2.1 Essential saddle characterization

Our results give insight into the way the transitions between ${\mathbf{e}}$ and ${\mathbf{o}}$ most likely occur in the low-temperature regime. This is usually described by identifying the optimal paths, saddles, and essential saddles that we define as follows.

• • $ \mathcal {S} ({\mathbf{e}}, {\mathbf{o}})$ is the communication level set between ${\mathbf{e}}$ and ${\mathbf{o}}$ defined by

  \begin{equation*} \mathcal {S} ({\mathbf{e}}, {\mathbf{o}}) \,:\!=\, \{ \sigma \in \mathcal {X} \,|\, \exists \, \omega \in ({\mathbf{e}} \to {\mathbf{o}})_{\mathrm {opt}}, \,:\, \sigma \in \omega \text { and } H(\sigma ) = \Phi _\omega = \Phi ({\mathbf{e}},{\mathbf{o}})\}, \end{equation*}
  where $({\mathbf{e}} \to {\mathbf{o}})_{\mathrm {opt}}$ is the set of optimal paths from ${\mathbf{e}}$ to ${\mathbf{o}}$ realizing the minimax in $\Phi ({\mathbf{e}}, {\mathbf{o}})$ , i.e.,
  \begin{equation*} ({\mathbf{e}} \to {\mathbf{o}})_{\mathrm {opt}} \,:\!=\, \{ \omega \,:\, {\mathbf{e}} \to {\mathbf{o}} \,|\, \Phi _\omega = \Phi ({\mathbf{e}},{\mathbf{o}})\}. \end{equation*}

• • The configurations in $\mathcal {S}({\mathbf{e}},{\mathbf{o}})$ are called saddles. Given an optimal path $\omega \in ({\mathbf{e}} \to {\mathbf{o}})_{\mathrm {opt}}$ , we define the set of its saddles $S(\omega )$ as $ S(\omega )\,:\!=\,\{ \sigma \in \omega \,|\, H(\sigma ) = \Phi _\omega = \Phi ({\mathbf{e}},{\mathbf{o}})\}.$ A saddle $\sigma \in \mathcal {S}({\mathbf{e}},{\mathbf{o}})$ is called essential if either

  1. (i) $\exists \, \omega \in ({\mathbf{e}} \to {\mathbf{o}})_{\mathrm {opt}}$ such that $S(\omega ) = \{\sigma \}$ , or

  2. (ii) $\exists \, \omega \in ({\mathbf{e}} \to {\mathbf{o}})_{\mathrm {opt}}$ such that $\sigma \in S(\omega )$ and $S(\omega ^{\prime}) \not \subseteq S(\omega ) \setminus \{\sigma \} \quad \forall \, \omega ^{\prime} \in ({\mathbf{e}} \to {\mathbf{o}})_{\mathrm {opt}}$ .

  A saddle $\sigma \in \mathcal {S}({\mathbf{e}},{\mathbf{o}})$ that is not essential is called unessential saddle or dead-end, i.e., for any $\omega \in ({\mathbf{e}} \to {\mathbf{o}})_{\mathrm {opt}}$ such that $\omega \cap \{\sigma \}\neq \emptyset$ we have that $S(\omega )\setminus \{\sigma \}\neq \emptyset$ and there exists $\omega ^{\prime} \in ({\mathbf{e}} \to {\mathbf{o}})_{\mathrm {opt}}$ such that $S(\omega ^{\prime})\subseteq S(\omega )\setminus \{\sigma \}$ .

• • The essential gate $\mathcal {G}({\mathbf{e}}, {\mathbf{o}})\subset \mathcal {X}$ is the collection of essential saddles for the transition ${\mathbf{e}} \to {\mathbf{o}}$ .

The aim of the present paper is to accurately identify the set $\mathcal {G}({\mathbf{e}},{\mathbf{o}})$ of essential saddles for the transition from ${\mathbf{e}}$ to ${\mathbf{o}}$ for the Metropolis dynamics of the hard-core model on a $L \times L$ grid with periodic boundary conditions.

Figure 2. An example of a configuration in $\mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})$ (on the left) and one in $\mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ (on the right).

Figure 3. An example of a configuration in $\mathcal {C}_{cr}({\mathbf{e}},{\mathbf{o}})$ (on the left) and one in $\mathcal {C}_{sb}({\mathbf{e}},{\mathbf{o}})$ (on the right).

Figure 4. An example of a configuration in $\mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ (on the left) and one in $\mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ (on the right).

The set $\mathcal {G}({\mathbf{e}},{\mathbf{o}})$ will be described as the union of six disjoint sets, each characterized by configurations with specific geometrical features. Although we refer the reader to section 4 for a precise definition of these sets (cf. Definitions 4.2–4.6), we provide here some intuitive descriptions of the geometrical features of the configurations in these sets. We denote by

• • $\mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})$ , $\mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ , and $\mathcal {C}_{cr}({\mathbf{e}},{\mathbf{o}})$ the collections of configurations with a unique cluster of particles in odd sites of rhomboidal shape with exactly two adjacent even empty sites as in Figure 2 and Figure 3 (left). Roughly speaking, $\mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})$ contains the configurations with $(\frac {L}{2}-1)^2$ occupied odd particles and $L^2+2$ empty even sites; $\mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ (resp. $\mathcal {C}_{cr}({\mathbf{e}},{\mathbf{o}})$ ) contains the configurations obtained from $\mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})$ (resp. $\mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ ) by removing some occupied even sites attached to the rhombus and growing along one (resp. the longest) side by adding some particles in the nearest odd sites of the rhombus.

• • $\mathcal {C}_{sb}({\mathbf{e}},{\mathbf{o}})$ , $\mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ , and $\mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ the collections of configurations with a unique cluster of particles at odd sites with at most two additional empty even sites as in Figure 3 (right) and in Figure 4. In particular, $\mathcal {C}_{sb}({\mathbf{e}},{\mathbf{o}})$ contains the configurations with $\frac {L}{2}-1$ particles arranged in an odd column with two other empty even sites; $\mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ contains the configurations obtained from $\mathcal {C}_{sb}({\mathbf{e}},{\mathbf{o}})$ such that there is at least one column or row with $\frac {L}{2}$ particles arranged in odd sites. $\mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ contains the configurations obtained from $\mathcal {C}_{sb}({\mathbf{e}},{\mathbf{o}})$ without having column or row with $\frac {L}{2}$ particles.

The following theorem characterizes the essential gate for the transition from ${\mathbf{e}}$ to ${\mathbf{o}}$ .

Theorem 2.1 (Essential saddles). Define the set

\begin{equation*} \mathcal {C}^*({\mathbf{e}},{\mathbf{o}})\,:\!=\,\mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})\cup \mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})\cup \mathcal {C}_{cr}({\mathbf{e}},{\mathbf{o}})\cup \mathcal {C}_{sb}({\mathbf{e}},{\mathbf{o}})\cup \mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})\cup \mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}}). \end{equation*}

The essential saddles for the transition from ${\mathbf{e}}$ to ${\mathbf{o}}$ of the hard-core model on a $L \times L$ toric grid graph $\Lambda$ are all and only the configurations in $\mathcal C^*({\mathbf{e}},{\mathbf{o}})$ , i.e.,

\begin{equation*} \mathcal G({\mathbf{e}},{\mathbf{o}})=\mathcal C^*({\mathbf{e}},{\mathbf{o}}). \end{equation*}

Furthermore, the possible transitions at energy not higher than $-\frac {L^2}{2}+L+1$ among the six subsets that form $\mathcal C^*({\mathbf{e}},{\mathbf{o}})$ are as detailed in Figure 5.

3.1 Odd clusters and regions

For any subset of sites $S \subseteq V$ we define the complement of $S$ as $S^c\,:\!=\,V\setminus S$ , the external boundary $\partial ^+ S$ as the subset of sites in $S^c$ that are adjacent to a site in $S$ , i.e.,

\begin{equation*} \partial ^+ S \,:\!=\, \{ v \in S^c \,|\, \exists \, w \in S \,:\, (v,w) \in E \}, \end{equation*}

and $\nabla S$ as the subset of edges connecting the sites in $S$ with those in $\partial ^+ S$ , i.e.,

\begin{equation*} \nabla S\,:\!=\,\{(v,w) \in E \,|\, v \in S, \, w \in \partial ^+ S\}. \end{equation*}

A (connected) odd cluster $C \subseteq V$ is a subset of sites that satisfies both the following conditions:

1. 1. If an odd site $v \in V_{\mathbf{e}}$ belongs to $C$ , then so do the four neighbouring even sites, i.e., $\partial ^+ \{v\} \subset C$ ;

2. 2. $C \cap V_{\mathbf{e}}$ is connected as a sub-graph of the graph $(V_{\mathbf{e}}, E^*)$ , with $E^*\,:\!=\,\{(v,w) \in V_{\mathbf{e}} \times V_{\mathbf{e}} \,|\, d(v,w)=2\}$ , where $d(\cdot, \cdot )$ denotes the usual graph distance on $\Lambda$ .

We denote by $\mathrm {C}_{\mathbf{o}}(\Lambda )$ the collection of odd clusters in $\Lambda$ .

