2.1 Contours and cluster expansion

A contour $\gamma $ is a collection of bonds $(e_i)_{i=1}^m$ in the dual lattice $(\mathbb {Z}^2)^*$ , where all bonds are distinct except possibly $e_1$ and $e_m$ may coincide, every two consecutive edges share a vertex, and the path formed is simple except in accordance with a splitting rule: If the pair $e_{i}, e_{i+1}$ and $e_j, e_{j+1}$ all intersect at a vertex $\mathsf {x}$ , then the two other end points of $e_i,e_{i+1}$ are on the same side of the line through $\mathsf {x}$ with slope $1$ (from southwest to northeast) and similarly for $e_j,e_{j+1}$ (this is the northeast splitting rule). We call $\gamma $ an open contour if $e_1\neq e_m$ .

In the context of the SOS model in a finite, connected $\Lambda \Subset \mathbb {Z}^2$ under $0$ boundary conditions, for any h, the h-level lines (recall that for any configuration $\varphi $ and integer h, these are the bonds dual to $\mathsf {x}\sim \mathsf {y}$ with $\varphi _{\mathsf {x}}<h$ and $\varphi _{\mathsf {y}}\geq h$ ) give rise to a collection of disjoint loops after applying the global splitting rule. In the presence of a boundary condition $\xi \in \{0,1\}^{\partial \Lambda }$ consisting of a connected stretch of $0$ ’s (and $1$ ’s elsewhere), this gives rise to a unique open contour among the height- $1$ level lines (accompanied by a collection of closed contours). We refer to this path as the open $1$ -contour. Let $\Delta _{\gamma }^+$ and $\Delta _{\gamma }^-$ denote the set of sites (of $\mathbb {Z}^2$ ) immediately above and below $\gamma $ , respectively, and define

$$\begin{align*}\Delta_{\gamma}:= \Delta_{\gamma}^+ \cup \Delta_{\gamma}^- . \end{align*}$$

Note that the sites in $\Delta _{\gamma }^+$ have height $\geq 1$ , while the sites in $\Delta _{\gamma }^-$ have height $\leq 0$ . Each $\gamma $ divides $\Lambda $ into two regions, $\Lambda _{\gamma }^+$ and $\Lambda _{\gamma }^-$ , where we write $\Lambda _{\gamma }^+$ to denote the region that contains $\Delta _{\gamma }^+$ as part of its inner boundary. The next proposition addresses the law of this unique SOS open contour $\gamma $ .

Proposition 2.1 [Reference Caputo, Lubetzky, Martinelli, Sly and Toninelli13, Lem. A.2].

Consider the SOS model $\widehat \pi _\Lambda ^\xi $ on any finite, connected $\Lambda \subset \mathbb {Z}^2$ and under any boundary condition $\xi \in \{0, 1\}^{\partial \Lambda }$ that induces a unique open $1$ -contour $\gamma $ . Then there exists a constant $\beta _0>0$ such that for all $\beta \geq \beta _0$ ,

(2.1)

for some ‘decoration functions’ $\{\phi (\mathcal {C}, \gamma )\}_{\mathcal {C} \subset \mathbb {Z}^2}$ satisfying the following properties:

1. (i) If $\mathcal {C}$ is not connected, then $\phi (\mathcal {C}; \gamma ) = 0$ .

2. (ii) The decoration function $\phi (\mathcal {C};\cdot )$ depends on $\gamma $ only through $\mathcal {C} \cap \Delta _{\gamma }$ .

3. (iii) For all ${\mathsf {v}} \in \mathbb {Z}^2$ , $\phi (\mathcal {C}; \gamma ) = \phi (\mathcal {C}+{\mathsf {v}}; \gamma +{\mathsf {v}})$ .

4. (iv) Letting $d(\mathcal {C})$ denote the cardinality of the smallest connected set of bonds of $\mathbb {Z}^2$ containing all boundary bonds of $\mathcal {C}$ (i.e., bonds connecting $\mathcal {C}$ to $\mathcal {C}^c$ ), we have the decay bound

   (2.2) $$ \begin{align} \sup_{\gamma} |\phi(\mathcal{C}; \gamma)| \leq \exp\Big(- (\beta-\beta_0) d(\mathcal{C})\Big). \end{align} $$

Furthermore, if $\widehat Z_{\Lambda }^\xi $ is the partition function of the SOS model $\widehat \pi _{\Lambda }^\xi $ and

$$\begin{align*}\mathcal{G}^{\xi}_{\Lambda} := \sum_{\gamma} e^{-\beta|\gamma| + \sum_{\mathcal{C} \subset \Lambda} \phi(\mathcal{C};\gamma)} \end{align*}$$

is the interface partition function corresponding to the distribution Equation (2.1), then

(2.3) $$ \begin{align} \mathcal{G}^{\xi}_{\Lambda} = \widehat{Z}_{\Lambda}^{\xi}/\widehat{Z}_{\Lambda}^{0}. \end{align} $$

The decoration functions come from cluster expansion applied to the partition functions in $\Lambda _{\gamma }^+$ and $\Lambda _{\gamma }^-$ . In Appendix A, we recall cluster expansion for the SOS model and provide the proof of Proposition 2.1, as the expression for the decoration function $\phi $ is needed to verify that it meets the criteria of modified Ising polymers (Definition 2.6). In light of Property (i), from now on we will write $\mathcal {C}$ to denote a connected subset (or cluster) of $\mathbb {Z}^2$ .

Next, we recall the notion of surface tension for the SOS model.

Definition 2.2 (Dobrushin boundary conditions, surface tension).

Fix $\vec {u} \in \mathbb {S}^{1}$ in the first quadrant, that is, with $\theta _{\vec {u}} \in [0,\pi /2)$ , where $\theta _{\vec {u}}$ is the angle $\vec {u}$ makes with the positive horizontal axis. Set , and let $\xi (\vec {u})$ denote the boundary condition defined by $\xi (\vec {u})_v = 0$ for all $v\in \partial \Lambda $ lying on or below $\mathrm {span}(\vec {u})$ , and $\xi (\vec {u})_v= 1$ otherwise. Our main focus is on the boundary condition $\xi (\mathsf {e}_1)$ , which we denote by $0,1,1,1$ (after the values induced by $\xi (\mathsf {e}_1)$ on the four sides of the box ). Define $d_{N,\vec {u}} := N/\cos (\theta _{\vec {u}})$ . The surface tension is defined by

(2.4)

The value of for the other quadrants is defined symmetrically so that when $\theta _{\vec u} = -\theta _{\vec v}$ or $\theta _{\vec u} = \pi -\theta _{\vec v}$ .

Finally, extends to an even function on all of $\mathbb {R}^2$ via homogeneity:

The proof of the existence of as well as many of its properties can be found in [Reference Dobrushin, Kotecký and Shlosman20, §1–2].

2.2 The free Ising polymer model

In this section, we define the class of Ising polymer models, as given by [Reference Ioffe, Shlosman and Toninelli32].Footnote ⁴ The reader is also referred to [Reference Dobrushin, Kotecký and Shlosman20] for many useful results on such polymer models. It will be shown that the SOS open contour from Equation (2.1) falls in this class.

Recall that we write $\mathcal {C}$ to denote a finite, connected subset of $\mathbb {Z}^2$ . For every contour $\gamma $ , consider any decoration function $\Phi (\mathcal {C}; \gamma )$ satisfying the following four properties:

1. (P1) Locality: $\Phi (\mathcal {C}; \cdot )$ depends on $\gamma $ only through $\mathcal {C}\cap \Delta _{\gamma }$ .

2. (P2) Decay: There exists some $\chi>1/2$ such that, for all $\beta>0$ sufficiently large,

   (2.5) $$ \begin{align} \sup_{\gamma} |\Phi(\mathcal{C},\gamma)| \leq \exp(-\chi \beta (d(\mathcal{C})+1)), \end{align} $$
   where $d(\mathcal {C})$ is defined as in Property (iv) of Proposition 2.1.

3. (P3) Translational symmetry: For all ${\mathsf {v}}\in \mathbb {Z}^2$ , $\Phi (\mathcal {C}; \gamma ) = \Phi (\mathcal {C}+{\mathsf {v}};\gamma +{\mathsf {v}})$ .

4. (P4) Symmetry of the surface tension: The surface tension $\tau _{\beta }(\mathsf {x})$ defined below in Equation (2.8) possesses all discrete symmetries of $\mathbb {Z}^2$ (rotations by $\pi /4$ and reflections w.r.t. axes and the diagonals $y=\pm x$ ).

Towards specifying the probability that the model assigns to each polymer – as well as the surface tension $\tau _\beta $ mentioned in Property (P4) – define the free (polymer) weight via

(2.6) $$ \begin{align} q(\gamma) = \exp\Big(-\beta|\gamma| + \sum_{\mathcal{C} \cap \Delta_{\gamma} \neq \emptyset} \Phi(\mathcal{C};\gamma)\Big), \end{align} $$

where, here and throughout the article, sums over $\mathcal {C}$ are assumed to only go over connected subsets $\mathcal {C} \subset \mathbb {Z}^2$ . Next, for any $\mathsf {x} \in (\mathbb {Z}^2)^*$ , consider the partition function going over all contours $\gamma $ with the start point given by the dual origin $\mathsf {o}^* := (1/2,1/2)$ and end point $\mathsf {x}$ :

$$ \begin{align*} \mathcal{G}(\mathsf{x}) := \sum_{\gamma: \mathsf{o}^* \to \mathsf{x}} q(\gamma). \end{align*} $$

For any set of contours E, consider also the partition function going over all contours in E with end points $0$ and $\mathsf {x}$ :

(2.7)

Lastly, define the Ising polymer surface tension $\tau _{\beta }(\cdot )$ viaFootnote ⁵

(2.8) $$ \begin{align} \tau_{\beta}(\vec{n}) := -\lim_{N\to\infty} \frac{1}{ N} \log \mathcal{G}(N\vec{n}) \qquad \text{ for } \vec{n} \in \mathbb{S}^1, \end{align} $$

where the limit is taken over N such that $N \vec {n}$ is in $(\mathbb {Z}^2)^*$ . By homogeneity, $\tau _{\beta }$ extends to all of $\mathbb {R}^2$ .

