1 Introduction↩︎

An Ising model in a disk \(\mathbb{D}\) consists of a planar graph \(G=(V,E,F)\) embedded in \(\mathbb{D}\) with \(n\) boundary vertices labeled \(b_1^\partial,\dots,b_n^\partial\) in clockwise cyclic order together with a function \(J:E \rightarrow \mathbb{R}_{>0}\) called the coupling constant. A spin configuration is a function \(\sigma : V \rightarrow \{ -1,1\}\) assigning to each vertex \(v \in V\) a spin \(\sigma_v\). The probability of a spin configuration is defined as \[\mathbf{P}(\sigma):= \frac{1}{Z} \prod_{e = uv \in E} e^{J_e \sigma_u \sigma_v}, \quad \text{where} \quad Z := \sum_{\sigma: V \rightarrow \{-1,1\} } \prod_{e = uv \in E} e^{J_e \sigma_u \sigma_v}\] is the partition function. We denote by \([n]\) the finite set \([n]:=\{1, \dots, n\}\) and by \(\binom{[n]}{k}\) the set of all subsets of \([n]\) of size \(k\). Given \(i,j \in [n]\), we define the boundary correlation \[\langle \sigma_i \sigma_j \rangle : = \sum_{\sigma : V \rightarrow \{-1,1\}} \mathbf{P}(\sigma) \sigma_{b_i^\partial} \sigma_{b_j^\partial}.\] These correlations are organized into an \(n \times n\) symmetric real matrix with \(1\)s on the diagonal \[M(G,J):=(\langle \sigma_i \sigma_j \rangle)_{ i,j \in [n]},\] called the boundary correlation matrix. We write \[\operatorname{Corr}_G: \mathcal{I}_G^{>0} \rightarrow \mathrm{Mat}_n^{\mathrm{sym}}(\mathbb{R},1), \qquad J\mapsto M(G,J),\] where \[\mathcal{I}_G^{>0} := \Big\{ J:E(G)\to\mathbb{R}_{>0}\Big\}\] is the space of Ising models on the graph \(G\) and \(\mathrm{Mat}_n^{\mathrm{sym}}(\mathbb{R},1)\) denotes the space of real symmetric \(n\times n\) matrices with \(1\)s on the diagonal. The inverse problem for the Ising model asks to recover the coupling constant \(J\) from the boundary correlation matrix \(M(G,J)\).

In order for this inverse problem to have a unique solution, one must impose a reducedness assumption on \(G\). This is analogous to the situation for electrical networks. Lam [1] showed that reduced electrical networks in a disk, modulo certain local transformations of the graph called Y-\(\Delta\) moves, are classified by matchings of \([2n]\). In the Ising setting, the same combinatorial object appears: to a reduced graph \(G\) one associates a matching \(\pi\) of \([2n]\). Throughout the paper, \(G\) will be assumed reduced, and all constructions will be compatible with Y-\(\Delta\) moves.

Galashin–Pylyavskyy [2] gave a recursive solution to the Ising inverse problem using operations called adjoining a boundary edge and adjoining a boundary spike. These operations are the Ising analogs of operations for electrical networks introduced by Curtis–Ingerman–Morrow [3]. The purpose of this paper is to give a direct and non-recursive solution to the Ising inverse problem using ideas that originate in the chamber ansatz of Berenstein–Fomin–Zelevinsky [4], later developed for the positive Grassmannian by Marsh–Scott [5], extended to arbitrary positroid cells by Muller–Speyer [6], and applied to electrical networks in the big cell by George [7]. In this paper, we develop an Ising analog of this chamber-ansatz framework.