Consider the dual graph $\Lambda ^{\prime}=(V^{\prime},E^{\prime})$ of the graph $\Lambda$ , which is a discrete torus of the same size. Given an odd cluster $C$ , consider the edge set $\nabla C$ that disconnects $C$ from its complement $C^c$ . We associate with $\nabla C$ the edge set $\gamma (C) \subset E^{\prime}$ on the dual graph $\Lambda ^{\prime}$ which consists of all the edges of $\Lambda ^{\prime}$ orthogonal to edges in $\nabla C$ . Such a set, to which we refer as the contour of the cluster $C$ , consists of one or more piecewise linear closed curves and, by construction, $|\gamma (C)|=|\nabla C|$ . Leveraging this fact, we define the perimeter $P(C)$ of the odd cluster $C$ as the total length of the contour $\gamma (C)$ , i.e.,

(3.2) \begin{align} P(C)\,:\!=\,|\gamma (C)|. \end{align}

As proved in [Reference Zocca, Borst, van Leeuwaarden and Nardi57], the perimeter of the odd cluster $C$ satisfies the following identity:

(3.3) \begin{align} P(C)=4(|C \cap V_{\mathbf{e}}|-|C \cap V_{\mathbf{o}}|). \end{align}

We call area of an odd cluster $C$ the number of odd occupied sites it comprises. We say that an odd cluster is degenerate if it has area 0 and non-degenerate otherwise.

We introduce a mapping $\mathcal {O}\,:\, \mathcal {X} \to 2^V$ that associates to a given hard-core configuration $\sigma \in \mathcal {X}$ the subset $\mathcal {O}(\sigma ) \subseteq V$ defined as

(3.4) \begin{align} \mathcal {O}(\sigma )\,:\!=\,\{ v \in V_{\mathbf{o}} \,|\, \sigma (v) =1 \} \cup \{ v \in V_{\mathbf{e}} \,|\, \sigma (v) =0 \}. \end{align}

In other words, $\mathcal {O}(\sigma )$ is the subset comprising all occupied odd sites and even empty sites of the configuration $\sigma$ . It is immediate to check that $\mathcal {O}$ is an injective mapping, and we will refer to the image $\mathcal {O}(\sigma )$ of a configuration $\sigma$ as its odd region.

The odd region $\mathcal {O}(\sigma )$ of a configuration $\sigma \in \mathcal {X}$ can be partitioned into its connected components, say $C_1(\sigma ), \ldots, C_m(\sigma ) \in \mathrm {C}_{\mathbf{o}}(\Lambda )$ , for some $m \in \mathbb N$ , which are, by definition, odd clusters, that is,

(3.5) \begin{align} \mathcal {O}(\sigma ) = \bigsqcup _{i=1}^m C_i(\sigma ). \end{align}

Using the partition (3.5) of the odd region $\mathcal {O}(\sigma )$ into odd clusters, the definitions of contour and perimeter can be extended to the whole odd region in an obvious way, so that we can ultimately define the contour $\gamma (\sigma )$ of a configuration $\sigma \in \mathcal {X}$ as

(3.6) \begin{align} \gamma (\sigma )\,:\!=\,\bigsqcup _{i=1}^m \gamma (C_i(\sigma )), \end{align}

and its perimeter $P(\sigma )$ as

(3.7) \begin{align} P(\sigma )\,:\!=\,\sum _{i=1}^m P(C_i(\sigma )). \end{align}

As shown in [Reference Zocca, Borst, van Leeuwaarden and Nardi57], starting from (3.3), a double counting argument yields the following identity that relates the perimeter $P(\sigma )$ of a hard-core configuration $ \sigma \in \mathcal {X}$ with its energy $H(\sigma )$ (recall (3.1)-(3.2))

(3.8) \begin{align} P(\sigma )= 4 \, \Delta H(\sigma ). \end{align}

Given a configuration $\sigma \in \mathcal {X}$ , we define the odd non-degenerate region $\mathcal {O}^{nd}(\sigma )$ as a subset of $\mathcal {O}(\sigma )$ containing only odd non-degenerate clusters. See Figure 7 for an example of an odd region and an odd non-degenerate region.

Figure 6. Example of four different rhombi, namely $\mathcal {R}_{1,2}$ , $\mathcal {R}_{4,2}$ , $\mathcal {R}_{0,2}$ and $\mathcal {R}_{1,0}$ (in clockwise order from the top-right corner).

3.2 Odd rhombi

Given an odd site $\eta =(\eta _1,\eta _2) \in V_{\mathbf{o}}$ and two positive integers $\ell _1,\ell _2\leq L$ , the odd rhombus $\mathcal {R}_{\ell _1,\ell _2}(\eta )$ with reference site $\eta$ and lengths $\ell _1$ and $\ell _2$ is the odd cluster defined as

(3.9) \begin{align} \mathcal {R}_{\ell _1,\ell _2}(\eta )\,:\!=\, S_{\ell _1,\ell _2}(\eta ) \cup \partial ^+ S_{\ell _1,\ell _2}(\eta ), \end{align}

where $S_{\ell _1,\ell _2}(\eta ) \subseteq V_{\mathbf{o}}$ is the subset of odd sites given by

(3.10) \begin{align} S_{\ell _1,\ell _2}(\eta ) &\,:\!=\, \bigcup _{0 \leq k \leq \ell _1-1, \, 0 \leq j \leq \ell _2-1} \{ (\eta _1 + k + j, \eta _2 + k - j )\} \nonumber \\[3pt] &= \{ v=(v_1,v_2) \in V \,|\, \exists \, k \in [[0,\ell _1]], \, j \in [[0,\ell _2]] \,:\, v_1 = \eta _1 + k + j, \, v_2 = \eta _2 + k - j\}. \end{align}

Figure 7. Example of a configuration $\sigma$ , in which the contour of the non-degenerate (degenerate) odd clusters is highlighted in black (red, respectively). The contour $\gamma (\sigma )$ of the configuration $\sigma$ is of the odd region $\mathcal {O}(\sigma )$ is the union of black lines (corresponding to $\mathcal {O}^{nd}(\sigma )$ ) and red lines.

Figure 8. Example of a odd cluster $C$ (on the left) and its surrounding rhombus $\mathcal {R}(C)=\mathcal {R}_{8,5}$ in red (on the right). On the left, the red squares contain the antiknobs and the decreasing broken diagonals are highlighted with blue rectangles. On the right, we highlight the decreasing shorter (resp. complete) diagonals with blue (resp. green) rectangles.

In the latter definition, the sums and subtractions of coordinates are taken modulo $L$ . In the case $\ell _1\ell _2=0$ , we can take $\eta \in V_{\mathbf{e}}$ and define the degenerate rhombus $\mathcal {R}_{\ell _1,\ell _2}(\eta )$ as the odd cluster

\begin{equation*} \mathcal {R}_{\ell _1,\ell _2}(\eta )= \begin{cases} \bigcup _{0 \leq j \leq \ell _2} \{ (\eta _1 + j, \eta _2 - j )\} &\hbox {if } \ell _1=0 \hbox { and } \ell _2\neq 0, \\[3pt] \bigcup _{0 \leq k \leq \ell _1} \{ (\eta _1 + k, \eta _2 - k )\} &\hbox {if } \ell _1\neq 0 \hbox { and } \ell _2=0, \\[3pt] (\eta _1,\eta _2) &\hbox {if } \ell _1=\ell _2=0. \end{cases} \end{equation*}

Note that, in this case, $\mathcal {R}_{\ell _1,\ell _2}(\eta )$ is a subset of even sites. The area of $\mathcal {R}_{\ell _1,\ell _2}(\eta )$ is the cardinality of $S_{\ell _1,\ell _2}(\eta )$ in the non-degenerate case, whereas in the degenerate case it is equal to zero. Some examples of rhombi and degenerate rhombi are shown in Figure 6. We observe that the non-degenerate rhombus $\mathcal {R}_{\ell _1,\ell _2}$ has $\ell _1$ diagonals of length $\ell _2$ and $\ell _2$ diagonals of length $\ell _1$ in the opposite direction, which we will refer to as complete diagonals. We denote by $\mathrm {R}_{\mathbf{o}}(\Lambda ) \subset \mathrm {C}_{\mathbf{o}}(\Lambda )$ the collection of all odd rhombi on $\Lambda$ including the degenerate ones. For every odd cluster $C \in \mathrm {C}_{\mathbf{o}}(\Lambda )$ , we define the surrounding rhombus $\mathcal {R}(C)$ as the minimal rhombus (by inclusion) in $\mathrm {R}_{\mathbf{o}}(\Lambda )$ such that $C \subseteq \mathcal {R}(C)$ ; see Figure 8 for an example.

Most of the results for odd rhombi that will be proved are translation-invariant, the reason why we will often refer to the rhombus $\mathcal {R}_{\ell _1,\ell _2}(\eta )$ simply as $\mathcal {R}_{\ell _1,\ell _2}$ , without explicitly specifying the reference site $\eta$ . The next two lemmas, Lemmas3.1 and 3.2, concern the properties of rhombi on a square $L \times L$ grid with periodic boundary conditions. Their proofs, being involved but not particularly insightful, are deferred to Appendix A.

For any subset of sites $A\subseteq V$ , we define the complement of $A$ as the complementary set of $A$ in $V$ , i.e., as $V \setminus A$ .

These two lemmas will now be used to prove the next proposition, which gives a formula for the perimeter of a rhombus $\mathcal {R}_{\ell _1,\ell _2}$ . To this end, we will use the fact that a rhombus $\mathcal {R}_{\ell _1,\ell _2}$ and its complement in $\Lambda$ have the same boundary for any $0\leq \ell _1,\ell _2\leq L$ , and, in particular the same perimeter. In addition, we will say that a rhombus $\mathcal {R}_{\ell _1,\ell _2}$ winds vertically (resp. horizontally) around the torus if there exists a set of $\frac {L}{2}$ odd sites $\eta _1,\ldots, \eta _{L/2}$ in $\mathcal {R}_{\ell _1,\ell _2}$ all on the same column (resp. row). If the direction is not relevant, we will simply say that the rhombus winds around the torus.