Definition 2.3 (Free Ising polymer).

The free Ising polymer model in a subset $D\subset \mathbb {R}^2$ is given by the probability measure over contours $\gamma :\mathsf {o}^* \to \mathsf {x}$ contained in D:

(2.9) $$ \begin{align} \mathbf{P}^{\mathsf{x}}( \cdot \;\big|\; \gamma \subset D) := \frac{\mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset D, \gamma \in \cdot )}{\mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset D)}, \end{align} $$

for a partition function $\mathcal {G}$ as above with a decoration function $\Phi $ satisfying Properties (P1) to (P4).

Below, we list some needed properties of $\tau _{\beta }$ , proven in [Reference Dobrushin, Kotecký and Shlosman20].

For our main application of Theorem 1.2, the SOS model, the following result relates the surface tension defined in Equation (2.4) to the Ising polymer surface tension $\tau _\beta $ defined above. We suspect it is known, though could not find an exact reference for it, and as it follows from our other arguments in Section 5, we include its proof in Section 5.3 for completeness.

Proposition 2.5. Fix $\beta \geq \beta _0$ . Let $q(\gamma )$ denote the free polymer weight as per Equation (2.6) with $\Phi (\mathcal {C};\gamma )$ taken to be $\phi (\mathcal {C};\gamma )$ from Proposition 2.1. Consider the Ising polymer surface tension $\tau _{\beta }$ corresponding to the free polymer weights $q(\gamma )$ as per Equation (2.8). For all $\mathsf {u} \in \mathbb {R}^2$ ,

Observe that the decoration function $\phi (\mathcal {C};\gamma )$ appearing in the law of the unique open $1$ -contour under $\widehat {\pi }_{\Lambda }^{\xi }$ satisfies Properties (P1) to (P4). Indeed, Proposition 2.1 states that the decoration function $\phi (\mathcal {C};\gamma )$ satisfies Properties (P1) to (P3). Property (P4) holds for because of the symmetry of the SOS model and thus for $\tau _{\beta }$ thanks to Proposition 2.5.

However, the open $1$ -contour in the SOS model is not a free Ising polymer due to the term appearing in Equation (2.1), which introduces an interaction of $\gamma $ with the boundary via the decoration function $\phi $ . This served as one of the motivations of [Reference Ioffe, Shlosman and Toninelli32] to study modified Ising polymers – the generalization of the free Ising polymers described above to allow domain-induced modifications on the ‘free’ weights (the SOS open contour does belong to that family of models: See Observation 2.7).

2.3 The modified Ising polymer model

For any $D \subset \mathbb {R}^2$ and for any decoration function $\Phi (\mathcal {C};\gamma )$ satisfying Properties (P1) to (P4), consider any function $\Phi _D(\mathcal {C};\gamma )$ satisfying

1. (M1) $\Phi _D(\mathcal {C}; \gamma ) = \Phi (\mathcal {C}; \gamma )$ for any $\mathcal {C} \subset D$ , and

2. (M2) $\Phi _D(\mathcal {C};\gamma )$ satisfies the same decay bound in Equation (2.5) for all $\mathcal {C}$ .

Call $\Phi _D(\mathcal {C};\gamma )$ a modified decoration function (with modifications outside of D), and define the modified polymer weight

$$\begin{align*}q_{D}(\gamma) = \exp\Big(-\beta|\gamma| + \sum_{\mathcal{C} \cap \Delta_{\gamma} \neq \emptyset} \Phi_D(\mathcal{C};\gamma)\Big) \end{align*}$$

as well as the partition function

Definition 2.6 (Modified Ising polymer).

The modified Ising polymer in a subset $D \subset \mathbb {R}^2$ is given by the probability measure on contours $\gamma :\mathsf {o}^*\to \mathsf {x}$ contained in D:

(2.10) $$ \begin{align} \mathbf{P}_{D}^{\mathsf{x}}(\cdot) := \frac{\mathcal{G}_D(\mathsf{x} \;\big|\; \gamma \in \cdot)}{\mathcal{G}_D(\mathsf{x})} , \end{align} $$

where the partition function $\mathcal {G}_D$ is defined for a decoration function $\Phi $ satisfying Properties (P1) to (P4) and a modified decoration function $\Phi _D$ satisfying Properties (M1) and (M2).

Note that the SOS open contour with law $\widehat {\pi }_{\Lambda }^{\xi }$ given by Equation (2.1) is of the form above, with $\Phi (\mathcal {C};\gamma )=\phi (\mathcal {C};\gamma )$ , $D=\Lambda $ , and for $\phi (\mathcal {C};\gamma )$ from Proposition 2.1 (in which this choice of $\Phi _D(\mathcal {C};\gamma )$ clearly satisfies Properties (M1) and (M2)). Namely, the following holds:

It will be very convenient to view Ising polymers as connected paths in the lattice $\mathbb {Z}^2$ rather than the dual lattice $(\mathbb {Z}^2)^*$ via the translation map $\iota : \mathsf {o}^* \mapsto 0$ , and this is the convention we will follow for the remainder of the article (for clarity, note also that, through this mapping, clusters $\mathcal {C}$ will henceforth denote finite, connected subsets of $(\mathbb {Z}^2)^*$ instead of $\mathbb {Z}^2$ ), excluding Section 3.

Let us now restate Theorem 1.2 in the above language, which is the form in which we will prove it (Section 5.7). For $N \in \mathbb {N}$ , define $Q:= [0,N]^2$ .

2.4 Nonnegative decoration functions and a product structure

Following [Reference Ioffe, Shlosman and Toninelli32, Section 3.1], we employ a construction going back to [Reference Dobrushin and Shlosman21] that allows us to consider decoration functions which are nonnegative. In this discussion, we allow for the case $D =\mathbb {R}^2$ , in which case $\Phi _D(\mathcal {C};\gamma ) = \Phi (\mathcal {C};\gamma )$ and $q_D(\gamma ) =q(\gamma )$ .

For a contour $\gamma $ , we consider the set of (not necessarily distinct) bonds in $\mathbb {Z}^2$ (recall we have applied the translation map $\iota : \mathsf {o}^* \mapsto 0$ ):

$$\begin{align*}\nabla_{\gamma} := \bigcup_{b = (\mathsf{y}, \mathsf{y}+\mathsf{e}_i) \in \gamma} \{ b, b+\mathsf{e}_i,b-\mathsf{e}_i\} . \end{align*}$$

To be clear, b is a bond in $\gamma $ , $\mathsf {e}_i$ denotes a standard basis vector, and $b\pm \mathsf {e}_i$ denotes the bond obtained by translating b by $\pm \mathsf {e}_i$ . Thus, $\nabla _{\gamma }$ contains three bonds for each bond of $\gamma $ . Define

$$ \begin{align*} \Phi_{D}'(\mathcal{C};\gamma) &:= \left|\mathcal{C} \cap \nabla_{\gamma}\right|e^{-\chi \beta (d(\mathcal{C})+1)} + \Phi_D(\mathcal{C};\gamma), \end{align*} $$

where $|\mathcal {C} \cap \nabla _{\gamma }|$ is equal to the number of bonds, counted with multiplicity, in $\nabla _{\gamma }$ that $\mathcal {C}$ intersects:

Observe that $\Phi ^{\prime }_D \geq 0$ by Equation (2.5) and Property (M2). For any fixed bond $b \in \mathbb {Z}^2$ , let $c(\beta )$ denote the value of

$$\begin{align*}c(\beta) := \sum_{\mathcal{C} \subset (\mathbb{Z}^2)^* ,\, \mathcal{C} \cap b \neq \emptyset} e^{-\chi\beta( d(\mathcal{C})+1)}. \end{align*}$$

Note that

(2.11) $$ \begin{align} c(\beta) = \sum_{m \geq 1} e^{-\chi\beta(m+1)} \left|\{\mathcal{C} \subset (\mathbb{Z}^2)^*: \mathcal{C} \cap b \neq \emptyset ,d(\mathcal{C}) = m \}\right| \leq e^{-\chi \beta}, \end{align} $$

where we used that $\left |\{\mathcal {C} \subset (\mathbb {Z}^2)^*: \mathcal {C} \cap b \neq \emptyset ,d(\mathcal {C}) = m \}\right | \leq e^{c m}$ for some $c>0$ . (To see this, replace the marked $e\in \mathcal {C}$ intersecting b by a marked boundary edge $e_0\in \partial \mathcal {C}$ with the same y-coordinate (say) at the cost of a factor of m; then, when enumerating the smallest connected set of edges of $\mathbb {Z}^2$ containing the marked $e_0$ and specified edges $\partial \mathcal {C}$ , regard $\partial \mathcal {C}$ as the vertices of a 6-regular graph (whose vertices are the $\mathbb {Z}^2$ bonds and two are adjacent if they share an end point; that is, the line graph of $\mathbb {Z}^2$ ), and recall that in a graph whose maximum degree is $\Delta $ , the number of m-vertex connected subgraphs rooted at a given vertex is at most $(e \Delta )^m$ .) Using the fact that $\mathcal {C}\cap \Delta _{\gamma } \neq \emptyset $ implies $\mathcal {C}\cap \nabla _{\gamma } \neq \emptyset $ , we have

$$ \begin{align*} \sum_{\mathcal{C} \cap \Delta_{\gamma} \neq \emptyset} \Phi_D(\mathcal{C};\gamma) = -3c(\beta)|\gamma| + \sum_{\mathcal{C} \cap \nabla_{\gamma} \neq \emptyset} \Phi_{D}'(\mathcal{C};\gamma). \end{align*} $$

Thus, we have

$$ \begin{align*} q_{D}(\gamma) = \exp \Big( -(\beta+3c(\beta))|\gamma| + \sum_{\mathcal{C} \cap \nabla_{\gamma} \neq \emptyset} \Phi_{D}'(\mathcal{C};\gamma)\Big). \end{align*} $$