We define two auxiliary functions \(s, c : E(G) \rightarrow (0,1)\) as follows: \[s_e := \operatorname{sech}(2J_e),\qquad c_e:= \operatorname{tanh}(2J_e).\] Rather than solving directly for the coupling constant \(J\), we solve for the functions \(s\) and \(c\) from which we can recover \(J\) as \(\frac{1}{2}\operatorname{arctanh}(c)\). The chamber ansatz for the Ising model asks that these functions factor through a collection of positive variables attached to the vertices and faces of \(G\). More precisely, a \(C\)-network on \(G\) is a function \[C:V(G) \sqcup F(G) \rightarrow \mathbb{R}_{>0}\] defined modulo rescaling by \(\mathbb{R}_{>0}\). Let \[\mathcal{C}_G^{>0} \cong \mathbb{R}_{>0}^{|V(G)| + |F(G)|-1}\] be the space of \(C\)-networks with underlying graph \(G\). In terms of the \(C\)-network, the functions \(s\) and \(c\) are given by \[\label{eq:intro95s95and95c} s_e := \sqrt{\frac{C_fC_g}{C_uC_v+C_fC_g}}, \qquad c_e := \sqrt{\frac{C_uC_v}{C_uC_v+C_fC_g}},\tag{1}\] where \(e=uv\) and \(f,g\) are the two faces adjacent to \(e\). In other words, 1 defines a map \[q_{G}:\mathcal{C}_G^{>0} \rightarrow \mathcal{I}_G^{>0}.\] There is a positive torus \[\label{pos:torus} T_\pi^{>0} \subset \mathbb{R}_{>0}^{2n},\tag{2}\] depending only on the matching \(\pi\), that acts on \(C_G^{>0}\). The map \(q_G\) is invariant under this action, and therefore descends to a map \[q_G:\mathcal{C}_G^{>0}/T_\pi^{>0}\longrightarrow \mathcal{I}_G^{>0}.\] In this framework, solving the inverse problem reduces to constructing the dotted map making the following diagram commute: \[\label{eq:dotted} \begin{tikzcd} \mathcal{C}_{G}^{>0}/T_\pi^{>0} \arrow[rr, "q_G"] & & \mathcal{I}_{G}^{>0} \\ &\{\substack{\mathrm{Boundary~correlation} \\ \mathrm{matrices}}\} \subset \mathrm{Mat}_n^{\mathrm{sym}}(\mathbb{R},1) \arrow[ur, "\operatorname{Corr}_{G}^{-1}"'] \arrow[ul, dotted ,"?"] & \end{tikzcd}\tag{3}\]

The terminology \(C\)-network is motivated by the transformation rule for the variables \(C\) under the Y-\(\Delta\) move. This transformation is the discrete CKP equation in the form introduced by Kashaev in his study of the Ising Y-\(\Delta\) move [8]; see also Schief [9]. It is Cayley’s \(2\times 2\times 2\) hyperdeterminant [10] and appears in the work of Holtz–Sturmfels [11] on the principal minor assignment problem for symmetric matrices. Related incarnations of the same equation appear in the work of Kenyon–Pemantle [12], [13], Leaf [14], Melotti [15], and Arthamonov–Harnad–Hurtubise [16], where it is connected to statistical mechanics, cluster algebras, and Grassmannians. It also appears in the work of Doliwa on discrete integrable systems [17] and in discrete differential geometry in the work of Bobenko–Schief [18], [19]. The chamber ansatz formula 1 appears in Melotti [15] and, up to a sign, in the formula (10) for the partial correlations in Sturmfels–Tsukerman–Williams [20].