Proposition 3.3 (Formula for rhombus perimeter). Given a $L \times L$ toric grid graph $\Lambda$ and any sizes $0\leq \ell _1,\ell _2 \leq L$ , the perimeter of the rhombus $\mathcal {R}_{\ell _1,\ell _2}$ satisfies the following identity

(3.13) \begin{align} P (\mathcal {R}_{\ell _1,\ell _2}) = 4 \times \begin{cases} \ell _1 + \ell _2+1 & \text { if } \min \{\ell _1,\ell _2\} \lt L/2\ \text{and} \max \{\ell _1,\ell _2\} \lt L,\\[3pt] 2L-(\ell _1+\ell _2+1) & \text { if } \min \{\ell _1,\ell _2\} \geq L/2 \text { and } \max \{\ell _1,\ell _2\} \lt L,\\[3pt] L & \text { if } \min \{\ell _1,\ell _2 \} \lt L/2 \text { and } \max \{\ell _1,\ell _2\} =L, \\[3pt] 0 & \text { if } \min \{\ell _1,\ell _2 \} \geq L/2 \text { and } \max \{\ell _1,\ell _2\} =L. \end{cases} \end{align}

Proof. First of all, we identify which of the conditions in (3.13) imply that a rhombus winds around the torus. Consider the rhombus $\mathcal {R}_{\ell _1,\ell _2}(\eta )$ with $\eta =(\eta _1,\eta _2) \in V_{\mathbf{o}}$ . Let $\sigma =(\sigma _1,\sigma _2) \in S_{\ell _1,\ell _2}(\eta )$ be such that $\sigma _2=\eta _2$ and $d(\eta _1,\sigma _1)$ is the maximal distance along that horizontal axis. Similarly, let $\xi =(\xi _1,\xi _2) \in S_{\ell _1,\ell _2}(\eta )$ be such that $d(\eta _2,\xi _2)$ is the maximal distance along the vertical axis. Recalling (3.10), since $\sigma _2=\eta _2$ , we have $k=j$ for any $k,j=1,\ldots, \ell _{\min }-1$ , where $\ell _{\min }=\min \{\ell _1,\ell _2\}$ . Thus, we obtain

(3.14) \begin{align} \sigma _1=\eta _1+k+j=\eta _1+2(\ell _{\min }-1), \end{align}

where the last equality follows from the fact that the maximal distance along the horizontal axis is precisely the distance between $\sigma _1$ and $\eta _1$ . We note that if $d(\eta _1,\sigma _1) \lt L-2$ then the rhombus does not wind horizontally around the torus. Thus,

(3.15) \begin{align} d(\eta _1,\sigma _1)=2(\ell _{\min }-1)\lt L-2 \iff \ell _{\min }\lt \frac {L}{2}. \end{align}

Now, consider the distance between $\eta _2$ and $\xi _2$ . Let $\ell _{\max }=\max \{\ell _1,\ell _2\}$ . In this case, if $d(\eta _2,\xi _2) \leq L-2$ then the rhombus does not wind vertically around the torus. Thus, we have

(3.16) \begin{align} d(\eta _2,\xi _2)=\max _{k,j}|\eta _2-(\eta _2-k+j)|=\ell _{\max }-1 \leq L-2. \end{align}

We conclude that if $\ell _{\min }\lt \frac {L}{2}$ and $\ell _{\max }\lt L$ , then the rhombus does not wind around the torus and the perimeter is the length of its external boundary. In view of (3.3), the claim follows. Otherwise, there are three cases, which will be treated separately:

1. (a) $\ell _{\min } \geq \frac {L}{2}$ and $\ell _{\max } \leq L-2$ ;

2. (b) $\ell _{\min } \geq \frac {L}{2}$ and $\ell _{\max } \gt L-2$ ;

3. (c) $\ell _{\min }\lt \frac {L}{2}$ and $\ell _{\max }=L$ .

1. (a) Consider the complement of the rhombus $\mathcal {R}_{\ell _1,\ell _2}(\eta )$ for some $\eta$ . By virtue of Lemma 3.2(i), we know that its complement in $V$ is a rhombus with side lengths $\tilde \ell _1=L-\ell _1-1$ and $\tilde \ell _2=L-\ell _1-1$ . We claim that this complementary rhombus does not wind around the torus. Using condition $\ell _{\min } \geq \frac {L}{2}$ , we find that the maximal side length of the complementary rhombus is

   (3.17) \begin{align} \max \{\tilde \ell _1, \tilde \ell _2\}=\max \{L-\ell _1-1,L-\ell _2-1\}=L-\ell _{\min }-1 \leq L-\frac {L}{2}-1 \lt L, \end{align}
   that is, $\max \{\tilde \ell _1, \tilde \ell _2\} \lt L$ . Moreover, the minimal side length is
   (3.18) \begin{align} \min \{\tilde \ell _1, \tilde \ell _2\}=\min \{L-\ell _1-1,L-\ell _2-1\}=L-\ell _{\max }-1 \leq L-\ell _{\min }-1 \leq L-\frac {L}{2}-1, \end{align}
   that is, $\min \{\tilde \ell _1, \tilde \ell _2\}\lt \frac {L}{2}$ . Since the perimeter of the rhombus $R_{\ell _1,\ell _2}$ is the same as that of $R_{L-\ell _1-1,L-\ell _2-1}$ , the claim follows from (3.3).

2. (b) The claim follows from Lemma 3.2(ii)–(iii) and (3.3).

3. (c) The claim follows from Lemma 3.2(iv) and (3.3).

We say that an odd cluster $C$ is monotone when its perimeter coincides with that of its surrounding rhombus $\mathcal {R}(C)$ if it does not wind around the torus, i.e., $P(C)=P(\mathcal {R}(C))$ . Otherwise, we say that an odd cluster $C$ is monotone when its perimeter coincides with that of a bridge, i.e., $P(C)=4L$ . Note that it immediately follows that a monotone odd cluster $C$ has no holes, i.e., empty odd sites with the four even neighbouring sites belonging to $C$ . An empty odd site $\eta \notin C$ is an antiknob for the cluster $C$ if it has at least three neighbouring even empty sites that belong to $C$ . Figure 8 (left) highlights in red the antiknobs of a hard-core configuration.

Given an odd cluster $C\in C_{\mathbf{o}}(\Lambda )$ and an integer $k\geq 1$ , we say that $C$ displays an increasing (resp. decreasing) diagonal broken in $k$ sites if there exist a sequence of sites $z_i=(x_i,y_i)\in V_{\mathbf{o}}\setminus C$ , $i=1,\ldots, k$ , such that

• • $x_{i+1}=x_i+1$ and $y_{i+1}=y_i+1$ (resp. $x_{i+1}=x_i+1$ and $y_{i+1}=y_i-1$ ) for any $i=1,\ldots, k-1$ , and

• • the two odd sites $(x_1-1,y_1-1)$ and $(x_k+1,y_k+1)$ (resp. $(x_1-1,y_1+1)$ and $(x_k+1,y_k-1)$ ) belong to the cluster $C$ .

By construction of a broken diagonal, the two sites $z_1$ and $z_k$ are always antiknobs. If it does not matter if an increasing or decreasing diagonal is broken, we simply say that a diagonal is broken. Broken diagonals are visualized in blue in Figure 8 (left). Given an odd cluster $C\in C_{\mathbf{o}}(\Lambda )$ and an integer $k\geq 1$ , we say that $C$ displays an increasing (resp. decreasing) shorter diagonal lacking in $k$ sites if there exist a sequence of sites $z_i=(x_i,y_i)\in (V_{\mathbf{o}}\cap \mathcal {R}(C))\setminus C$ , $i=1,\ldots, k$ , such that

• • $x_{i+1}=x_i+1$ and $y_{i+1}=y_i+1$ (resp. $x_{i+1}=x_i+1$ and $y_{i+1}=y_i-1$ ) for any $i=1,\ldots, k-1$ , and

• • the two odd sites $(x_1-1,y_1-1)$ and $(x_k+1,y_k+1)$ (resp. $(x_1-1,y_1+1)$ and $(x_k+1,y_k-1)$ ) do not belong to $\mathcal {R}(C)$ .

Figure 8 (right) highlights the shorter diagonals in blue.

3.3 Expanding an odd cluster: the filling algorithms

We now describe an iterative procedure that builds a path $\omega$ in $\mathcal {X}$ from a configuration $\sigma$ with a unique odd cluster to another configuration $\sigma ^{\prime}$ that (i) displays a rhombus, and (ii) whose energy $H(\sigma ^{\prime})$ is equal to or lower than $H(\sigma )$ .

The path $\omega$ can be described as the concatenation of two paths, each obtained by means of a specific filling algorithm. The reason behind this name is that, along the generated paths, any incomplete diagonal in the odd cluster of the starting configuration is gradually filled by adding particles at odd sites until a rhombus is obtained.

The two paths can be intuitively described as follows. The first path, denoted as $\tilde \omega$ , starts from a configuration with at least one broken diagonal and, by filling one by one all broken diagonals in lexicographic order, arrives at a configuration without broken diagonals. Each broken diagonal is progressively filled by removing a particle in the even site at distance $1$ from an antiknob that lies on that diagonal and adding a particle in the odd site where the antiknob is. The antiknobs on the same diagonal are processed in lexicographic order. The second path, denoted as $\bar \omega$ , starts from a configuration without broken diagonals (such as any ending configuration of the path $\tilde \omega$ ) and arrives at a configuration displaying an odd rhombus. The construction of this second path is similar to that of the first path, but in this case the particles are added in odd sites to fill all the shorter diagonals.