Since $f(\beta ):= \beta +3c(\beta )$ is strictly increasing for all $\beta $ large enough, we will henceforth redefine $\beta = f(\beta )$ so that we may drop the laborious $3c(\beta )$ from our weights:

(2.12) $$ \begin{align} q_{D}(\gamma) = \exp \Big( -\beta|\gamma| + \sum_{\mathcal{C} \cap \nabla_{\gamma} \neq \emptyset} \Phi_{D}'(\mathcal{C};\gamma)\Big) . \end{align} $$

A useful comparison to record at this stage is that, for any $D \subset \mathbb {R}^2$ ,

(2.13) $$ \begin{align} \left| \log \frac{q_D(\gamma)}{q(\gamma)}\right| \leq 6 e^{-\chi \beta}|\gamma|. \end{align} $$

We next uncover the product structure of $q_D$ . Defining , we may write

$$\begin{align*}\exp\Big(\sum_{\mathcal{C} \cap \nabla_{\gamma} \neq \emptyset} \Phi_{D}'(\mathcal{C};\gamma)\Big) = \prod_{\mathcal{C} \cap \nabla_{\gamma} \neq \emptyset} \Big(\big(e^{ \Phi_{D}'(\mathcal{C};\gamma)} -1\big) +1\Big) = \sum_{\underline{\mathcal{C}} = \{\mathcal{C}_i\}} \prod_i \Psi_{D}(\mathcal{C}_i;\gamma), \end{align*}$$

where the sum goes over all possible finite collections $\underline {\mathcal {C}}$ of clusters. Given this, for any contour $\gamma $ and any collection $\underline {\mathcal {C}}$ of clusters, we define the animal weight $q_{D}(\Gamma )$ of the animal $\Gamma = [\gamma , \underline {\mathcal {C}}]$ by

(2.14) $$ \begin{align} q_{D}(\Gamma) = q_{D}([\gamma,\underline{\mathcal{C}}]) := e^{-\beta|\gamma|}\prod_{\mathcal{C} \in \underline{\mathcal{C}}} \Psi_{D}(\mathcal{C}; \gamma) , \end{align} $$

and observe

$$\begin{align*}q_{D}(\gamma) = \sum_{\Gamma = [\gamma,\underline{\mathcal{C}}]}q_{D}(\Gamma). \end{align*}$$

The above allows us to consider the free and modified Ising polymer measures as probability measures on animals:

(2.15) $$ \begin{align} \mathbf{P}^{\mathsf{x}}(\cdot \;\big|\; \gamma \subset D) = \frac{\mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset D, \Gamma \in \cdot)}{\mathcal{G}(\mathsf{x}\;\big|\; \gamma \subset D)} \qquad \text{and}\qquad \mathbf{P}_{D}^{\mathsf{x}}(\cdot) := \frac{\mathcal{G}_D(\mathsf{x} \;\big|\; \Gamma \in \cdot)}{\mathcal{G}_D(\mathsf{x})} . \end{align} $$

Due to the product structure of $q_{D}(\Gamma )$ , it is often convenient to consider animals rather than contours; indeed, we will see in Section 4.1 that the product structure begets a connection with a random walk, which is crucial to our analysis.

When D is taken to be $\mathbb {R}^2$ , we will omit D from the notation laid out above.

2.5 Notation for Ising polymers and animals

Below, we set notation that will be used throughout the article in the context of Ising polymers and animals. Recall that we regard Ising polymers as connected paths in the lattice $\mathbb {Z}^2$ rather than the dual lattice $(\mathbb {Z}^2)^*$ via the translation map $\iota : \mathsf {o}^* \mapsto 0$ (the convention hereafter, excluding Section 3).

• • We write $Q:= [0,N]^2$ .

• • For a point $\mathsf {u} \in \mathbb {Z}^2$ , we will write $\mathsf {u}_{1}$ and $\mathsf {u}_{2}$ to denote the x-coordinate and y-coordinate of $\mathsf {u}$ , respectively.

• • For a subset $E \subset \mathbb {Z}^2$ , we’ll write $\mathsf {u}+ E$ to denote the translation of E by the vector defined by $\mathsf {u}$ .

• • For a contour $\gamma $ , we write $|\gamma |$ to denote the number of bonds in $\gamma $ . We write $(\gamma (0), \dots , \gamma (|\gamma |))$ to denote the ordered vertices of $\gamma $ .

• • For an animal $\Gamma := [\gamma , \underline {\mathcal {C}}]$ , we write $|\Gamma | := |\gamma |$ , and $\mathsf {X}(\Gamma ) := \mathsf {X}(\gamma )$ to denote the displacement of $\gamma $ ; that is, the vector in $\mathbb {Z}^2$ given by the end point of $\gamma $ minus the start point of $\gamma $ .

• • For a pair of contours $\gamma := (\gamma (0), \dots , \gamma (|\gamma |))$ and $\gamma ':= (\gamma '(0), \dots , \gamma '(|\gamma '|))$ , let $\Delta := \gamma (|\gamma |) - \gamma '(0)$ . We define their concatenation to be

  $$\begin{align*}\gamma \circ \gamma' := \big( \gamma(0), \dots, \gamma(|\gamma|),\Delta+ \gamma'(1), \dots, \Delta+ \gamma'(|\gamma'|) \big ) . \end{align*}$$
  Similarly, for a pair of animals $\Gamma := [\gamma , \underline {\mathcal {C}}]$ and $\Gamma ':= [\gamma ', \underline {\mathcal {C}}']$ , we define their concatenation to be
  $$\begin{align*}\Gamma \circ \Gamma' := [ \gamma \circ \gamma', ~\underline{\mathcal{C}} \cup \underline{\mathcal{C}}']. \end{align*}$$

• • For ${\mathsf {u}}, {\mathsf {v}} \in \mathbb {Z}^2$ , we write $\gamma : {\mathsf {u}} \to {\mathsf {v}}$ to indicate that $\gamma (0) = {\mathsf {u}}$ and $\gamma (|\gamma |) = {\mathsf {v}}$ . We write $\Gamma : {\mathsf {u}} \to {\mathsf {v}}$ to indicate that $\Gamma = [\gamma , \underline {\mathcal {C}}]$ for some contour $\gamma : {\mathsf {u}} \to {\mathsf {v}}$ .

• • For a set $D\subset \mathbb {R}^2$ , we say $\Gamma = [\gamma , \underline {\mathcal {C}}] \subset D$ if $\gamma $ as well as all clusters $\mathcal {C} \in \underline {\mathcal {C}}$ are contained in D.

• • For any $D \subset \mathbb {R}^2$ , we define the set of contours

  $$\begin{align*}\mathcal P_D({\mathsf{u}},{\mathsf{v}}):= \{\gamma: \gamma \subset D ,\, \gamma : {\mathsf{u}} \to {\mathsf{v}} \}. \end{align*}$$
  When ${\mathsf {u}} = 0$ , we will simply write $\mathcal P_D({\mathsf {v}})$ . We’ll write $\Gamma \in \mathcal P_D({\mathsf {u}},{\mathsf {v}})$ for animals $\Gamma $ to mean $\Gamma = [\gamma , \underline {\mathcal {C}}]$ for some contour $\gamma \in \mathcal P_D({\mathsf {u}},{\mathsf {v}})$ .

2.6 The Wulff shape

Now, define the Wulff shape

$$\begin{align*}\mathcal W := \bigcap_{\mathsf{y} \in \mathbb{R}^2} \{\mathsf{h} \in \mathbb{R}^2: \mathsf{h} \cdot \mathsf{y} \leq \tau_\beta(\mathsf{y}) \}, \end{align*}$$

which is clearly closed and convex (as it is the intersection of half-spaces). Observe that

$$\begin{align*}\mathcal W = \overline{\left\{\mathsf{h} \in \mathbb{R}^2: \sum_{\mathsf{y} \in \mathbb{Z}^2} e^{\mathsf{h}\cdot \mathsf{y}} \mathcal{G}(\mathsf{y}) < \infty\right\}}. \end{align*}$$

Indeed, from Equation (2.8), we have

(2.16) $$ \begin{align} \log \mathcal{G}(\mathsf{y}) = - \tau_\beta(\mathsf{y}) \big(1+o_{\left\Vert \mathsf{y} \right\Vert{}_1}(1)\big) . \end{align} $$

It follows that the sum in the second expression for $\mathcal W$ converges if and only if $\mathsf {h} \cdot \mathsf {y} < \tau _\beta (\mathsf {y})$ for all $\mathsf {y}\in \mathbb {Z}^2$ large, which by homogeneity and continuity of $\tau _{\beta }$ is equivalent to the same holding for all $\mathsf {y} \in \mathbb {R}^2$ . Including the equality case $\mathsf {h}\cdot \mathsf {y} = \tau _{\beta }(\mathsf {y})$ in the first expression for $\mathcal W$ is equivalent to taking the closure of the set in the second expression.

2.7 Cone points, the irreducible decomposition of animals and weight factorization

Let us define the forward cone $\mathcal {Y}^\blacktriangleleft := \{(x,y) \in \mathbb {Z}^2 : |y|\leq x \}$ and the backward cone $\mathcal {Y}^\blacktriangleright := -\mathcal {Y}^\blacktriangleleft $ . We will also need $\mathcal {Y}^\blacktriangleleft _{\delta }:= \{ (x,y) \in \mathbb {Z}^2 : |y|\leq \delta x\}$ for $\delta>0$ . Given a contour $\gamma $ and an animal $\Gamma =[\gamma ,\underline {\mathcal {C}}]$ , we say that ${\mathsf {u}}\in \gamma $ is a cone point for $\gamma $ if

$$\begin{align*}\gamma \subset {\mathsf{u}}+\mathcal{Y}^\blacktriangleright \cup {\mathsf{u}}+\mathcal{Y}^\blacktriangleleft, \end{align*}$$

and we say ${\mathsf {u}}\in \gamma $ is a cone point for $\Gamma $ if

$$\begin{align*}\Gamma \subset {\mathsf{u}}+\mathcal{Y}^\blacktriangleright \cup {\mathsf{u}}+\mathcal{Y}^\blacktriangleleft. \end{align*}$$

Recall from Section 2.5 that the previous display means that the forward and backward cones emanating from ${\mathsf {u}}$ fully contain $\gamma $ as well as all clusters $\mathcal {C}$ in $\underline {\mathcal {C}}$ . Of course if ${\mathsf {u}} \in \gamma $ is a cone point for $\Gamma $ , then ${\mathsf {u}}$ is a cone point for $\gamma $ as well. Note that for any ${\mathsf {u}}, {\mathsf {v}} \in \mathbb {Z}^2$ , ${\mathsf {u}} \in {\mathsf {v}} + \mathcal {Y}^\blacktriangleleft $ if and only if ${\mathsf {v}} \in {\mathsf {u}} +\mathcal {Y}^\blacktriangleright $ .