We will construct the dotted map in 3 via a modification of the twist map of Muller–Speyer [6] for positroid cells. We first recall how boundary correlation matrices are related to the totally nonnegative Grassmannian. The Grassmannian \(\mathrm{Gr}_{n,2n}\) is the space of \(n\)-dimensional subspaces of \(\mathbb{R}^{2n}\). Such a subspace \(V\subset \mathbb{R}^{2n}\) can be represented as the rowspan of a full-rank \(n\times 2n\) matrix. Its maximal minors \[\Delta_I(V), \qquad I\in \binom{[2n]}{n},\] are homogeneous coordinates on \(\mathrm{Gr}_{n,2n}\), called Plücker coordinates. The totally nonnegative Grassmannian is the subset \[\mathrm{Gr}_{n,2n}^{\geq 0} := \left\{ V\in \mathrm{Gr}_{n,2n}: \Delta_I(V)\geq 0 \text{ for all } I\in \binom{[2n]}{n} \right\}.\] Postnikov [21] showed that \(\mathrm{Gr}_{n,2n}^{\geq 0}\) decomposes into positroid cells \[\mathrm{Gr}_{n,2n}^{\geq 0} = \bigsqcup_{\mathscr M} \Pi_{\mathscr M}^{>0}, \quad \text{where} \quad \Pi_{\mathscr M}^{>0} := \left\{ V\in \mathrm{Gr}_{n,2n}^{\geq 0}: \Delta_I(V)>0 \text{ if and only if } I\in \mathscr M \right\}.\] The indexing sets \(\mathscr M\subset \binom{[2n]}{n}\) that arise in this way are called positroids. Equivalently, positroid cells are indexed by (decorated) permutations of \([2n]\); we denote the positroid corresponding to a permutation \(\pi\) by \(\mathscr{M}_\pi\).

Galashin–Pylyavskyy [2] define a map \[\phi_n: \mathrm{Mat}_n^{\mathrm{sym}}(\mathbb{R},1) \rightarrow \mathrm{Gr}_{n,2n},\] called the doubling map, whose image lies in the orthogonal Grassmannian \[\mathrm{OGr}_{n,2n} := \left\{ V\in \mathrm{Gr}_{n,2n}: \Delta_I(V)=\Delta_{[2n]\setminus I}(V) \text{ for all } I\in \binom{[2n]}{n} \right\}.\] Moreover, if \(G\) is reduced and \(\pi\) is the associated matching of \([2n]\), then \(\pi\) determines a positroid \(\mathscr M_{\pi}\) (viewing the matching \(\pi\) as a permutation of \([2n]\)), and \[\phi_n(M(G,J)) \in \mathrm{OGr}_{n,2n} \cap \Pi_{\mathscr M_{\pi}}^{>0}.\]

Next, we relate \(C\)-networks to the totally nonnegative Grassmannian. \(C\)-networks on graphs \(G\) with the maximal matching (i.e., \(i \mapsto i+n ~(\text{mod}~2n))\) were first defined by Kenyon–Pemantle [13], who used them to parameterize the space of positive-definite symmetric matrices. Here, the \(C\)-variables are principal and almost-principal minors of the matrix. Equivalently, this space can be identified with the positive part \({\rm LG}^{> 0}_{n,2n}\) of a Lagrangian Grassmannian \(\text{LG}_{n,2n} \subset \mathrm{Gr}_{n,2n}\) with respect to the symplectic form \[\omega (x,y) := \sum_{i=1}^{n} (-1)^{i-1} (x_{i}y_{i+n} - x_{i+n}y_{i}), \quad \text{for } x,y \in \mathbb{R}^{2n}.\] This semialgebraic set appears in work of Shevchenko [22], and differs from the totally positive Lagrangian Grassmannian (in Lusztig’s sense [23]) studied by Karpman [24] and the totally positive Lagrangian Grassmannian arising in the theory of electrical networks in work of Lam [1], Chepuri–George–Speyer [25] and Bychkov–Gorbounov–Kazakov–Talalaev [26], which use other symplectic forms. Under this identification with \({\rm LG}^{>0}_{n,2n}\), the \(C\)-variables, i.e., the principal and almost-principal minors, are Plücker coordinates of the point in \(\text{LG}^{>0}_{n,2n}\) and this construction can be viewed as a specialization of a construction of Scott [27] for the totally nonnegative Grassmannian. Generalizing this, we define subsets \[\Lambda_{\pi}^{>0} \subset \Pi_{\mathscr M_{\pi}}^{>0},\] which we call locally Lagrangian positroid cells, together with maps \[\boldsymbol{H}_G: \Lambda_\pi^{>0} \rightarrow \mathcal{C}_G^{>0},\] where each \(C\)-variable is given by a Plücker coordinate of the point in \(\Lambda_{\pi}^{>0}\). We show that the conditions defining \(\Lambda_{\pi}^{>0}\) inside \(\Pi_{\mathscr M_{\pi}}^{>0}\) may be viewed as local Lagrangian conditions, thereby justifying the terminology. We then prove that \(\boldsymbol{H}_G\) is a bijection and that these bijections are compatible with Y-\(\Delta\) moves.