The filling algorithms that generate the two paths $\tilde \omega$ and $\bar \omega$ are designed so that the maximum energy along the resulting path $\omega =\tilde \omega \cup \bar \omega$ , i.e., $\Phi _{\omega }$ , is never larger than $H(\sigma )+1$ . More specifically, the perimeter of the odd cluster either decreases or does not change along $\tilde \omega$ , whereas it can increase and sequentially decrease by the same amount along $\bar \omega$ . Proposition3.4, whose proof is postponed to subsection 5.1, specifies the requirement for the starting configuration and summarizes the properties of the path generated by the filling algorithm.

To formally define these two algorithms, we introduce the following notation. Given two configurations $\sigma, \sigma ^{\prime} \in \mathcal {X}$ and a subset of sites $W \subset \Lambda$ , we write $\sigma _{|W} = \sigma ^{\prime}_{|W}$ if $\sigma (v) = \sigma ^{\prime}(v)$ for every $v \in W$ . Given a configuration $\sigma \in \mathcal {X}$ , we let

• • $\sigma ^{(v,0)}$ be the configuration $\sigma ^{\prime} \in \mathcal {X}$ such that $\sigma ^{\prime}_{|V \setminus \{v\}} = \sigma _{|V \setminus \{v\}}$ and $\sigma ^{\prime}(v)=0$ ; and

• • $\sigma ^{(v,1)}$ be the configuration $\sigma ^{\prime}$ such that $\sigma ^{\prime}_{|V \setminus \{v\}} = \sigma _{|V \setminus \{v\}}$ and $\sigma ^{\prime}(v)=1$ .

In general, $\sigma ^{(v,1)}$ may not be a hard-core configuration in $\mathcal {X}$ , since $\sigma$ may already have a particle residing in one of the four neighbouring sites of $v$ .

Algorithm 1 (resp. Algorithm 2) provide the detailed pseudocode for the filling algorithm that yields $ \tilde{\omega}$ (resp. $ \tilde{\omega}$ ).

Algorithm 1: Filling algorithm to build path $ \tilde{\omega}$

Algorithm 2: Filling algorithm to build path $ \tilde{\omega}$

We note that conditions $(i),(ii)$ , and $(iii)$ mean that there is a unique odd cluster in $\sigma$ different from a rhombus, i.e., there exists at least one broken or shorter diagonal.

Thanks to Proposition3.4, we are able to characterize the configurations with minimal perimeter for a fixed number of occupied odd sites. This finding is formalized in the following two results, Proposition3.6 and Corollary3.7, whose proofs are deferred to subsection 5.1.

To state the precise results, we first introduce the notion of bars as follows. We define a vertical (resp. horizontal) bar $B$ of length $k$ as the union of particles arranged in odd sites $x_1,\ldots, x_k$ belonging to the same column (resp. row) such that $d(x_i,x_{i+1})=2$ for any $i=1,\ldots, k-1$ . In the case of $k=1$ , we will refer to it as protuberance. If in the same column (resp. row) there are $m$ disjoint vertical (resp. horizontal) bars, each of them of length $k_i$ , we say that the total length of the bars is $k=k_1+\ldots +k_m$ . Similarly, we can define a diagonal bar and note that it can correspond to a shorter or complete diagonal. If it does not matter whether the bar is vertical, horizontal, or diagonal, we simply refer to it as a bar. Finally, we will say that a bar $B$ is attached to a cluster $C$ when all the particles belonging to $B$ are at distance two from $C$ . Note that in Figure 8 (right) the shorter diagonals of lengths three and one are diagonal bars. See Figure 8 (left) for examples of vertical bars.

Corollary 3.7 (Minimal perimeter). Consider $n\leq L(L-2)$ and let $s,k$ be the unique integers as in Lemma 3.5. The perimeter $P$ of an odd cluster with area $n$ satisfies the following inequalities:

\begin{equation*} \Bigl (\frac {P}{4}-1\Bigr )^2\geq \begin{cases} 4n &\hbox { if } s\lt L/2, \\[3pt] 2(L^2-2n) &\hbox { if } L/2\leq s\lt L. \end{cases} \end{equation*}

In addition, for all $s\lt L$ we have that

(3.19) \begin{align} P\geq 4(2\sqrt {n}+1) \end{align}

and for $s\lt \frac {L}{2}$ the equality holds if and only if the odd cluster is the rhombus $\mathcal {R}_{s,s}$ .

Lastly, in the next lemma, we derive an isoperimetric inequality assuming the total number of odd and even occupied sites is fixed. To this end, we first define the real area of a configuration $\sigma$ as

(3.20) \begin{align} \tilde n(\sigma )= o(\sigma )+\tilde e(\sigma ), \end{align}

where $o(\sigma )$ (resp. $\tilde e(\sigma )$ ) denotes the number of occupied odd (resp. empty even) sites of the configuration $\sigma$ .

We use this lemma to characterize the critical configurations with an odd cluster of rhomboidal shape that does not wind around the torus. Indeed, we identify the shape of protocritical configurations $\sigma$ with minimal perimeter and fixed real area $\frac {L^2}{2}-L+1$ such that $\mathcal {R}(\mathcal {O}^{nd}(\sigma ))$ does not wind around the torus, and we show that if the trajectory visits another type of configuration with such a real area, then the corresponding path would not be optimal. The proof is given in subsection 5.1.

4.1 Preliminaries

We say that a configuration $\sigma \in \mathcal {X}$ has a odd (resp. even) vertical bridge if there exists a column in which the configuration $\sigma$ perfectly agrees with ${\mathbf{o}}$ (resp. ${\mathbf{e}}$ ). We define odd (resp. even) horizontal bridge in an analogous way and we say that a configuration $\sigma \in \mathcal {X}$ has an odd (resp. even) cross if it has both vertical and horizontal odd (resp. even) bridges (see Figure 9). In addition, we say that a configuration displays an odd (resp. even) vertical $m$ -uple bridge, with $m\geq 2$ , if there exist $m$ contiguous columns in which the configuration perfectly agrees with ${\mathbf{o}}$ (resp. ${\mathbf{e}}$ ). Similarly, we can define an odd (resp. even) vertical $m$ -uple bridge (see Figure 10). We refer to [Reference Nardi, Zocca and Borst47] for more details.

Figure 9. Examples of configurations displaying an odd horizontal bridge (on the left) and an odd cross (on the right).

Figure 10. Examples of configurations displaying an odd horizontal double (2-uple) bridge (on the left) and an odd vertical triple (3-uple) bridge (on the right).

The next lemma states that all configurations $\sigma \in \mathcal {X}$ with $\Delta H(\sigma )\lt L$ must belong to one of the two initial cycles.

See Figure 2 (left) for an example of configurations in $\mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})$ .

See Figure 2 (right) and Figure 3 (left) for an example of configurations in $\mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ and $\mathcal {C}_{cr}({\mathbf{e}},{\mathbf{o}})$ , respectively. Note that for any $\sigma \in \mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})\cup \mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ the surrounding rhombus $R(\mathcal {O}(\sigma ))$ is $R_{\frac {L}{2}-1,\frac {L}{2}}$ , while for any $\sigma \in \mathcal {C}_{cr}({\mathbf{e}},{\mathbf{o}})$ the surrounding rhombus $R(\mathcal {O}(\sigma ))$ is $R_{\frac {L}{2},\frac {L}{2}}$ .

Note that the unique possibilities are that (i) $D$ consists of two degenerate rhombi $\mathcal {R}_{0,0}$ as in Figure 3 (right) or (ii) $D\in \{\mathcal {R}_{0,1},\mathcal {R}_{1,0}\}$ as in Figure 11 (left).

Figure 11. An example of a configuration in $\mathcal {C}_{sb}({\mathbf{e}},{\mathbf{o}})$ that communicates with $\mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ and not with $\mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ (on the left) and an example of a configuration in $\mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ (on the right).

See Figure 4 (left) for an example of a configuration in $\mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ . Note that any configuration $\sigma \in \mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ is such that $\mathcal {R}(\mathcal {O}(\sigma ))$ winds around the torus. Thus, the assumption that the non-degenerate cluster $C$ is monotone implies that in condition 3 not every choice of the bars is allowed. Indeed, in order for the cluster to be monotone, the bars on the right (resp. left) of the $m$ -uple bridge can be adjacent only to a longer (resp. shorter) bar in lexicographic order. This implies that all bridges are in contiguous rows or columns. Furthermore, the length of a bar is inversely proportional to its distance from the nearest bar composing the bridge.

See Figure 4 (right) and Figure 11 (right) for examples of configurations in $\mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ .

Next, we provide several results investigating the main properties of the optimal paths connecting ${\mathbf{e}}$ to ${\mathbf{o}}$ . To this end, we introduce the following partition in manifolds of the state space $\mathcal {X}$ : for every $m=-\frac {L^2}{2},\ldots, \frac {L^2}{2}$ we define the manifold as the subset of configurations where the difference $m(\sigma )\,:\!=\,-e(\sigma )+o(\sigma )$ between odd and even occupied sites is equal to $m \in \mathbb {N}$ , i.e.,

(4.1) \begin{align} \mathcal {V}_m \,:\!=\, \{\sigma \in \mathcal {X} \,|\, m(\sigma ) = m\}, \end{align}

where $e(\sigma )\,:\!=\,\sum _{v \in V_{\mathbf{e}}} \sigma (v)$ (resp. $o(\sigma )\,:\!=\,\sum _{v \in V_{\mathbf{o}}} \sigma (v)$ ) is the number of the even (resp. odd) occupied sites in $\sigma$ .