If $\Gamma $ has two cone points ${\mathsf {u}}$ and ${\mathsf {v}}$ , the contour part of $\Gamma $ between ${\mathsf {u}}$ and ${\mathsf {v}}$ as well as all associated clusters are entirely contained within the ‘diamond’ ${\mathsf {u}}+\mathcal {Y}^\blacktriangleleft \cap {\mathsf {v}}+\mathcal {Y}^\blacktriangleright $ . An animal is called left-irreducible if it contains no cone points and if it is entirely contained in the backwards cone emanating from its end point. Similarly, an animal is called right-irreducible if it contains no cone points and it is contained in the forward cone emanating from its start point. An animal is called irreducible if it is both left- and right-irreducible. We let $\mathsf {A_L}, \mathsf {A_R}$ and $\mathsf {A}$ be the sets of left-irreducible, right-irreducible and irreducible animals, respectively, with start point at the origin.

Consider now an animal $\Gamma $ with at least two cone points, and decompose it into (left-/right-) irreducible animals as follows:

(2.17) $$ \begin{align} \Gamma = \Gamma^{(L)} \circ \Gamma^{(1)} \circ \cdots \circ \Gamma^{(n)} \circ \Gamma^{(R)}, \end{align} $$

for some $n\geq 1$ , $\Gamma ^{(L)} \in \mathsf {A_L}$ , $\Gamma ^{(R)} \in \mathsf {A_R}$ , and $\Gamma ^{(i)} \in \mathsf {A}$ for $i =1,\dots ,n$ . For $i\in \{L, 1,\dots ,n, R\}$ , we will write

$$\begin{align*}\Gamma^{(i)} =: [\gamma^{(i)} , \underline{\mathcal{C}}_i]. \end{align*}$$

From the definition of a cone point, any cluster $\mathcal {C} \in \underline {\mathcal {C}}_i$ will be such that $\mathcal {C} \cap \nabla _{\gamma ^{(j)}} = \emptyset $ for $j \neq i$ . Informally, this means that any cluster of $\Gamma ^{(i)}$ will not be a cluster of $\Gamma ^{(j)}$ . Thus, the weights $q_D(\Gamma )$ ‘factorize’ into a product of the weights of the irreducible pieces:

(2.18) $$ \begin{align} q_D(\Gamma) = \prod_{i \in \{L, 1,\dots, n,R\}} e^{- \beta |\gamma^{(i)}|} \prod_{\mathcal{C} \in \underline{\mathcal{C}}_i} \Psi_D(\mathcal{C};\gamma) = q_D(\Gamma^{(L)}) q_D(\Gamma^{(R)})\prod_{i=1}^n q_D(\Gamma^{(i)}). \end{align} $$

When $\gamma \subset Q$ , note that the shape of the forward and backward cones necessitates

$$\begin{align*}\Gamma^{(1)} \circ \cdots \Gamma^{(n)} \subset [1,N-1] \times (-\infty, N]\, \end{align*}$$

so that Equation (2.18) becomes

(2.19) $$ \begin{align} q_Q(\Gamma) = q_Q(\Gamma^{(L)}) q_Q(\Gamma^{(R)}) \prod_{i=1}^n q_{{\mathbb H}}(\Gamma^{(i)}). \end{align} $$

We write $\mathsf {Cpts}(\gamma )$ and $\mathsf {Cpts}(\Gamma )$ to denote the set of cone points of $\gamma $ and the set of cone points of $\Gamma $ , respectively. Let $n:= n(\Gamma ) = |\mathsf {Cpts}(\Gamma )|-1$ as above so that n is equal to the number of irreducible pieces of $\Gamma $ . We label the cone points of $\Gamma $ by $\zeta ^{(1)}, \dots , \zeta ^{(n+1)}$ , where the ordering is strictly increasing in the x-coordinates, that is,

$$\begin{align*}\zeta_{1}^{(i)} < \zeta_1^{(i+1)} . \end{align*}$$

The set $\mathsf {Cpts}(\Gamma )$ therefore takes on a natural meaning as an ordered $n(\Gamma )-$ tuple, and so we will abuse notation and write

(2.20) $$ \begin{align} \mathsf{Cpts}(\Gamma) = \big(\zeta^{(i)}\big)_{i=1}^{n+1} = \big( \mathsf{X}(\Gamma^{(L)}), \mathsf{X}(\Gamma^{(L)}) + \mathsf{X}(\Gamma^{(1)}), \dots, \mathsf{X}(\Gamma^{(L)})+ \cdots + \mathsf{X}(\Gamma^{(R)}) \big ) . \end{align} $$

The following result of [Reference Ioffe, Shlosman and Toninelli32] states that the free weight of contours in $\mathbb {Z}^2$ that have large length and contain a sublinear number of cone points compared to the distance of the end point is exponentially small compared to $\mathcal {G}$ .

Lemma 2.11 ([Reference Ioffe, Shlosman and Toninelli32, Eq. (4.5)]Footnote ⁷ ).

Fix any $\epsilon ,\delta \in (0,1)$ . There exist $\beta _0 \in (0, \infty )$ , $\nu _0>0$ , $\delta _0>0$ and $c>0$ such that the following bounds hold uniformly over $\beta \geq \beta _0$ , $\mathsf {y} \in \mathcal {Y}^\blacktriangleleft _{\delta }\setminus \{0\}$ and $r \geq 1+\epsilon $ :

(2.21) $$ \begin{align} \mathcal{G}\big(\mathsf{y} \;\big|\; |\gamma| \geq r \left\Vert \mathsf{y} \right\Vert{}_1\big) &\leq ce^{-\nu_0 \beta r \left\Vert \mathsf{y} \right\Vert{}_1} \mathcal{G}(\mathsf{y})\end{align} $$

(2.22) $$ \begin{align} \mathcal{G}\big(\mathsf{y} \;\big|\; |\mathsf{Cpts}(\gamma)| < 2\delta_0 \left\Vert \mathsf{y} \right\Vert{}_1 \big) &\leq c e^{- \nu_0 \beta \left\Vert \mathsf{y} \right\Vert{}_1} \mathcal{G}(\mathsf{y}). \end{align} $$

In [Reference Ioffe, Shlosman and Toninelli32],Footnote ⁸ it is claimed without proof that the an analogue of Lemma 2.11 also holds for animals. For completeness, we supply the result as well as a proof below.

Proposition 2.12. For any $\delta \in (0,1)$ , there exist $\nu>0$ and $c>0$ such that the following bounds hold uniformly over $\beta \geq \beta _0$ , and $\mathsf {y} \in \mathcal {Y}^\blacktriangleleft _{\delta }\setminus \{0\}$ :

$$\begin{align*}\mathcal{G}(\mathsf{y} \;\big|\; |\mathsf{Cpts}(\Gamma)| < \delta_0\|\mathsf{y}\|_1) \leq ce^{-\nu\beta \|\mathsf{y}\|_1}\mathcal{G}(\mathsf{y}) , \end{align*}$$

where $\delta _0$ is as in Lemma 2.11.

Proof. The idea is to show that, for a typical $\Gamma = [\gamma ,\underline {\mathcal {C}}]$ , many of the cone points for $\gamma $ are also cone points for $\Gamma $ . We do this by showing that typically, $\mathcal {C}$ does not contain many clusters (more precisely, the total $d(\cdot )$ -size is not too big). The result then follows from Equation (2.22).

Our starting point is the following, which comes from Property (P2) and Remark 2.10 and states that for some constant $c_1>0$ , for all $\beta>0$ sufficiently large, and for any animal $[\gamma , \underline {\mathcal {C}}]$ ,

(2.23) $$ \begin{align} q([\gamma, \underline{\mathcal{C}}]) \leq e^{-\beta|\gamma|}\exp\bigg({-c_1\beta\sum_{\mathcal{C} \in \underline{\mathcal{C}}} d(\mathcal{C})}\bigg). \end{align} $$

Now, define the events

$$ \begin{align*} A &= \{\Gamma = [\gamma,\underline{\mathcal{C}}] \;\big|\; |\mathsf{Cpts}(\Gamma)| < \delta_0\|\mathsf{y}\|_1\} \\ B &= \{\Gamma = [\gamma,\underline{\mathcal{C}}] \;\big|\; |\mathsf{Cpts}(\gamma)| \geq 2\delta_0\|\mathsf{y}\|_1, \,|\gamma| \leq 1.1 \|\mathsf{y}\|_1 \}. \end{align*} $$

Then, from Equations (2.21) and (2.22), we have

$$\begin{align*}\frac{\mathcal{G}(\mathsf{y} \;\big|\; A)}{\mathcal{G}(\mathsf{y})} = \frac{\mathcal{G}(\mathsf{y} \;\big|\; A, B)}{\mathcal{G}(\mathsf{y})} + O\big( e^{-\nu_0 \beta \left\Vert \mathsf{y} \right\Vert{}_1}\big). \end{align*}$$