The final ingredient is a modification of the twist map. For an arbitrary positroid \(\mathscr M\), Muller–Speyer [6] define right and left twist maps \[\vec{\tau}_{\mathscr{M}},\reflectbox{\vec{\reflectbox{\tau}}}_{\mathscr{M}}: \Pi_{\mathscr M}^{>0} \rightarrow \Pi_{\mathscr M}^{>0},\] which are inverse to each other and which relate the \(A\)- and \(X\)-cluster coordinates of the positroid cell. In the Ising setting, the twist maps must be modified by the natural action of \(\mathbb{R}_{>0}^{2n}\) rescaling the columns of a matrix representative to be compatible with \(q_G\); a similar modification was required for electrical networks in George [7].

The positive torus 2 acts on \(\Lambda_{\pi}^{>0}\), and the map \(\boldsymbol{H}_G\) is equivariant and therefore descends to a map \[\boldsymbol{H}_G: \Lambda_{\pi}^{>0}/T_{\pi}^{>0} \rightarrow \mathcal{C}_G^{>0}/T_{\pi}^{>0}.\] The Ising twist and inverse twist maps \[\hat{\tau}_{\mathscr M_\pi}: \Lambda_{\pi}^{>0}/T_{\pi}^{>0} \to \mathrm{OGr}_{n,2n} \cap \Pi_{\mathscr M_{\pi}}^{>0}, \qquad \hat{\tau}_{\mathscr M_{\pi}}^{-1}: \mathrm{OGr}_{n,2n} \cap \Pi_{\mathscr M_{\pi}}^{>0} \to \Lambda_{\pi}^{>0}/T_{\pi}^{>0}.\] defined after passing to the quotient \(\Lambda_\pi^{>0}/T_\pi^{>0}\). We can now state our main theorem.

In particular, the dotted map in 3 which solves the inverse problem is given by the composition \[\boldsymbol{H}_G \circ \hat{\tau}^{-1}_{\mathscr{M}_{\pi}} \circ \phi_n.\]

Open problems↩︎

We end the introduction with some open problems that we believe are interesting for further study.

Organization of the paper↩︎

In 2, we review the necessary background on the positive Grassmannian and its connections to the dimer model and \(A\)-networks. In 3, we recall the corresponding background on the positive orthogonal Grassmannian and its relation to the Ising model. In 4, we introduce \(C\)-networks and locally Lagrangian positroid cells. Finally, in 5, we define the maps appearing in the diagram of the main result, 1, and prove the theorem.

2 Background on the positive Grassmannian, the dimer model and \(A\)-networks↩︎

In this section we collect some background on the totally nonnegative Grassmannian which we shall need throughout this article.