The proof of (a) is immediate by noticing that at every step of the dynamics either $e(\sigma )$ or $o(\sigma )$ can change value and at most by $\pm 1$ and that of (b) follows from the fact that $\Delta H(\sigma )=-e(\sigma )-o(\sigma )+\frac {L^2}{2}$ and that $L$ is even.

A special role in our analysis will be played by the non-backtracking paths, i.e., those paths that visit each manifold exactly once. Lemmas4.8–4.10 below ensure the existence of a optimal path connecting ${\mathbf{e}}$ to ${\mathbf{o}}$ , which can be constructed as a suitable composition of the paths described below, and passing through the six sets that define $\mathcal {C}^*({\mathbf{e}},{\mathbf{o}})$ . The optimality of this path is guaranteed by the conditions on the height of each path described below. In addition, Lemma 4.11 below shows that some of the sets composing $\mathcal {C}^*({\mathbf{e}},{\mathbf{o}})$ do not communicate directly. Later, in subsection 4.2.2, combining these lemmas, we will show that the communication structure of these six sets at energy not higher than $H({\mathbf{e}})+L+1$ is the one illustrated in Figure 5. The proof of these lemmas is deferred to subsection 5.2.

4.2 Proof of Theorem 2.1

This section is entirely devoted to the proof of Theorem 2.1. More specifically, we prove that any essential saddle belongs to the set $\mathcal {C}^*({\mathbf{e}},{\mathbf{o}})$ in subsection 4.2.1, we describe how the transitions between essential gates can take place in subsection 4.2.2, and we prove that all the saddles in $\mathcal {C}^*({\mathbf{e}},{\mathbf{o}})$ are essential in subsection 4.2.3.

4.2.1 Every essential saddle belongs to $\mathcal {C}^*({\mathbf{e}},{\mathbf{o}})$

In this subsection, we will show that any essential saddle $\sigma$ belongs to the subset $\mathcal {C}^*({\mathbf{e}},{\mathbf{o}})$ . This readily follows from Proposition4.12 below.

Proposition 4.12. Let $\sigma$ be an essential saddle. Then, the following statements hold:

1. (i) If $\mathcal {R}(\mathcal {O}^{nd}(\sigma ))$ and $\mathcal {R}(\mathcal {O}(\sigma ))$ do not wind around the torus, then $\sigma \in \mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})\cup \mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ .

2. (ii) If $\mathcal {R}(\mathcal {O}^{nd}(\sigma ))$ does not wind around the torus, $\mathcal {R}(\mathcal {O}(\sigma ))$ does, and $\sigma$ belongs to $\omega \in ({\mathbf{e}}\to {\mathbf{o}})_{opt}$ that crosses the set $\mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})\cup \mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ , then $\sigma \in \mathcal {C}_{cr}({\mathbf{e}},{\mathbf{o}})$ .

3. (iii) If $\mathcal {R}(\mathcal {O}^{nd}(\sigma ))$ does not wind around the torus, $\mathcal {R}(\mathcal {O}(\sigma ))$ does, and $\sigma$ belongs to $\omega \in ({\mathbf{e}}\to {\mathbf{o}})_{opt}$ that does not cross the set $\mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})\cup \mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ , then $\sigma \in \mathcal {C}_{sb}({\mathbf{e}},{\mathbf{o}})\cup \mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ .

4. (iv) If $\mathcal {R}(\mathcal {O}^{nd}(\sigma ))$ winds around the torus, then $\sigma \in \mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ .

We observe that these four cases (i)–(iv) listed in Proposition4.12 cover all possibilities and thus form a partition of the set $\mathcal {G}({\mathbf{e}},{\mathbf{o}})$ of essential saddles for the transitions ${\mathbf{e}} \rightarrow {\mathbf{o}}$ .

Before presenting the proof, note that given a configuration $\sigma \in \mathcal {S}({\mathbf{e}},{\mathbf{o}})$ , if we know on which manifold $\mathcal {V}_m$ it lies, then the quantities $e(\sigma )$ and $o(\sigma )$ are uniquely determined and can be explicitly calculated as

(4.2) \begin{align} & o(\sigma )=\frac {m}{2}+\frac {L^2}{4}-\frac {L+1}{2} \qquad \text {and} \qquad e(\sigma )=-\frac {m}{2}+\frac {L^2}{4}-\frac {L+1}{2}. \end{align}

This readily follows from the fact that $\Delta H(\sigma ) = L+1=-e(\sigma )-o(\sigma )+\frac {L^2}{2}$ and $m=-e(\sigma )+o(\sigma )$ . Furthermore, since $\sigma \in \mathcal {S}({\mathbf{e}},{\mathbf{o}})$ , $\Delta H(\sigma ) = L+1$ and $m$ have to be odd integers by Lemma 4.7(b).

To prove Proposition4.12, we make use of an additional lemma (whose proof is deferred to subsection 5.3), which characterizes the intersection between any optimal path and a specific manifold, namely $\mathcal {V}_{m^*}$ with $m^*\,:\!=\,3-L$ .

Lemma 4.13 (Geometrical properties of the saddles on the manifold $\mathcal {V}_{m^*}$ ). Any non-backtracking optimal path $\omega \,:\, {\mathbf{e}} \to {\mathbf{o}}$ visits a configuration $\sigma \in \mathcal {V}_{m^*}$ that satisfy one of the following properties:

1. (i) If both $\mathcal {R}(\mathcal {O}^{nd}(\sigma ))$ and $\mathcal {R}(\mathcal {O}(\sigma ))$ do not wind around the torus, then $\sigma \in \mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})$ .

2. (ii) If $\mathcal {R}(\mathcal {O}^{nd}(\sigma ))$ does not wind around the torus but $\mathcal {R}(\mathcal {O}(\sigma ))$ does, then $\sigma \in \mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ .

3. (iii) If both $\mathcal {R}(\mathcal {O}^{nd}(\sigma ))$ and $\mathcal {R}(\mathcal {O}(\sigma ))$ wind around the torus, then $\sigma \in \mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ .

Proof of Proposition 4.12. Case (i). Let $\sigma$ be an essential saddle such that $\mathcal {R}(\mathcal {O}^{nd}(\sigma ))$ and $\mathcal {R}(\mathcal {O}(\sigma ))$ do not wind around the torus. If $\sigma \in \mathcal {V}_{m^*}$ , by Lemma 4.13(i) we know that $\sigma \in \mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})$ . Otherwise, we assume that $\sigma \notin \mathcal {V}_{m^*}$ . First, we observe that $\sigma \not \in \mathcal {V}_{m}$ with $m\lt m^*$ , otherwise the saddle $\sigma$ is not essential. Indeed, every optimal path from ${\mathbf{e}}$ to ${\mathbf{o}}$ has to cross a configuration $\bar \sigma \in \mathcal {V}_{m^*}$ in $\mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})$ due to Lemma 4.13(i). Thus, we can write a general optimal path $\omega =({\mathbf{e}},\omega _1,\ldots, \omega _k,\sigma, \ldots, \bar \sigma, \omega _{k+1},\ldots, \omega _{k+m},{\mathbf{o}})$ and define the path $\omega ^{\prime}=({\mathbf{e}},\tilde \omega _1,\ldots, \tilde \omega _n,\bar \sigma, \omega _{k+1},\ldots, \omega _{k+m},{\mathbf{o}})$ , where $\arg \max _{\xi \in \{{\mathbf{e}},\tilde \omega _1,\ldots, \tilde \omega _n,\bar \sigma \}}H(\xi )=\{\bar \sigma \}$ . This path $\omega ^{\prime}$ exists thanks to Lemma 4.8(i). Thus, we are left to analyse the case $\sigma \in \mathcal {V}_{m}$ with $m\gt m^*$ . We need to show that any essential saddle crossed afterward belongs to the set $\mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ . Using again Lemma 4.13, the path $\omega$ crosses a configuration $\sigma \in \mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})$ to reach ${\mathbf{o}}$ . Starting from it, there is a unique possible move to lower the energy towards ${\mathbf{o}}$ along the path $\omega$ , that is adding a particle in the unique unblocked empty odd site. Afterward, the unique possible move is to remove a particle from an even site. If this site is at a distance greater than one from an antiknob, then there is no more allowed move. Otherwise, the resulting configuration belongs to the set $\mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ . By iterating this pair of moves until the shorter diagonal of the rhombus is completely filled, we find that all the saddles that are crossed belong to $\mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ . Moreover, from this point onward, it is only possible to remove a particle from an even site at distance one from the antiknob, obtaining a configuration in $\mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})\cap \mathcal {V}_{1}$ .