Now, consider a contour $\gamma $ with at least $2\delta _0\|\mathsf {y}\|_1$ cone points and an animal $\Gamma $ such that $\Gamma = [\gamma , \underline {\mathcal {C}}]$ for some set of clusters $\underline {\mathcal {C}}$ . Note that if $\Gamma $ has less than $\delta _0\|\mathsf {y}\|_1$ cone points, then the clusters of $\underline {\mathcal {C}}$ must intersect $\gamma $ in such a way that at least $\delta _0\|\mathsf {y}\|_1$ cone points of $\gamma $ are not cone points of $\Gamma $ . This necessitates that the sum of $d(\mathcal {C})$ over $\mathcal {C} \in \underline {\mathcal {C}}$ exceeds $\delta _0 \|\mathsf {y}\|_1$ . Hence, we can write

$$ \begin{align*} &\mathcal{G}(\mathsf{y} \;\big|\; A, B) \\ \leq &\sum_{\substack{\gamma:0\to \mathsf{y} \\ |\gamma|\leq 1.1 \|\mathsf{y}\|_1}}\sum_{m \geq \delta_0 \left\Vert \mathsf{y} \right\Vert{}_1} \sum_{n=1}^{m}\sum_{k=1}^{n}\binom{3|\gamma|}{k} \sum_{\substack{\underline{n} = (n_1,\dots,n_k) \\ n_1+\dots+n_k=n}} \sum_{\substack{\underline{m} = (m_1,\dots,m_n) \\ m_1+\dots+m_n=m}} \bigg(\prod_{i=1}^k|G_{i, \underline{n}, \underline{m}}|\bigg) e^{-\beta|\gamma|-c_1\beta m}, \end{align*} $$

where: the second sum accounts for the possible values of $m =\sum _{\mathcal {C}\in \underline {\mathcal {C}}} d(\mathcal {C})$ ; the third sum accounts for the possible values of $n=|\underline {\mathcal {C}}|$ given m; the fourth sum and the binominal coefficient account for the number of possible bonds of $\nabla _{\gamma }$ that the clusters of $\underline {\mathcal {C}}$ intersect; the fifth sum accounts for the number of clusters that intersect a particular bond of $\nabla _{\gamma }$ (given k and an arbitrary labeling of the bonds $b_1, \dots , b_k$ ); the sixth sum accounts for the possible $d(\cdot )$ -value of each of the n clusters given that their total $d(\cdot )$ -sum is m, and where $\underline {m}=(m_1,\dots ,m_n)$ is ordered such that $m_1,\dots ,m_{n_1}$ represent the $d(\cdot )$ -values of the $n_1$ clusters intersecting $b_1$ , $m_{n_1+1},\dots ,m_{n_1+n_2}$ represent the $d(\cdot )$ -values of the $n_2$ clusters intersecting $b_2$ and so on; each $G_{i, \underline {n}, \underline {m}}$ denotes the set of all possible collections of the $n_i$ clusters intersecting $b_i$ given the aforementioned $\underline {n}$ and $\underline {m}$ ; and lastly, the exponential is the cumulative upper bound on $q([\gamma ,\underline {\mathcal {C}}])$ given by Equation (2.23) (given that the cumulative $d(\cdot )$ is m).

Now, similar to Equation (2.11), the number of clusters $\mathcal {C}$ with $d(\mathcal {C})= r$ and intersecting some fixed bond b is bounded from above by $e^{c_2 r}$ , and therefore,

$$\begin{align*}\prod_{i=1}^k|G_{i, \underline{n}, \underline{m}}| \leq e^{c_2 (m_1+\dots+m_n)} = e^{c_2 m}. \end{align*}$$

Hence, $\mathcal {G}(\mathsf {y} \;\big |\; A, B)$ is upper bounded by

$$ \begin{align*} &\sum_{\substack{\gamma:0\to \mathsf{y} \\ |\gamma|\leq 1.1 \|\mathsf{y}\|_1}}e^{-\beta|\gamma|}\sum_{m \geq \delta_0 \left\Vert \mathsf{y} \right\Vert{}_1} \sum_{n=1}^{m}\sum_{k=1}^{n}\binom{3|\gamma|}{k} \sum_{\substack{\underline{n} = (n_1,\dots,n_k) \\ n_1+\dots+n_k=n}} \sum_{\substack{\underline{m} = (m_1,\dots,m_n) \\ m_1+\dots+m_n=m}} e^{-c_1\beta m}e^{c_2 m} \\& \quad \leq\sum_{\substack{\gamma:0\to \mathsf{y} \\ |\gamma|\leq 1.1 \|\mathsf{y}\|_1}}e^{-\beta|\gamma|}\sum_{m \geq \delta_0 \left\Vert \mathsf{y} \right\Vert{}_1}e^{-c_3\beta m} \sum_{n=1}^{m}\sum_{k=1}^{n}\binom{3|\gamma|}{k} \binom{n-1} {k-1} \binom{m-1}{n-1} \\& \quad \leq\sum_{\substack{\gamma:0 \to \mathsf{y} \\ |\gamma|\leq 1.1 \|\mathsf{y}\|_1}}e^{-\beta|\gamma|} \sum_{m \geq \delta_0 \left\Vert \mathsf{y} \right\Vert{}_1}e^{-c_3\beta m} 2^m (2^m-1) 2^{3|\gamma|} \leq \ e^{-c_4\beta \left\Vert \mathsf{y} \right\Vert{}_1}\sum_{\gamma:0\to\mathsf{y}}e^{-\beta|\gamma|}. \end{align*} $$

Finally, the nonnegativity of the decorations $\Phi '$ implies

$$\begin{align*}\mathcal{G}(\mathsf{y}) = \sum_{\substack{\Gamma = [\gamma, \underline{\mathcal{C}}] \\ \gamma:0 \to\mathsf{y}}}q([\gamma, \underline{\mathcal{C}}]) = \sum_{\gamma:0 \to\mathsf{y}}\exp\bigg({-\beta|\gamma|+\sum_{\substack{\mathcal{C} \subset \Lambda \\ \mathcal{C} \cap \Delta_{\gamma} \neq \emptyset}} \Phi'(\mathcal{C};\gamma)}\bigg) \geq \sum_{\gamma:0 \to\mathsf{y}}e^{-\beta|\gamma|}, \end{align*}$$

and thus

$$\begin{align*}\frac{\mathcal{G}(\mathsf{y} \;\big|\; A, B)}{\mathcal{G}(\mathsf{y})} \leq e^{-c_5\beta \left\Vert \mathsf{y} \right\Vert{}_1}, \end{align*}$$

concluding the proof.

2.8 Ornstein–Zernike theory and its applications

For $\mathsf {h} \in \mathbb {R}^2$ , we define $W^{\mathsf {h}}(\cdot )$ by

$$ \begin{align*} W^{\mathsf{h}}(\Gamma) := e^{\mathsf{h} \cdot \mathsf{X}(\Gamma)}q(\Gamma) \end{align*} $$

for any animal $\Gamma $ (not necessarily irreducible). For $\mathsf {y}\neq 0$ , define the dual parameter $\mathsf {h}_{\mathsf {y}}$ by

$$\begin{align*}\mathsf{h}_{\mathsf{y}} = \nabla \tau_\beta(\mathsf{y}). \end{align*}$$

The homogeneity of $\tau _\beta (\mathsf {y})$ as per Equation (2.8) implies that $\mathsf {h}_{\mathsf {y}} \in \partial \mathcal W$ (since $\mathsf {h}_{\mathsf {y}} \cdot \mathsf {y} = \nabla \tau _\beta (\mathsf {y})\cdot \mathsf {y} = \tau _\beta (\mathsf {y})$ and recalling the first definition of $\mathcal W$ ).

Proposition 2.14 below, which forms the main result of the Ornstein–Zernike theory for the Ising polymer models, was stated in [Reference Ioffe, Shlosman and Toninelli32, Thm. 5] in a slightly different setting (namely, the cones are defined differently here), pointing to [Reference Ioffe and Velenik34] as its relevant input. Indeed, one may infer this result via an Ornstein–Zernike analysis as in [Reference Ioffe and Velenik34, Sec 3.3 and 3.4], but since (a) the models considered in that work do not include Ising polymers, and (b) our setting differs somewhat from that of [Reference Ioffe, Shlosman and Toninelli32], we include a full proof of this proposition in Appendix B.

Proposition 2.14. For any $\delta \in (0,1)$ , there exists $\beta _0>0$ such that for all $\beta>\beta _0$ and for any $\mathsf {y} \in \mathcal {Y}^\blacktriangleleft _{\delta }\setminus \{0\}$ , the collection of weights $W^{\mathsf {h}_{\mathsf {y}}}$ defines a probability distribution on the set $\mathsf {A}$ of irreducible animals. To emphasize that $W^{\mathsf {h}_{\mathsf {y}}}$ defines a probability distribution (on irreducible animals), and for consistency with [Reference Ioffe, Shlosman and Toninelli32], we use the notation

(2.24) $$ \begin{align} \mathbb{P}^{\mathsf{h}_{\mathsf{y}}}(\Gamma) := e^{\mathsf{h}_{\mathsf{y}}\cdot \mathsf{X}(\Gamma)} q(\Gamma) = W^{\mathsf{h}_{\mathsf{y}}}(\Gamma) . \end{align} $$

Let $\mathbb {E}^{\mathsf {h}_{\mathsf {y}}}$ denote expectation under $\mathbb {P}^{\mathsf {h}_{\mathsf {y}}}$ . Then

(2.25) $$ \begin{align} \mathbb{E}^{\mathsf{h}_{\mathsf{y}}}[\mathsf{X}(\Gamma)] = \alpha \mathsf{y} , \end{align} $$

for some constant $\alpha := \alpha (\beta , \mathsf {y})>0$ – in particular, the expectation of $\mathsf {X}(\Gamma )$ under $\mathbb {P}^{\mathsf {h}_{\mathsf {y}}}$ is collinear to $\mathsf {y}$ . Finally, there exists a ‘mass-gap’ constant $\nu _g>0$ such that for all $\beta>0$ large, $\mathsf {y} \in \mathcal {Y}^\blacktriangleleft _{\delta } \setminus \{0\}$ , and $k \geq 1$ ,

(2.26)

where $C:= C(\beta )>0$ is a positive constant.