2.1 The Grassmannian and positivity↩︎

Let \(1 \leq k \leq n\) be positive integers. The real Grassmannian \(\mathrm{Gr}_{k,n} := \mathrm{Gr}_{k,n}(\mathbb{R})\) is the moduli space of \(k\)-dimensional subspaces of the vector space \(\mathbb{R}^n\). A \(k\)-dimensional subspace \(V \subset \mathbb{R}^{n}\) can be represented as the rowspan of a full-rank \(k \times n\) matrix \(M\), and the map \[\mathrm{Gr}_{k,n} \rightarrow \mathbb{P}^{\binom{n}{k}-1} , \quad V = {\rm rowspan}(M) \mapsto \big(\Delta_I(V) = \det(M_I)\big)_{I \in \binom{[n]}{k}},\] is the Plücker embedding of the Grassmannian \(\mathrm{Gr}_{k,n}\) into the projective space \(\mathbb{P}^{\binom{n}{k}-1}\). Here \(M_I\) denotes the \(k \times k\)-submatrix of \(M\) with all rows but columns only from the index set \(I\). The Grassmannian \(\mathrm{Gr}_{k,n}\) is a projective variety and the \(\Delta_I(V)\) are called Plücker coordinates. We will often omit the vector space in the notation and simply write \(\Delta_I\). The totally positive Grassmannian \(\mathrm{Gr}_{k,n}^{> 0}\) is the semialgebraic subset of points in \(\mathrm{Gr}_{k,n}\) such that \(\Delta_{I} > 0\) for all \(I \in \binom{[n]}{k}\). Its topological closure in \(\mathbb{P}^{\binom{n}{k}-1}\) is the totally nonnegative Grassmannian consisting of points in \(\mathrm{Gr}_{k,n}^{\geq 0}\) such that \(\Delta_{I} \geq 0\) for any \(I \in \binom{[n]}{k}\). Given a point \(V \in \mathrm{Gr}_{k,n}^{\geq 0}\), its matroid \(\mathscr{M}_V\) is defined by \[\mathscr{M}_V := \left\{I \in \binom{[n]}{k}: \Delta_I(V) \neq 0\right\}.\] In this article, we shall identify any rank \(k\) matroid \(\mathscr{M}\) on the ground set \([n]\) with its set of bases and write \(\mathscr{M} \subset \binom{[n]}{k}\). Any matroid \(\mathscr{M} \subset \binom{[n]}{k}\) such that there exists a \(V \in \mathrm{Gr}_{k,n}^{\geq 0}\) with \(\mathscr{M} = \mathscr{M}_V\) is called a positroid. The corresponding positroid cell is \[\Pi_{\mathscr{M}}^{>0}:= \left\{V \in \mathrm{Gr}_{k,n}^{\geq 0}: \mathscr{M}_V = \mathscr{M} \right\}.\] By definition of positroid cells, we have \[\mathrm{Gr}_{k,n}^{\geq 0} = \bigsqcup_{\text{positroids}~\mathscr{M}} \Pi_{ \mathscr{M}}^{>0}, \label{eq:positroidstratification}\tag{4}\] where the union is taken over all positroids of rank \(k\) on the ground set \([n]\).

Torus actions will play an important role in this paper and we will systematically keep track of actions on the various objects we define. In particular, we note that \(\mathbb{R}_{>0}^n\) acts on \(\mathrm{Gr}_{k,n}^{\geq 0}\) by rescaling the basis vectors of \(\mathbb{R}^n\), i.e., by rescaling the columns of any matrix representative. More explicitly, we have \[(t \cdot \Delta)_I = \left(\prod_{i \in I} t_i\right) \Delta_I, \quad \text{for } \Delta \in \mathrm{Gr}_{k,n}^{\geq 0} \text{ and } t = (t_1, \dots, t_n) \in \mathbb{R}_{>0}^n,\] and this action preserves each positroid cell \(\Pi^{>0}_{\mathscr{M}}\).

2.2 Plabic graphs in the disk↩︎

A plabic graph \(\Gamma\) is a bipartite graph embedded in the disk \(\mathbb{D}\) such that:

1. \(\Gamma\) has exactly \(n\) degree-\(1\) boundary vertices, labeled \(u_1^\partial, \dots, u_{n}^\partial\) in clockwise cyclic order, each of which is adjacent to exactly one internal vertex.