Case (ii). By assumption, the path $\omega$ crosses the set $\mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})\cup \mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ . Without loss of generality, we may consider $\omega$ as a non-backtracking path. If this is not the case, we can apply the following argument to the last configuration visited by the path in the manifold $\mathcal {V}_{m^*}$ . In particular, in view of the properties of the path $\omega$ shown in case (i), we know that the last configuration crossed in $\mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ belongs to $\mathcal {V}_1$ and it is composed of a unique non-degenerate cluster $\mathcal {R}_{\frac {L}{2}-1,\frac {L}{2}}$ with a degenerate cluster $\mathcal {R}_{0,0}$ at distance one from the antiknob. Starting from it, there is a unique possible move to lower the energy towards ${\mathbf{o}}$ along the path $\omega$ , that is, adding a particle in the unique unblocked empty odd site. Afterward, the unique possible move is to remove a particle from an even site. The resulting configuration is in $\mathcal {C}_{cr}({\mathbf{e}},{\mathbf{o}})$ . By iterating this pair of moves until the shorter diagonal of the rhombus is completely filled, we find that all the saddles that are crossed belong to $\mathcal {C}_{cr}({\mathbf{e}},{\mathbf{o}})$ .

Case (iii). As in case (ii), we assume that the path $\omega$ is non-backtracking. By assumption, the path $\omega$ does not cross the set $\mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})\cup \mathcal {C}_{gr}({\mathbf{e}},{\mathbf{o}})$ , thus for Lemma 4.13(ii) the path crosses the set $\mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ , say in configuration $\bar \eta$ . Let $\mathcal {V}_{\bar m}$ the first manifold containing a configuration in $\mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ .

Consider first the case in which the saddle $\sigma$ is crossed by the path $\omega$ on the manifolds $\mathcal {V}_m$ , with $\bar m\leq m\lt m^*$ . Since $\omega$ crosses the set $\mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ , we will show that the saddle $\sigma$ belongs to the set $\mathcal {C}_{sb}({\mathbf{e}},{\mathbf{o}})\cup \mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ . To this end, we need to consider the time-reversal of the path $\omega =({\mathbf{e}},\omega _1,\ldots, \omega _k,\bar \eta, \ldots, {\mathbf{o}})$ , where $\bar \eta \in \mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ . Starting from $\bar \eta$ , since $\Delta H(\bar \eta )=L+1$ , the energy of configuration $\omega _k$ is less than that of $\bar \eta$ . Moreover, the unique possible move is to add a particle in the unique empty even site at distance one from an antiknob. Then, the unique possible move from $\omega _k$ to $\omega _{k-1}$ is to remove a particle from an occupied odd site. The configuration $\omega _{k-1}$ belongs to the set $\mathcal {C}_{sb}({\mathbf{e}},{\mathbf{o}})$ if it contains at least one bridge, otherwise, it belongs to the set $\mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ . By iterating this argument, we obtain the desired claim.

Consider now the case in which the saddle $\sigma$ is crossed by the path $\omega$ on the manifolds $\mathcal {V}_m$ , with $ m\gt m^*$ . Since $\omega$ crosses the set $\mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ , we will show that the saddle $\sigma$ belongs to the set $\mathcal {C}_{sb}({\mathbf{e}},{\mathbf{o}})\cup \mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ . Note that the unique admissible moves are the following: add a particle in the antiknob and then remove a particle from an even site at distance one from an antiknob. By iterating this couple of moves, we get that the resulting configuration belongs to the set $\mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ since $\mathcal {R}(\mathcal {O}^{nd}(\sigma ))$ does not wind around the torus.

It remains to consider the case $\sigma \in \mathcal {V}_m$ , with $m\lt \bar m$ . We will prove that any such $\sigma$ is not essential. Indeed, any non-backtracking optimal path that visits a configuration in $\mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ has crossed a configuration in $\mathcal {C}_{sb}({\mathbf{e}},{\mathbf{o}})$ before, by using the same argument as in case (iii) of this proof for $\bar m\leq m\lt m^*$ . Thus, we can write $\omega =({\mathbf{e}},\omega _1,\ldots, \omega _k,\sigma, \ldots, \tilde \sigma, \ldots, \bar \sigma, \omega _{k+1},\ldots, \omega _{k+m},{\mathbf{o}})$ and define the path $\omega ^{\prime}=({\mathbf{e}},\omega _1^{\prime},\ldots, \omega _n^{\prime},\tilde \sigma, \ldots, \bar \sigma, \omega _{k+1},\ldots, \omega _{k+m},{\mathbf{o}})$ , where $\tilde \sigma$ is a configuration in $\mathcal {C}_{sb}({\mathbf{e}},{\mathbf{o}})$ , $\bar \sigma \in \mathcal {V}_{\bar m}$ is a configuration in $\mathcal {C}_{ib}({\mathbf{e}},{\mathbf{o}})$ thanks to Lemma 4.13(ii) and $\arg \max _{\xi \in \{{\mathbf{e}},\omega _1^{\prime},\ldots, \omega _n^{\prime},\tilde \sigma \}}H(\xi )=\{\tilde \sigma \}$ . This path $\omega ^{\prime}$ exists thanks to Lemma 4.8(ii) and this concludes case (iii).

Case (iv). By Lemma 4.13(iii), we know that any optimal path $\omega \in ({\mathbf{e}}\rightarrow {\mathbf{o}})_{\mathrm {opt}}$ crosses the manifold $\mathcal {V}_{m^*}$ in a configuration $\bar \sigma$ belonging to the set $\mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ .

If the essential saddle $\sigma \in \mathcal {V}_{m^*}$ , then we deduce that $\sigma \in \mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ .

Suppose now that the saddle $\sigma$ belongs to the manifold $\mathcal {V}_m$ , with $m\gt m^*$ . Starting from such a saddle $\bar \sigma$ , there is a unique possible move to lower the energy towards ${\mathbf{o}}$ along the path $\omega$ , that is, add a particle to the unique unblocked empty odd site. Afterward, the unique possible move is to remove a particle from an even site. If this site is at a distance greater than one from an antiknob, then there is no more possible move to reach ${\mathbf{o}}$ in such a way that the path $\omega$ is optimal. Otherwise, the resulting configuration belongs to the set $\mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ . Indeed, by construction, we deduce that the resulting non-degenerate odd cluster is still monotone due to the properties of the bars attached to each bridge. By iterating this pair of moves until there is a row or a column that is not a bridge, we obtain that all the saddles that are crossed belong to $\mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ . Finally, from this point onward, it is only possible to remove a particle from an even site at distance one from the antiknob, obtaining the last configuration in $\mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ . Afterward, the energy only decreases and therefore no more saddles are crossed.

Suppose now that the saddle $\sigma$ belongs to the manifold $\mathcal {V}_m$ , with $m\lt m^*$ . Note that, starting from such $\bar \sigma \in \mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ , the unique admissible moves to get $\sigma$ are the following: add a particle to the unique empty even site at distance one from an antiknob and afterward remove a particle from an occupied odd site. By iterating this couple of moves, we get that the resulting configuration belongs to the set $\mathcal {C}_{mb}({\mathbf{e}},{\mathbf{o}})$ as long as $\mathcal {R}(\mathcal {O}^{nd}(\sigma ))$ winds around the torus, otherwise the saddle $\sigma$ does not satisfy the properties in the statement.

4.2.3 All the saddles in $\mathcal {C}^*({\mathbf{e}},{\mathbf{o}})$ are essential

In this first part of the proof, we will prove that every $\sigma \in \mathcal C^*({\mathbf{e}},{\mathbf{o}})$ is an essential saddle by constructing a non-backtracking optimal path $\omega \,:\, {\mathbf{e}} \to {\mathbf{o}}$ that visits $\sigma$ .

Leveraging the fact that $\sigma \in \mathcal C^*({\mathbf{e}},{\mathbf{o}})$ , we construct the desired non-backtracking path $\omega$ as a concatenation of two paths as follows. First, using a suitable concatenation of the paths described in Lemmas4.9–4.10, we can define a path $\omega _1$ that starts from the considered configuration $\sigma$ to the initial cycle $\mathcal {C}_{\mathbf{e}}$ . We then construct another path $\omega _2$ that goes from $\sigma$ to the target cycle $\mathcal {C}_{\mathbf{o}}$ as a suitable concatenation of the paths described in Lemmas4.8–4.9. The desired non-backtracking path $\omega$ is the time-reversal of $\omega _1$ concatenated with $\omega _2$ and it is easy to show that it is also optimal.

Assume now by contradiction that $\sigma$ is not essential, which means that there must exist another optimal path $\omega ^{\prime} \in ({\mathbf{e}} \to {\mathbf{o}})_{\textrm {opt}}$ such that $S(\omega ^{\prime}) \subset S(\omega ) \setminus \{ \sigma \}$ . Recall that by Lemma 4.7(a), such a path $\omega ^{\prime}$ that avoids $\sigma$ still needs to visit the manifold $\mathcal {V}_{m(\sigma )}$ where $\sigma$ lives at least once. Let $\eta$ be any such configuration in $\mathcal {V}_{m(\sigma )}\cap \omega ^{\prime}$ . We claim that such a configuration $\eta$ must satisfy

\begin{equation*} \Delta H(\eta )\equiv 1 \pmod 2. \end{equation*}

This claim readily follows from Lemma 4.7(b) in combination with the facts that $L$ is even and $\Delta H(\sigma ) = L+1$ by construction.

If $\Delta H(\eta ) \geq L+3$ , then $\omega ^{\prime}$ is not an optimal path, since $\Phi _{\omega ^{\prime}}- H({\mathbf{e}}) \geq L+3 \gt L+1 = \Phi ({\mathbf{e}},{\mathbf{o}})-H({\mathbf{e}})$ .

