $1\le b\le d$ we have

(21) \begin{equation} q^{-\left (\frac{1}{a}+\frac{1}{b}\right )} Z_{K_{a, b}}(q,\beta )^{\frac{1}{a b}} \le q^{-\left (\frac{1}{c}+\frac{1}{d}\right )} Z_{K_{c,d}}(q,\beta )^{\frac{1}{c b}}. \end{equation}

To see this, apply the inequality to each term of the product and collect the factors of $q$ to obtain

\begin{equation*} Z_G(q,\beta ) \le q^{t + \sum _{uv\in E(G)}\left (\frac {1}{d_u}+\frac {1}{d_v}\right ) - \frac {2m}{\Delta }}Z_{K_{\Delta,\Delta }}(q,\beta )^{\frac {m}{\Delta ^2}}. \end{equation*}

Some simple counting gives

\begin{equation*} \sum _{uv\in E(G)}\left (\frac {1}{d_u}+\frac {1}{d_v}\right ) = n - t \end{equation*}

since each vertex $u$ appears as the endpoint of precisely $d_u$ edges, and the result follows.

It remains to prove inequality (21), which follows from Hölder’s inequality and Jensen’s inequality as notedFootnote 2 in [Reference Sah, Sawhney, Stoner and Zhao40]. For concreteness, we observe that inequality (21) is a direct consequence of a generalised Hölder inequality stated as Theorem 3.1 in [Reference Lubetzky and Zhao32]. We can write $\Omega =[q]$ , and let $\mu$ be the uniform probability measure on $\Omega$ and $\mu ^a$ be the uniform probability measure on $\Omega ^a$ such that for the non-negative functions $f\,{:}\,\Omega ^2\to \mathbb{R}$ and $g\,{:}\,\Omega ^a\to \mathbb{R}$ given by

\begin{align*} f(x,y) &= \begin{cases}e^\beta & x = y \\[5pt] 1 & \text{otherwise},\end{cases} \\[5pt] g(x_1,\dotsc,x_a) &= \int _{\Omega } \prod _{i=1}^a f(x_i,y) \,\mathrm d\mu. \end{align*}

Then to prove (21) in this notation, we have

\begin{equation*} \mathrm {LHS} \,{:\!=}\, q^{-\left (\frac {1}{a}+\frac {1}{b}\right )} Z_{K_{a, b}}(q,\beta )^{\frac {1}{a b}} = \left (\int _{[q]^a}|g|^b\,\mathrm d\mu ^a\right )^{\frac {1}{ab}}. \end{equation*}

When $b\le d$ we can apply the generalised Hölder inequality (or in this somewhat special case iterated applications of the usual Hölder inequality followed by Jensen’s inequality) to the integral over $[q]^a$ above to obtain the upper bound

\begin{equation*} \mathrm {LHS} \le \left (\int _{\Omega ^a} g^d \,\mathrm d\mu ^a \right )^{\frac {1}{ad}} = q^{-\left (\frac {1}{a}+\frac {1}{d}\right )} Z_{K_{a, d}}(q,\beta )^{\frac {1}{a d}}. \end{equation*}

A symmetric application of this argument for any integer $c$ such that $a\le c$ gives inequality (21).

Proof. We will use the fact that both $K_{d+1}$ and $K_{d,d}$ are $\eta$ -expanders with $\eta = d/2$ . We will state the proof for $K_{d+1}$ , and proof for $K_{d,d}$ follows similarly. Let $\sigma$ be any colouring of the vertices of $K_d$ , and let $c(\sigma )$ denote the number of colours that appear in $\sigma$ .

We first observe that the proof of Lemma 16 also extends to all $\sigma$ such that $c(\sigma ) \geq 2$ . Indeed, the only place where $S$ was used was to establish Lemma 17, and (20). Both of these also hold for $K_{d+1}$ whenever there are at least two colours. So we have

\begin{align*} \frac{\sum _{\sigma :c(\sigma ) \geq 2}e^{\beta \left (\binom{d+1}{2} - \textrm{nm}\!(\sigma )\right )}}{q \cdot e^{\beta \binom{d+1}{2}}} & \leq \frac{1}{q}\sum _{k \geq d}\sum _{\substack{\sigma :c(\sigma ) \geq 2 \\ \textrm{nm}\!(\sigma ) = k}}q^{-(2+\epsilon )\frac{k}{d}} \\[5pt] & \leq \frac{d^4}{q}\sum _{k \geq d}q^{-\Omega \left (\frac{\epsilon k}{d}\right )} \\[5pt] & \leq q^{-\Omega \left (\epsilon \right )}. \end{align*}

The proofs of Lemmas 11 and 12 are now straightforward calculations.

Proof of Lemma 11. By Corollary 20 and Lemma 23,

\begin{align*} Z_{G[\gamma ]}(q-1, \beta ) &\leq (q-1)^{v_\gamma -\frac{2e_\gamma }{d}} Z_{K_{d+1}}(q-1, \beta )^{\frac{2e_\gamma }{d(d+1)}}\\[5pt] &\leq (q-1)^{v_\gamma -\frac{2e_\gamma }{d}}\left (2(q-1) e^{\beta \binom{d+1}{2}}\right )^{\frac{2e_\gamma }{d(d+1)}}\\[5pt] &\lt q^{v_\gamma - \frac{2e_\gamma }{d}+\frac{e_\gamma }{\binom{d+1}{2}}} e^{\beta e_\gamma }2^{\frac{e_\gamma }{\binom{d+1}{2}}}. \end{align*}

Proof of Lemma 12. By Corollary 22 and Lemma 23,

\begin{align*} Z_{G[\gamma ]}(q-1, \beta )&\leq (q-1)^{v_\gamma - \frac{2e_\gamma }{d}}Z_{K_{d,d}}(q-1,\beta )^{\frac{e_\gamma }{d^2}}\\[5pt] &\leq (q-1)^{v_\gamma - \frac{2e_\gamma }{d}} \cdot (2(q-1)e^{\beta d^2})^{\frac{e_\gamma }{d^2}}\\[5pt] &\leq q^{v_\gamma - \frac{2e_\gamma }{d}+\frac{e_\gamma }{d^2}} \cdot 2^{\frac{e_\gamma }{d^2}} \cdot e^{\beta e_\gamma }. \end{align*}