2. The internal vertices are colored black or white and adjacent vertices have opposite colors.

Although there is a unique choice of coloring of boundary vertices that makes \(\Gamma\) bipartite, it will be more convenient to treat the boundary vertices as uncolored. We denote the vertices, edges and faces of \(\Gamma\) by \(V(\Gamma), E(\Gamma)\) and \(F(\Gamma)\) respectively. We write \(B(\Gamma), W(\Gamma)\) and \(\partial V(\Gamma)\) for the black, white and boundary vertices of \(\Gamma\) respectively, so \(V(\Gamma) = B(\Gamma) \sqcup W(\Gamma) \sqcup \partial V(\Gamma).\)

We denote by \(f^\partial_{i,i+1}\) the boundary face between \(u_{i}^\partial\) and \(u_{i+1}^\partial\). We denote by \[k:= |W(\Gamma)|-|B(\Gamma)|+ \Big|\{b \in B(\Gamma): b~\text{is adjacent to the boundary}\}\Big|\] the helicity of \(\Gamma\) and say that \(\Gamma\) is of type \((k,n)\).

Let \(\Gamma^\times\) denote the directed medial graph of \(\Gamma\) constructed as illustrated in 1. The boundary vertices of \(\Gamma^\times\) naturally correspond to the boundary vertices of \(\Gamma\); we therefore label them by \(u_1^\partial,\dots,u_n^\partial\) as well. A strand of \(\Gamma\) is a maximal directed path in \(\Gamma^\times\). Each strand is either a closed loop or a directed path connecting two boundary vertices. The decorated permutation \(\pi_\Gamma^: =(\pi_\Gamma,\text{col})\) consists of:

1. A permutation \(\pi_\Gamma : [n] \rightarrow [n]\) defined by \[\pi_\Gamma(i) := j, \quad \text{if the strand starting at \( u_i^\partial \) ends at \( u_j^\partial \)}.\]

2. When \(i\) is a fixed point of \(\pi_\Gamma\), a color \(\text{col}(i) \in \{ \bullet, \circ\}\). The color will not play a role for the plabic graphs that arise from Ising models that we treat in this paper; hence, we omit the details of its construction.

We say that the graph \(\Gamma\) is reduced if the following hold:

1. No strand is a closed loop.

2. No strand has a self-intersection.

3. No two strands form a parallel bigon, i.e., there are no two edges \(e_1, e_2 \in E(\Gamma)\) such that both strands go through \(e_1\) and \(e_2\) with the same orientation.

Hereafter, we assume that our plabic graphs are reduced.

Given a reduced plabic graph \(\Gamma\) of type \((k,n)\), we define two maps \[\boldsymbol{S}, \boldsymbol{T}: F(\Gamma) \rightarrow \binom{[n]}{k},\] called source and target labels respectively, as follows. Each strand of \(\Gamma\) is assigned a label \(i \in [n]\), determined by its source vertex or its target vertex, depending on the chosen convention. Since \(\Gamma\) is reduced, there are exactly \(n\) such strands labeled by \([n]\). Each face \(f\) of \(\Gamma\) is labeled by the subset of strand labels such that \(f\) lies on the left side of the strand. It is easy to see that each face of \(\Gamma\) thus obtains a label \(I \subset [n]\) of the same cardinality \(k\).

A Grassmann necklace \(\boldsymbol{I} = (I_1,\dots,I_n)\) is a sequence of \(k\)-element subsets of \([n]\) such that for each \(i \in [n]\), there is a \(j_i \in [n]\) such that \(I_{i+1}=I_i \setminus \{i\} \cup \{j_i\}\). The Grassmann necklace \(\boldsymbol{I}_\Gamma = (I_1,\dots,I_n)\) of \(\Gamma\) is defined by \(I_{i} := \boldsymbol{T}(f^\partial_{i-1,i})\). Here \(f^\partial_{i-1,i} \in F(\Gamma)\) denotes the boundary face of \(\Gamma\) between the boundary vertices \(u_{i-1}^\partial\) and \(u_i^\partial\). Similarly, there is a dual object called the reverse Grassmann necklace defined by \(\boldsymbol{I}^{\text{rev}}_\Gamma = (I_1^\text{rev},\dots,I_n^\text{rev})\) where \(I_i^{\text{rev}} := \boldsymbol{S}(f_{i,i+1}^\partial)\).