Note that, since $\mathsf {h}_{\mathsf {y}} \cdot \mathsf {y} = \tau _{\beta }(\mathsf {y})$ (by the homogeneity of $\tau _\beta $ as mentioned above), we have the following from Equation (2.18)

(2.27) $$ \begin{align} q(\Gamma) = e^{-\tau_{\beta}(\mathsf{y})} \mathbb{P}^{\mathsf{h}_{\mathsf{y}}}(\Gamma^{(L)}) \mathbb{P}^{\mathsf{h}_{\mathsf{y}}}(\Gamma^{(R)}) \prod_{i=1}^n \mathbb{P}^{\mathsf{h}_{\mathsf{y}}}(\Gamma^{(i)}) , \end{align} $$

for any $\Gamma $ with at least two cone points. For ${\mathsf {u}}, {\mathsf {v}} \in \mathbb {Z}^2$ and a set of animals E, introduce

(2.28)

When E is taken to be the set of all possible animals, we’ll simply write $\mathcal {A}({\mathsf {u}},{\mathsf {v}}; E) = \mathcal {A}({\mathsf {u}},{\mathsf {v}})$ . Then, for $D = Q$ or ${\mathbb H}$ , Equation (2.27) allows us to write

(2.29) $$ \begin{align} &\mathcal{G}(\mathsf{y} \;\big|\; \gamma \subset D, |\mathsf{Cpts}(\Gamma)|\geq 2) \nonumber\\ &\quad= \ e^{-\tau_{\beta}(\mathsf{y})} \sum_{\substack{\Gamma^{(L)} \in \mathsf{A_L} \\ \gamma^{(L)} \subset D}} \mathbb{P}^{\mathsf{h}_{\mathsf{y}}}(\Gamma^{(L)}) \ \sum_{\substack{\Gamma^{(R)} \in \mathsf{A_R} \\ \gamma^{(R)} \subset D}} \mathbb{P}^{\mathsf{h}_{\mathsf{y}}}(\Gamma^{(R)}) \ \mathcal{A}( \mathsf{X}(\Gamma^{(L)}), \mathsf{y}- \mathsf{X}(\Gamma^{(R)}) ). \end{align} $$

Note that the sum over n in Equation (2.28) above closely resembles the probability of a random walk event. This product structure of probability weights $\mathbb {P}^{\mathsf {h}_{\mathsf {y}}}$ naturally leads to considering the random walk model related to the law of $\mathsf {Cpts}(\Gamma )$ described in Section 4.1.

4.1 The random walk model and free Ising polymers

Consider the random walk $(\mathsf {S}(i))_{i \in \mathbb {Z}_{\geq 0}}$ , whose law will be denoted by $\mathbb {P}$ , with i.i.d. increments $\{X(i) = (X_1(i), X_2(i))\}_{i \geq 1}$ of step distribution X, where for any $\vec {v} \in \mathbb {Z}^2$ ,

(4.1)

We will denote by $(\mathsf {S}_1(\cdot ),\mathsf {S}_2(\cdot ))$ the two coordinates of $\mathsf {S}(\cdot )$ . For ${\mathsf {u}} \in \mathbb {Z}^2$ , we will write $\mathbb {P}_{{\mathsf {u}}}$ to denote the law of $\mathsf {S}(\cdot )$ started from $\mathsf {S}(0) = {\mathsf {u}}$ . From Equation (2.25), we have $\mathbb {E}[X] = \alpha \mathsf {x}$ , and from Equation (2.26), we inherit exponential tail decay: For all $\beta $ sufficiently large, there exists a constant $C':= C'(\beta )>0$ such that for all $k \geq 1$ , we have

(4.2) $$ \begin{align} \sum_{\vec{v} \in \mathbb{Z}^2: \left\Vert \vec{v} \right\Vert{}_1 \geq k} \mathbb{P}(X = \vec{v}) \leq C' e^{-\nu_g \beta k}. \end{align} $$

We remark that the step distribution of X need not be symmetric in the y-coordinate $X_2$ due to the northeast splitting rule (this is the key difference between our random walk and the random walk considered in [Reference Ioffe, Ott, Velenik and Wachtel31] and why we cannot use their random walk results here). Also note that $\mathbb {P}(X \in \mathcal {Y}^\blacktriangleleft ) = 1$ , and thus the x-coordinate $X_1>0$ a.s.

We will write $H_{E}^A$ to denote the first hitting time of a set $E \subset \mathbb {R}^2$ by a stochastic process $A_.$ , omitting the A from the notation when the stochastic process is clear. For singleton sets, we will simply write $H_{{\mathsf {u}}}^A$ rather than $H_{\{{\mathsf {u}}\}}^A$ . Let ${\mathbb H}_{-} := \mathbb {R} \times \mathbb {R}_{<0}$ . In light of the factorization (2.27), for fixed $\Gamma ^{(L)}$ and $\Gamma ^{(R)}$ , we will be interested in our random walk $\mathsf {S}(\cdot )$ started from $\Gamma ^{(L)}$ and conditioned on the event

$$\begin{align*}\{ H_{\mathsf{x} - \mathsf{X}(\Gamma^{(R)})} < H_{{\mathbb H}_{-}} \} . \end{align*}$$

It will therefore be helpful to obtain an expression for $\mathbb {P}_{{\mathsf {u}}}(H_{{\mathsf {v}}} < H_{{\mathbb H}_-})$ in terms of animal weights, for any ${\mathsf {u}}, {\mathsf {v}} \in \mathbb {Z}^2$ . Towards this end, define the sets

(4.3) $$ \begin{align} \mathsf{V}^+_{{\mathsf{u}}, {\mathsf{v}}} &:= \{ (\vec{v}_1, \dots, \vec{v}_n) \in (\mathbb{Z}^2)^n : n \geq 1, {\mathsf{u}}+\vec{v}_1+ \dots+ \vec{v}_m = {\mathsf{v}}, {\mathsf{u}}+\vec{v}_1+\dots+\vec{v}_i \in {\mathbb H} \ ~\forall 1 \leq i \leq n \} \nonumber \\ \mathsf{A}^+_{{\mathsf{u}},{\mathsf{v}}} &:= \{ (\Gamma^{(1)}, \dots, \Gamma^{(n)}) \in \mathsf{A}^n : n \geq 1 , \big(\mathsf{X}(\Gamma^{(1)}), \dots, \mathsf{X}(\Gamma^{(n)})\big) \in \mathsf{V}^+_{{\mathsf{u}}, {\mathsf{v}}} \}, \end{align} $$

and, for a set of animals E, introduce

When E is taken to be the set of all possible animals, we’ll simply write $\mathcal {A}^+({\mathsf {u}},{\mathsf {v}}; E) = \mathcal {A}^+({\mathsf {u}},{\mathsf {v}})$ . Observe that

(4.4) $$ \begin{align} \mathbb{P}_{{\mathsf{u}}}\big(H_{{\mathsf{v}}} < H_{{\mathbb H}_{-}} \big) = \sum_{n \geq 1} \sum_{(\vec{v}_1, \dots, \vec{v}_n) \in \mathsf{V}^+_{{\mathsf{u}}, {\mathsf{v}}}} \prod_{i=1}^n \mathbb{P}(X=\vec{v}_i) = \mathcal{A}^+({\mathsf{u}}, {\mathsf{v}}). \end{align} $$

In words, the above equation states that the probability of the random walk hitting ${\mathsf {v}}$ before leaving ${\mathbb H}_{-}$ is equal to a sum over all possible products of weights of irreducible animals whose concatenation has all cone points in ${\mathbb H}$ .

Recall $\mathcal {A}(\cdot ,\cdot ;\cdot )$ from Equation (2.28), and note that for any ${\mathsf {u}}, {\mathsf {v}} \in \mathbb {Z}^2$ and any set of animals E,

(4.5) $$ \begin{align} \mathcal{A}({\mathsf{u}},{\mathsf{v}};E) \leq \mathcal{A}^+({\mathsf{u}},{\mathsf{v}};E) , \end{align} $$

since the (only) difference between $\mathcal {A}^+({\mathsf {u}}, {\mathsf {v}};E)$ and $\mathcal {A}({\mathsf {u}}, {\mathsf {v}};E)$ is that the sum defining $\mathcal {A}({\mathsf {u}}, {\mathsf {v}};E)$ restricts to tuples of irreducible animals such that the entire concatenation ${\mathsf {u}}+ \Gamma ^{(1)} \circ \cdots \circ \Gamma ^{(n)}$ is contained in ${\mathbb H}$ , while the sum defining $\mathcal {A}^+({\mathsf {u}}, {\mathsf {v}};E)$ requires only the cone points to stay in ${\mathbb H}$ . In particular, $\mathsf {Cpts}(\Gamma )$ (viewed as an ordered tuple, see Equation (2.20)) under the conditioned law $\mathbf {P}^{\mathsf {x}}(\cdot \;\big |\; \Gamma ^{(L)}, \Gamma ^{(R)}, \gamma \subset {\mathbb H})$ and the trajectory of the random walk $\mathbb {P}_{\mathsf {X}(\Gamma ^{(L)})}(\cdot \;\big |\; H_{\mathsf {x}- \mathsf {X}(\Gamma ^{(R)})} < H_{{\mathbb H}_-})$ only differ due to these requirements.

Ultimately, we will achieve a coupling between $\mathsf {Cpts}(\Gamma )$ with the trajectory of the random walk because of an entropic repulsion result (Proposition 5.11): The cone points of $\Gamma $ in the ‘bulk’ of the strip $[0,N] \times [0,\infty )$ stay far away from the boundary of ${\mathbb H}$ with high probability so that the aforementioned difference between $\mathcal {A}^+$ and $\mathcal {A}$ becomes negligible. A major step towards showing entropic repulsion is is the following crucial result from [Reference Ioffe, Shlosman and Toninelli32].Footnote ⁹

Proposition 4.1 [Reference Ioffe, Shlosman and Toninelli32, Theorem 7].