On the other hand, if $\Delta H(\eta )\leq L-1$ , then from Lemma 4.1 it follows that $\eta$ belongs to one of the two initial cycles. Proposition4.14 ensures that every non-backtracking optimal path crosses $\mathcal {C}^*({\mathbf{e}},{\mathbf{o}})$ in one of the three ways (i)–(iii) described therein, so that also the optimal path $\omega ^{\prime}$ passing through $\eta$ has to visit $\mathcal {C}^*({\mathbf{e}},{\mathbf{o}})$ . Since, by assumption, the path $\omega ^{\prime}$ has to avoid the saddle $\sigma$ , we deduce that there exists another saddle $\tilde \eta$ obtained starting from $\eta$ that does not belong to $\mathcal {V}_{m(\sigma )}$ . In particular, the two paths cross the set $\mathcal {C}^*({\mathbf{e}},{\mathbf{o}})$ in three different ways according to Proposition4.14(i)–(iii). Thus, we deduce that $S(\omega ^{\prime}) \not \subset S(\omega ) \setminus \{ \sigma \}$ .

Thus, in view of the parity of $\Delta H(\eta )$ , we must have $\Delta H(\eta ) = L+1$ , but then $\eta$ is a saddle and, by construction, it did not belong to $S(\omega )$ and thus $S(\omega ^{\prime}) \not \subset S(\omega ) \setminus \{ \sigma \}$ .

5.1 Results on the perimeter of an odd region

Proof of Proposition 3.4 . The proof revolves around the simple idea that using the filling algorithms introduced in subsection 3.3 we can expand the odd cluster $C$ (i.e., progressively increase the number of the occupied odd sites in $C$ ) in such a way that $\mathcal {R}$ remains the surrounding rhombus and the energy of all configurations along such a path never exceeds $H(\sigma )+1$ . Since by assumption $\sigma \neq \sigma ^{\prime}$ , the odd cluster $C$ cannot coincide with $\mathcal {R}$ , in view of conditions (i), (ii), and (iii). Thus, $C$ contains at least a broken diagonal or a shorter diagonal than those of the surrounding rhombus.

We can define the desired path $\omega$ as the concatenation of the two paths returned by the filling algorithms $\tilde \omega$ and $\bar \omega$ . If $C$ has no broken diagonal, we take $\tilde \omega$ empty. If $C$ has no shorter diagonal, then $C$ already has the shape of a rhombus, and therefore we take $\bar \omega$ empty. By the definition of these two paths, the energy increases by one only if an even site has to be emptied, but all these moves are followed by the addition of a particle in an antiknob. Therefore, the energy along the path $\omega$ increases by at most one with respect to the starting configuration $\sigma$ . This procedure ends when $C$ coincides with $\mathcal {R}$ , which implies that the resulting configuration is $\sigma ^{\prime}$ .

To conclude the proof, we need to show the properties claimed for the perimeter of $\sigma$ . If $C$ contains $m\geq 1$ broken diagonals, we argue as follows. Since the cluster $C$ is connected, all the empty odd sites in which the diagonals are broken are antiknobs, i.e., they have $n \in \{3,4\}$ neighbouring even sites belonging to $C$ , see Figure 12. We distinguish the following two cases. If $n=3$ , then we first need to remove a particle from the unique neighbouring occupied even site, like the site $v$ represented in Figure 12 on the left.

Figure 12. Example of a configuration $\sigma$ as in the statement of Proposition3.4 (on the left), where we highlight in red the site containing the target antiknob, and the configuration obtained from $\sigma$ by filling it after removing a particle from the even site $v$ (on the right), with highlighted in red the site containing the next target antiknob.

After that move, when the particle is added in the antiknob, the perimeter does not change in view of (3.3), otherwise if $n=4$ then the perimeter decreases (see Figure 12 on the right). This also occurs when we add a particle in the target antiknobs except the last antiknob to obtain the complete diagonal, for which by construction, we have that the perimeter decreases by 4. By iterating this argument for every broken diagonal, we get

\begin{equation*} P(\sigma )\geq P(\tilde \sigma )+4m\gt P(\tilde \sigma ). \end{equation*}

If $C$ does not contain any broken diagonal, we argue as follows. By construction, we deduce that the first odd site we will fill, which is the nearest neighbour of the first shorter diagonal, has three neighbouring even sites belonging to $C$ . Thus, we need to remove a particle from the unique occupied neighbouring even site and then, when the first particle is added to the unique possible antiknob, the perimeter does not change thanks to (3.3). This also occurs when we iterate this argument.

Proof of Corollary 3.7 . We first consider the case in which the area $n$ is of the form $n=s(s-1)+k$ . If $s\lt L/2$ , we have that the perimeter of the odd cluster is $P=4(2s+1)$ . Since the cluster is contained in a rhombus $\mathcal {R}_{s,s}$ , we have that $n\leq s^2$ and, hence, $\bigl (\frac {P}{4}-1\bigr )^2\geq 4n$ . If $L/2\leq s\lt L$ , then the perimeter of the odd cluster is $P=4(2L-2s-1)$ . Since the cluster is contained in a rhombus $\mathcal {R}_{s,s}$ and therefore the complement in $V$ is a rhombus $\mathcal {R}_{L-s-1,L-s-1}$ , we have that $n\geq \frac {L^2}{2}-(L-s-1)^2$ . Hence

\begin{equation*} \left(\frac {P}{4}-1 \right)^2=4(L-s-1)^2\geq 2(L^2-2n). \end{equation*}

Consider now the other case, when the area $n$ is of the form $n=s^2+k$ . If $s\lt L/2$ we have that the perimeter of the odd cluster is $P=4(2s+2)$ . Since the cluster is contained in a rhombus either $\mathcal {R}_{s+1,s}$ or $\mathcal {R}_{s,s+1}$ , we have that $n\leq s^2+s$ . Thus, it holds

\begin{equation*} \left(\frac {P}{4}-1\right)^2=(2s+1)^2=4(s^2+s)+1\geq 4n+1. \end{equation*}

If $L/2\leq s\lt L$ , then the perimeter of the odd cluster is $P=4(2L-2s-2)$ . Since the cluster is contained in either a rhombus $\mathcal {R}_{s+1,s}$ or $\mathcal {R}_{s,s+1}$ and therefore the complement in $V$ is either a rhombus $\mathcal {R}_{L-s-2,L-s-1}$ or $\mathcal {R}_{L-s-1,L-s-2}$ , we have that $n\geq \frac {L^2}{2}-(L-s-1)(L-s-2)$ . Hence,

\begin{equation*} \left(\frac {P}{4}-1 \right)^2=(2L-2s-3)^2=4(L-s-1)(L-s-2)+1\geq 4\left(\frac {L^2}{2}-n \right)+1=2(L^2-2n)+1. \end{equation*}

Using condition $n\leq L(L-2)$ , inequality (3.19) directly follows.

Proof of Lemma 3.8 . Let $\sigma$ be a configuration with real area $\tilde n=2\ell ^2+2\ell +1$ . First, we suppose that the set of odd clusters in $\sigma$ is composed only of $j \geq 2$ non-degenerate clusters. Each of them has area $n_i$ and perimeter $p_i$ for $i=1,\ldots, j$ . Suppose by contradiction that $\sigma$ has minimal perimeter so that the area of the configuration $\sigma$ is $n_\sigma =\sum _{i=1}^j n_i$ and its perimeter is $p_\sigma = 4(2\sqrt {n_\sigma }+1)$ . By (3.19), we have $p_i \geq 4(2\sqrt {n_i}+1)$ for any $i=1,\ldots, j$ . Then we obtain that

(5.1) \begin{align} p_{\sigma }=\sum _{i=1}^j p_i\geq \sum _{i=1}^j 4(2\sqrt {n_i}+1)\geq 8\sqrt {\sum _{i=1}^j n_i}+4j\geq 8\Bigg (\sqrt {\sum _{i=1}^j n_i}+1\Bigg )=4(2\sqrt {n_\sigma }+2), \end{align}

that is a contradiction.

Second, we suppose that the set of odd clusters in $\sigma$ is composed of $k \geq 1$ degenerate clusters and of $j \geq 1$ non-degenerate clusters. Each of these non-degenerate clusters has area $n_i$ and perimeter $p_i$ for $i=1,\ldots, j$ , so that $n_\sigma =\sum _{i=1}^j n_i$ . We denote by $\tilde p_i$ for $i=1,\ldots, k$ the perimeter of a degenerate cluster. Suppose by contradiction that $\sigma$ has minimal perimeter. By (3.19), we have $p_i \geq 4(2\sqrt {n_i}+1)$ for any $i=1,\ldots, j$ and $p_\sigma = 4(2\sqrt {n_\sigma }+1)$ . Thus, we obtain

(5.2) \begin{align} p_{\sigma } & = \sum _{i=1}^{j} p_i+\sum _{i=1}^{k} \tilde p_i \geq \sum _{i=1}^j 4(2\sqrt {n_i}+1)+4k\geq 8\sqrt {\sum _{i=1}^j n_i}+4j+4k\geq 8 \nonumber \\ & \quad \sqrt {\sum _{i=1}^j n_i}+4+4=4(2\sqrt {n_\sigma }+2), \end{align}

that is a contradiction. Thus, we obtain that $k=0$ and $j=1$ . Since the real area $\tilde n$ of the configuration $\sigma$ is fixed, then also the area $n_\sigma$ is fixed. Thus, given that $\sigma$ contains only one non-degenerate cluster with minimal perimeter and with fixed area, by Corollary3.7 we obtain that the non-degenerate cluster is the rhombus $\mathcal {R}_{\ell, \ell }$ , which has precisely real area $\tilde n$ .