Recall $\nu _g$ from Proposition 2.14. There exists $\overline {\delta } \in (0,\nu _g/4)$ and $\beta _0>0$ such that for all $\beta> \beta _0$ , there exists a constant $C:=C(\beta )> 0$ such that

$$\begin{align*}\sup_{\substack{{\mathsf{u}}, \mathsf{x} - {\mathsf{v}} \in \mathcal{Y} \\ {\mathsf{u}}, {\mathsf{v}} \in {\mathbb H}}} e^{-\overline{\delta} \beta ( \left\Vert {\mathsf{u}} \right\Vert{}_1 + \left\Vert \mathsf{x}-{\mathsf{v}} \right\Vert{}_1 )} \mathbb{P}_{{\mathsf{u}}}(H_{{\mathsf{v}}} < H_{{\mathbb H}_{-}}) \leq C e^{\tau_{\beta}(\mathsf{x})} \mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset {\mathbb H}). \end{align*}$$

Proposition 4.1 combined with the exponential tail decay of $\Gamma ^{(L)}$ and $\Gamma ^{(R)}$ (2.26) (note the significance of $\bar {\delta } < \nu _g/4$ ) will allow us to eliminate bad events for $\mathsf {Cpts}(\Gamma )$ under the free Ising polymer law via estimates for random walks in a half-space, which are significantly easier to obtain compared to directly analyzing the polymer law.

4.2 Limit theorems for our random walk, confined to the half-space

We now state two important limit theorems, Theorems 4.2 and 4.3, for our random walk $\mathsf {S}(\cdot )$ confined to positive half-space, though our results apply to a more general class of 2D random walks. Our random walk $\mathsf {S}(\cdot )$ is on $\mathbb {Z}^2$ and has i.i.d. increments X with mean $(\mu ,0)$ for some $\mu>0$ , exponential tails and satisfies $X_1>0$ a.s. Let us also write $\operatorname {Var}(X) = (\sigma _1^2, \sigma _2^2)$ .

Let $V_1$ denote the unique positive harmonic function for the one-dimensional random walk $\mathsf {S}_2$ killed upon leaving $(0,\infty )$ satisfying

$$\begin{align*}\lim_{a \to \infty} \frac{V_1(a)}{a} = 1 . \end{align*}$$

The uniqueness of $V_1$ was established by Doney in [Reference Doney22]. Similarly, let $V_1'$ denote the analogous harmonic function for the random walk $-\mathsf {S}_2$ .

Next, we describe the diffusive limit of our random walk. For $\ell , n\in \mathbb {N}$ and any ordered $\ell $ -tuple of points $\mathcal S$ in $\mathbb {R}^2$ , enumerated as

$$\begin{align*}\mathcal S = \big( (s_1(1), s_2(1)), (s_1(2), s_2(2)) , \dots, (s_1(\ell),s_2(\ell))\big) \end{align*}$$

and satisfying $s_1(i) < s_1(i+1)$ for each $i \in [1,\ell -1]$ , we define $\mathfrak {J}_n(\mathcal S)$ to be the linear interpolation through the diffusively rescaled points

(4.8) $$ \begin{align} \mathfrak{J}_n(\mathcal S) := \Big(\frac{1}{n}s_1(i), \frac{1}{\sigma\sqrt{n}}s_2(i) \Big)_{i=1}^k \end{align} $$

where $\sigma ^2 := \sigma _2^2/\mu $ (c.f. Remark 2.9). Consider now the linear interpolation corresponding to our random walk $\mathsf {S}(\cdot )$ up to time $H_{(N,v)}$ :

$$\begin{align*}\mathfrak{e}_N^{\mathsf{S},v} := \mathfrak{J}_N\Big( (\mathsf{S}(i))_{i =0}^{H_{(N,v)}} \Big). \end{align*}$$

Viewing $\mathfrak {e}_N^{\mathsf {S},v}$ as a random element of $C[0,1]$ , Theorem 4.3 below gives a convergence result to the Brownian excursion on $[0,1]$ .

Theorem 4.3. Fix any $\delta \in (0,1/2)$ . Uniformly over $u, v \in (0,N^{1/2-\delta }] \cap \mathbb {N}$ , we have the following. The family of conditional laws

$$ \begin{align*} \mathbf{Q}_{u,v}^N (\cdot) := \mathbb{P}_{(0,u)}\Big(\big(\mathfrak{e}_N^{\mathsf{S},v}(t))_{t\in [0,1]} \in \cdot \;\big|\; H_{(N,v)} < H_{{\mathbb H}_{-}} \Big) \, \end{align*} $$

converges weakly as $N \to \infty $ to the law of the standard Brownian excursion in $(C[0,1], \|\cdot \|_{\infty })$ .

Theorems 4.2 and 4.3 are proved in Section 6.3. These results are modifications of a ballot-type theorem from [Reference Denisov and Wachtel17] and an invariance principle from [Reference Duraj and Wachtel23], respectively. Those important results were proved for a much broader class of random walks in very general cones; however, they were only proved for fixed start points and end points. The two results above have been modified to hold uniformly in appropriate ranges of start points and end points, which is crucial for our application. We remark that [Reference Ioffe, Ott, Velenik and Wachtel31, Section 5] states and proves results that similarly modify the results of [Reference Denisov and Wachtel17] and [Reference Duraj and Wachtel23]. However, first, the random walk they consider is symmetric in the y-coordinate, which simplifies their analysis, and second, their proofs do not always make explicit the aforementioned uniformity, in particular for the x-coordinate of the end point of their random walk. In Section 6, where our random walk estimates are proved, we take care to describe explicitly how we modify the proofs in [Reference Denisov and Wachtel17] and [Reference Duraj and Wachtel23] to handle uniformity in a broad range of start and end points.

5.1 Comparing Ising polymers with modified and unmodified weights

We begin with the main theorem of [Reference Ioffe, Shlosman and Toninelli32], which states that the partition functions of the Ising polymers in ${\mathbb H}$ , with and without modifications, are equivalent up to a $\beta $ -dependent constant.

Theorem 5.1 [Reference Ioffe, Shlosman and Toninelli32, Theorem 1].

For all $\beta>0$ large enough, we have

(5.1) $$ \begin{align} \tau_{\beta}(\mathsf{x}) = - \lim_{N\to\infty} \frac{1}{\left\Vert \mathsf{x} \right\Vert} \log \mathcal{G}_{{\mathbb H}}(\mathsf{x}). \end{align} $$

In fact, there exist constants $C_1:=C_1(\beta ), C_2:=C_2(\beta )>0$ such that for all $\beta $ large enough,

(5.2) $$ \begin{align} C_1\, \mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset {\mathbb H}) \leq \mathcal{G}_{{\mathbb H}}(\mathsf{x}) \leq C_2\, \mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset {\mathbb H}). \end{align} $$

Though we only consider ${\mathbb H}$ in this article, [Reference Ioffe, Shlosman and Toninelli32, Theorem 1] is stated more generally for half-spaces whose interior normal has argument lying in $[-\pi /4, 3\pi /4]$ .

The next proposition can be seen as an extension of Theorem 5.1 to address polymers in Q.

Proposition 5.2. For all $\beta>0$ sufficiently large, there exists a constant $C:= C(\beta )>0$ such that for any $N\in \mathbb {N}$ ,

(5.3) $$ \begin{align} C^{-1}\,\mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset {\mathbb H} ) \leq \mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset Q) \leq C\,\mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset {\mathbb H} ) , \end{align} $$

and

(5.4) $$ \begin{align} C^{-1}\,\mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset Q ) \leq \mathcal{G}_Q(\mathsf{x}) \leq C\mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset Q ) . \end{align} $$

The proof of Proposition 5.2 is given in Section 5.2.

One can readily imply from Equations (5.2) and (5.4) the following, relating $\mathbf {P}^{\mathsf {x}}(\cdot \mid \gamma \subset D)$ and $\mathbf {P}_D^{\mathsf {x}}$ , exactly as [Reference Ioffe, Shlosman and Toninelli32] derived its analogue from Equation (5.2) (see the discussion below Equation (2.7) in that paper):

Corollary 5.3. Let $D = {\mathbb H}$ or Q. There exists a constant $C:= C(\beta )>0$ such that for any set of contours A contained in D,

$$\begin{align*}\mathbf{P}_D^{\mathsf{x}}(A) \leq C \sqrt{\mathbf{P}^{\mathsf{x}}(A \;\big|\; \gamma \subset D)} \quad \text{ and } \quad \mathbf{P}^{\mathsf{x}}(A \;\big|\; \gamma \subset D) \leq C\sqrt{\mathbf{P}_D^{\mathsf{x}}(A)} . \end{align*}$$

For instance, if $Y_\gamma := \sum _{\mathcal {C} \cap \nabla _{\gamma } \neq \emptyset } \Phi _D'(\mathcal {C};\gamma ) - \Phi '(\mathcal {C};\gamma )$ and $\mathbf {E}[\cdot ]$ denotes expectation under $\mathbf {P}^{\mathsf {x}}(\cdot \mid \gamma \subset D)$ , then $\mathbf {E}[e^{Y_{\gamma }}]$ is just $\mathcal {G}_D(\mathsf {x})/\mathcal {G}(\mathsf {x} \mid \gamma \subset D)$ , and therefore . Notice $C^{-1} \leq \mathbf {E}[e^{Y_{\gamma }}] \leq C$ by Equation (5.2) (for $D={\mathbb H}$ ) and Equation (5.4) (for $D=Q$ ). By Cauchy–Schwarz, $\mathbf {P}_D^{\mathsf {x}}(A) \leq C \sqrt {\mathbf {P}^{\mathsf {x}}(A\mid \gamma \subset D )}\sqrt {\mathbf {E}[e^{2Y_\gamma }]}$ , and the last expectation is uniformly bounded (as $\mathbf {E}[e^{2Y_{\gamma }}] =\tilde {\mathcal {G}}_D(\mathsf {x})/\mathcal {G}(\mathsf {x} \mid \gamma \subset D)$ , where $\tilde {\mathcal {G}}_D(\mathsf {x})$ is defined w.r.t. $\tilde {\Phi }^{\prime }_D(\mathcal {C};\gamma ) := 2\Phi _D'(\mathcal {C};\gamma )-\Phi '(\mathcal {C};\gamma )$ ).

A particular consequence of Corollary 5.3 is that contour events that hold with probability tending to $1$ in the unmodified models (which are much easier to study) still hold with probability tending to $1$ in the modified models. The most significant application of this fact is in the proof of entropic repulsion under $\mathbf {P}_Q^{\mathsf {x}}(\cdot )$ (Proposition 5.11), which, as explained at the start of this Section, is a crucial step towards the random walk coupling achieved in Section 5.6.

5.2 Proof of Proposition 5.2

We will need two auxiliary lemmas. The first, Lemma 5.5, states that in any model where at least two cone points exist, the partition function is dominated by animals whose first (left) and last (right) irreducible pieces have size of order $1$ .

Lemma 5.5. Let D be either Q or ${\mathbb H}$ . For any $\epsilon \in (0,1)$ , there exists $K_{\epsilon } := K_{\epsilon ,\beta }>0$ such that

(5.5) $$ \begin{align} \mathcal{G}_{D}(\mathsf{x} \;\big|\; |\mathsf{Cpts}(\Gamma)|\geq 2 ,\, \max(|\Gamma^{(L)}|, |\Gamma^{(R)}|)> K_{\epsilon}) \leq \epsilon \mathcal{G}_{{\mathbb H}}(\mathsf{x}) \end{align} $$

and

(5.6) $$ \begin{align} \mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset D ,\, |\mathsf{Cpts}(\Gamma)|\geq 2 ,\, \max(|\Gamma^{(L)}|, |\Gamma^{(R)}|)> K_{\epsilon}) \leq \epsilon \mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset {\mathbb H}). \end{align} $$

The second auxiliary result, Lemma 5.7, extends Lemma 2.11, and thus Proposition 2.12 thanks to Remark 2.13, to the modified models on Q and ${\mathbb H}$ .

Lemma 5.7. Let D be either Q or ${\mathbb H}$ . Fix any $\epsilon ,\delta \in (0,1)$ . There exist positive constants $\beta _0, \delta _0$ and $\nu _0>0$ such that for all $\beta \geq \beta _0$ , there exists $C:= C(\beta )>0$ such that the following hold uniformly over all $N \in \mathbb {N}$ and $r \geq 1+\epsilon $ :

(5.7) $$ \begin{align} \mathcal{G}_D\big(\mathsf{x} \;\big|\; |\gamma| \geq r \left\Vert \mathsf{x} \right\Vert{}_1\big) &\leq Ce^{-\nu_0 \beta r \left\Vert \mathsf{x} \right\Vert{}_1} \mathcal{G}_{D}(\mathsf{x})\end{align} $$

(5.8) $$ \begin{align} \mathcal{G}_D\big(\mathsf{x} \;\big|\; |\mathsf{Cpts}(\gamma)|< 2\delta_0 \left\Vert \mathsf{x} \right\Vert{}_1\big) &\leq C e^{- \nu_0 \beta\left\Vert \mathsf{x} \right\Vert{}_1} \mathcal{G}_D(\mathsf{x}). \end{align} $$

In particular, there exists $\nu>0$ such that for all $\beta \geq \beta _0$ , there exists $C:= C(\beta )>0$ such that the following holds uniformly over all $N\in \mathbb {N}$ :

(5.9) $$ \begin{align} \mathcal{G}_D(\mathsf{x} \;\big|\; |\mathsf{Cpts}(\Gamma)| < \delta_0\|\mathsf{x}\|_1) \leq Ce^{-\nu \beta \|\mathsf{x}\|_1}\mathcal{G}_D(\mathsf{x}) . \end{align} $$

We postpone the proof of Lemma 5.5 to the end of the subsection. We prove Proposition 5.2 and Lemma 5.7 in tandem, proceeding as follows:

1. (a) We begin by proving Lemma 5.7 for $D={\mathbb H}$ .

2. (b) We then prove the first part of Proposition 5.2 (Equation (5.3)), as well as its analog for $\mathcal {G}_{{\mathbb H}}$ , that is,

   (5.10) $$ \begin{align} C^{-1}\mathcal{G}_{{\mathbb H}}(\mathsf{x})\leq \mathcal{G}_{{\mathbb H}}(\mathsf{x} \;\big|\; \gamma \subset Q) \leq C\mathcal{G}_{{\mathbb H}}(\mathsf{x}). \end{align} $$

3. (c) These results are then used to prove Lemma 5.7 for $D=Q$ .

4. (d) Finally, we prove Equation (5.4), thereby completing the proof of Proposition 5.2.

Proof of Part (a). For each bound, we will first apply Lemma 2.11, then Theorem 5.1. Recall that Lemma 2.11 states that the analogous bounds to Equation (5.7) and Equation (5.8) hold for $\mathcal {G}(\mathsf {x})$ . Equation (5.1) implies that $\log \mathcal {G}(\mathsf {x})$ and $\log \mathcal {G}(\mathsf {x}\;\big |\; \gamma \subset {\mathbb H})$ have the same leading order; thus, for some $\nu _0' \in (0, \nu _0]$ , we have

$$ \begin{align*} \mathcal{G}\big(\mathsf{x} \;\big|\; \gamma \subset {\mathbb H} ,\, |\gamma| \geq r \left\Vert \mathsf{x} \right\Vert{}_1\big) &\leq \mathcal{G}\big(\mathsf{x} \;\big|\; |\gamma| \geq r \left\Vert \mathsf{x} \right\Vert{}_1\big) \leq Ce^{-\nu_0' \beta r \left\Vert \mathsf{x} \right\Vert{}_1} \mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset {\mathbb H}), \text{ and} \\ \mathcal{G}\big(\mathsf{x} \;\big|\; \gamma \subset {\mathbb H} ,\, \mathsf{Cpts}(\gamma) < 2\delta_0 \left\Vert \mathsf{x} \right\Vert{}_1 \big) &\leq \mathcal{G}\big(\mathsf{x} \;\big|\; |\mathsf{Cpts}(\gamma)| < 2\delta_0 \left\Vert \mathsf{x} \right\Vert{}_1 \big) \leq C e^{-\nu_0' \beta \left\Vert \mathsf{x} \right\Vert{}_1} \mathcal{G}(\mathsf{x} \;\big|\; \gamma \subset {\mathbb H}). \end{align*} $$

Equation (5.7) and Equation (5.8) for $D ={\mathbb H}$ now follow from Corollary 5.3, and Equation (5.9) from Remark 2.13.

Proof of Part (b). The upper bounds in both Equations (5.3) and (5.10) hold with $C = 1$ , since $Q \subset {\mathbb H}$ .

We now show the lower bound for Equation (5.10). The lower bound in Equation (5.3) is obtained simply by replacing $q_{{\mathbb H}}$ with q in what follows. By Lemmas 5.5 and 5.7 (for $D={\mathbb H}$ ), we have the following: for any $\epsilon \in (0,1)$ , there exists $K:=K(\beta ,\epsilon )>0$ such that for $\|\mathsf {x}\|_1$ large,

(5.11) $$ \begin{align} (1-\epsilon)\mathcal{G}_{{\mathbb H}}(\mathsf{x}) \leq \mathcal{G}_{{\mathbb H}}\big(\mathsf{x} \;\big|\; |\mathsf{Cpts}(\Gamma)| \geq \delta_0 \left\Vert \mathsf{x} \right\Vert{}_1 ,\, \max(|\Gamma^{(L)}|, |\Gamma^{(R)}|)\leq K \big) . \end{align} $$

Now, using the weight factorization of $q_{{\mathbb H}}(\Gamma )$ in Equation (2.18), we may express the right-hand side of the above (for $N=\left \Vert \mathsf {x} \right \Vert {}_1 \geq 2/\delta _0$ ) as

For any ${\mathsf {u}} \in \mathcal {Y}^\blacktriangleleft $ , let $\gamma ^{(L)}({\mathsf {u}})$ denote an arbitrary up-right path from $0$ to ${\mathsf {u}}$ , and let $\gamma ^{(R)}({\mathsf {u}})$ denote an arbitrary down-right path from $\mathsf {x}-{\mathsf {u}}$ to $\mathsf {x}$ (if one exists). Since the backwards cone of $\Gamma ^{(L)}$ must contain the origin and the forwards cone of $\mathsf {x}-\Gamma ^{(R)}$ must contain $\mathsf {x}$ , it follows that

$$\begin{align*}\gamma^{(L)}(\mathsf{X}(\Gamma^{(L)})) \circ \gamma_1 \circ \cdots \circ \gamma_m \circ \gamma^{(R)}(\mathsf{X}(\Gamma^{(R)})) \subset Q \end{align*}$$

for any collection $\{\Gamma ^{(L)}, \Gamma ^{(1)} ,\dots , \Gamma ^{(m)}, \Gamma ^{(R)}\}$ contributing to a nonzero term in the second-to-last display. In particular, for any $\Gamma ^{(L)} \in \mathsf {A_L}$ and $\Gamma ^{(R)} \in \mathsf {A_R}$ such that $\{|\Gamma ^{(L)}|, |\Gamma ^{(R)}|\leq K\}$ , we have

(5.12)

Since $ q_{{\mathbb H}}(\Gamma ^{(L)}) \leq q_{{\mathbb H}}(\gamma ^{(L)}(\mathsf {X}(\Gamma ^{(L)}))) $ and $q_{{\mathbb H}}(\Gamma ^{(R)}) \leq q_{{\mathbb H}}(\gamma ^{(R)}(\mathsf {X}(\Gamma ^{(R)}))) $ , it follows from the last three displays that the right-hand side of Equation (5.11) is bounded above by

The prefactor in the last display is a constant depending only $K:=K(\beta ,\epsilon )$ , finishing the proof.