5.2 Results on optimal reference paths

Proof of Lemma 4.8 . We start by proving (i). The desired path $\omega =({\mathbf{e}},\omega _1,\ldots, \omega _{k(L)},\eta )$ is obtained as follows. Starting from ${\mathbf{e}}$ , define the configuration $\omega _1$ as that in which one particle is removed from an empty even site, say $v_1\in V_{\mathbf{e}}$ . Similarly, we define $\omega _2$ as the configuration in which a particle is removed from a site $v_2\in V_{\mathbf{e}}$ such that $d(v_1,v_2)=2$ . Similarly, by removing the particles in $v_3,v_4\in V_{\mathbf{e}}$ in such a way $d(v_i,v_j)=2$ for any $i,j=1,\ldots, 4$ and $i\neq j$ , we define the configurations $\omega _3$ and $\omega _4$ . Note that $\Delta H(\omega _4)=4\lt L+1$ . Thus, we can define the configuration $\omega _5$ as that obtained from $\omega _4$ by adding a particle to the unique unblocked odd site, i.e., the one at distance one from $v_i$ for any $i=1,\ldots, 4$ . See Figure 13 on the left.

Figure 13. Example of the configurations $\omega _5$ (on the left), $\omega _7$ (in the middle) and $\omega _{13}$ (on the right) visited by the path described in the proof of Lemma 4.8(i).

We obtain that $\Delta H(\omega _5)=3\lt L+1$ and $\omega _5$ is composed of a unique non-degenerate odd cluster, which is $\mathcal {R}_{1,1}$ . To define the configuration $\omega _6$ , we remove a particle from a site $v_5\in V_{\mathbf{e}}$ such that $d(v_i,v_5)=2$ for two indices $i=1,\ldots, 4$ . Similarly, we define $\omega _7$ in such a way there is an empty odd site with all the neighbouring even sites that are empty, see Figure 13 in the middle. Note that $\Delta H(\omega _7)=5\lt L+1$ . Then, we define $\omega _8$ by adding a particle in the unique unblocked odd site, so that $\Delta H(\omega _8)=4\lt L+1$ . Note that $\omega _8$ contains an odd cluster $\mathcal {R}_{1,2}$ . By growing the odd cluster in a spiral fashion, emptying the even sites that are strictly necessary, note that the configuration $\omega _{13}$ contains only the non-degenerate cluster $\mathcal {R}_{2,2}$ (see Figure 13 on the right). Then, our path visits all the configurations which have a unique non-degenerate odd cluster $C$ such that $C=\mathcal {R}_{\ell, \ell }$ and $C=\mathcal {R}_{\ell, \ell +1}$ for any $3\leq \ell \leq \frac {L}{2}-1$ up to the configuration $\omega _{{k(L)}-1}$ , whose unique non-degenerate odd cluster is $C=\mathcal {R}_{\frac {L}{2}-1,\frac {L}{2}-1}$ . Then, the configuration $\omega _{k(L)}$ is defined by removing a particle from an even site $w$ at distance two from $C$ . Finally, the configuration $\eta$ is obtained by removing a particle from an even site at distance two from $C$ and from $w$ . Since the procedure we defined is invariant by translation, given a fixed configuration $\eta \in \mathcal {C}_{ir}({\mathbf{e}},{\mathbf{o}})$ is possible to choose the position of the odd clusters in such a way the final configuration of the path we described coincides with the desired $\eta$ . It remains to show that $\arg \max _{\xi \in \omega }H(\xi )=\{\eta \}$ . To this end, since $\Delta H(\eta )=L+1$ and therefore $\Delta H(\omega _{k(L)})=L$ and $\Delta H(\omega _{{k(L)}-1})=L-1$ , we need only to show that

\begin{equation*} \max _{\xi \in \{{\mathbf{e}},\omega _1,\ldots, \omega _{{k(L)}-2}\}}H(\xi )\lt L+1. \end{equation*}

First, we will show that $\Delta H(\eta )=3+2(\ell -1)$ by induction over the dimension $\ell =1,\ldots, \frac {L}{2}-1$ of the rhombus $\mathcal {R}_{\ell, \ell }$ composing the unique odd cluster of the configurations $\eta$ visited by $\omega$ . We have already proven the desired property in the case $\ell =1$ . Suppose now that the claim holds for $\ell$ , with $1\leq \ell \leq \frac {L}{2}-2$ , thus we will prove that it holds also for $\ell +1$ . To reach the configuration displaying the rhombus $\mathcal {R}_{\ell, \ell +1}$ starting from $\mathcal {R}_{\ell, \ell }$ , we need first to remove particles from two even sites by increasing the energy by two. Then, we sequentially add a particle in an odd site and remove a particle in an even site until the length of the shorter diagonal is $\ell -1$ . Finally, the last move is the addition of a particle in an odd site without the need to remove any particle from an even site. Starting from $\mathcal {R}_{\ell, \ell +1}$ , to obtain $\mathcal {R}_{\ell +1,\ell +1}$ we follow the same sequence of moves. Thus, for a configuration $\eta$ containing as unique odd cluster a rhombus $\mathcal {R}_{\ell +1,\ell +1}$ we deduce that $\Delta H(\eta )=3+2(\ell -1)+2$ , which proves our claim for $\ell +1$ . Along the sequence of moves from $\mathcal {R}_{\ell, \ell }$ to $\mathcal {R}_{\ell +1,\ell +1}$ , the energy is at most $3+2(\ell -1)+3$ , which is strictly less than $L+1$ for $\ell \leq \frac {L}{2}-1$ . Note that we do not to consider the case $\ell =\frac {L}{2}-2$ , indeed the path $\omega$ stops before reaching the rhombus $\mathcal {R}_{\frac {L}{2}-1,\frac {L}{2}}$ . This concludes the proof of (i).

Now, we prove (ii). The desired path $\omega =({\mathbf{e}},\omega _1,\ldots, \omega _{k(L)},\eta )$ is obtained as follows. Starting from ${\mathbf{e}}$ , define the configuration $\omega _1$ as that in which one particle is removed from an empty even site, say $v_1\in V_{\mathbf{e}}$ . Similarly, we define $\omega _2$ as the configuration in which a particle is removed from a site $v_2\in V_{\mathbf{e}}$ such that $d(v_1,v_2)=2$ . Similarly, by removing particles in $v_3,v_4\in V_{\mathbf{e}}$ in such a way $d(v_i,v_j)=2$ for any $i,j=1,\ldots, 4$ and $i\neq j$ , we define the configurations $\omega _3$ and $\omega _4$ . Note that $\Delta H(\omega _4)=4\lt L+1$ . Thus, we can define the configuration $\omega _5$ as the one obtained from $\omega _4$ by adding a particle in the unique unblocked odd site, i.e., the one at distance one from $v_i$ for any $i=1,\ldots, 4$ . We obtain that $\Delta H(\omega _5)=3\lt L+1$ and $\omega _5$ is composed of a unique non-degenerate odd cluster, which is $\mathcal {R}_{1,1}$ , see Figure 14 on the left.

Figure 14. Example of the configurations $\omega _5$ (on the left), $\omega _7$ (in the middle) and $\omega _{9}$ (on the right) visited by the path described in the proof of Lemma 4.8(ii).

Next, we describe four steps that are used in the following iteration. The first step is to define the configuration $\omega _6$ by removing a particle from a site $v_5\in V_{\mathbf{e}}$ such that $d(v_i,v_5)=2$ for two indices $i=1,\ldots, 4$ . The second step is to remove a particle from a site $v_6\in V_{\mathbf{e}}$ in such a way there is an empty odd site with two neighbouring empty even sites and the other one is occupied. In this way, we obtain the configuration $\omega _7$ (see Figure 14 in the middle). The third step is to obtain the configuration $\omega _8$ by removing the particle in the site $v_7\in V_{\mathbf{e}}$ such that $d(v_7,v_5)=d(v_7,v_6)=2$ . Note that $\Delta H(\omega _8)=6\lt L+1$ . Then, the last step is to define $\omega _9$ by adding a particle in the unique unblocked odd site, so that $\Delta H(\omega _9)=5\lt L+1$ and the energy cost of these four steps is $2$ . Note that $\omega _9$ is composed of a non-degenerate odd cluster with two odd particles along either the same column or the same row, see Figure 14 on the right.

From this point on, we iterate these four steps for other $\frac {L}{2}-3$ times, until we obtain the configuration $\omega _{k(L)-1}$ in which either a column or a row contains $\frac {L}{2}-1$ odd particles. Note that $\Delta H(\omega _{k(L)-1})=5+2(\frac {L}{2}-3)=L-1$ . Then, we repeat the first two steps described above and reach the configuration $\eta \in \mathcal {C}_{sb}({\mathbf{e}},{\mathbf{o}})$ with $\Delta H(\eta )=L+1$ . It is easy to check that $\arg \max _{\xi \in \omega }=\{\eta \}$ . Indeed, we get

\begin{equation*} \begin{array}{lll} \Delta H (\omega _{i})&=\Delta H (\omega _{i-1})+1 \qquad &\text { if } 1\leq i\leq 4, \\[3pt] \Delta H (\omega _{2i-1})&=\Delta H (\omega _{2i})-1 \qquad &\text { if } 3\leq i \leq \frac {k(L)}{2}, \\[3pt] \Delta H (\omega _{2i})&=\Delta H (\omega _{2i-1})+3 \qquad &\text { if } 3\leq i \leq \frac {k(L)}{2}-1, \\[3pt] \Delta H (\omega _{k(L)})&=\Delta H (\omega _{k(L)-1})+2, \end{array} \end{equation*}

which concludes the proof of (ii).