A dimer cover or almost perfect matching of \(\Gamma\) is a subset of the edges that uses every internal vertex exactly once and every boundary vertex at most once. The boundary of a dimer cover \(M\) is defined as follows: \[\partial M := \left\{ i \in [n] \colon \begin{matrix} u_i^\partial \text{ is adjacent to a white vertex and covered by M}, \\ \text{or is adjacent to a black vertex and \emph{not} covered by M} \end{matrix} \right\}.\] The positroid \(\mathscr{M}_\Gamma\) of \(\Gamma\) is then defined as \[\mathscr{M}_\Gamma := \left\{ I \in \binom{[n]}{k}: \text{there is a dimer cover M with \partial M=I }\right\}.\]

The local modifications of plabic graphs shown in 4 are called moves. Two plabic graphs \(\Gamma\) and \(\Gamma'\) are said to be move-equivalent if they are related by a sequence of moves.

2.3 Dimer models↩︎

An edge weight on a plabic graph \(\Gamma\) is a function \[{\mathrm{wt}}: E(\Gamma) \rightarrow \mathbb{R}_{>0}.\]

We say that two edge weights \({\mathrm{wt}}_1\) and \({\mathrm{wt}}_2\) are gauge equivalent if there exists a function \[g: V(\Gamma) \rightarrow \mathbb{R}_{>0}\] satisfying \(g(u_i^\partial)=1\) for \(i \in [n]\) such that \[{\mathrm{wt}}_1(e) = g(b)^{-1} {\mathrm{wt}}_2(e) g(w) \qquad \text{for all } e = bw \in E(\Gamma).\] We denote the gauge equivalence class of a weight \({\mathrm{wt}}\) by \([{\mathrm{wt}}]\). For any face \(f\) of \(\Gamma\) delimited by a sequence of edges \[b_1 \xrightarrow{e_1} w_1 \xrightarrow{e_2} b_2 \xrightarrow{e_3} \cdots \xrightarrow{e_{2k-2}} b_k \xrightarrow{e_{2k-1}} w_k \xrightarrow{e_{2k}} b_1,\] the function \[X_f({\mathrm{wt}}):= \frac{{\mathrm{wt}}(e_1) \cdots {\mathrm{wt}}(e_{2k-1})}{{\mathrm{wt}}(e_2) \cdots {\mathrm{wt}}(e_{2k})}\] is gauge-invariant. These functions are called \(X\)-cluster variables.

A dimer model is a pair \((\Gamma,[{\mathrm{wt}}])\) where \(\Gamma\) is a reduced plabic graph and \([{\mathrm{wt}}]\) is a gauge equivalence class of edge weights on \(\Gamma\). The \(X\)-cluster variables provide coordinates on the space of edge weights modulo gauge equivalence up to the unique relation \(\prod_{f \in F(\Gamma)}X_f=1\). Let \[\mathcal{X}_\Gamma^{>0} \cong \mathbb{R}_{>0}^{|F(\Gamma)|-1}\] denote the space of dimer models with underlying graph \(\Gamma\). Each move \(\Gamma \to \Gamma'\) induces a bijection between the corresponding spaces of dimer models (see 3) \[\mathcal{X}_\Gamma^{>0} \xrightarrow[]{\sim} \mathcal{X}_{\Gamma'}^{>0}.\] Given a positroid \(\mathscr{M}\), the space of dimer models with positroid \(\mathscr{M}\) is defined as \[\mathcal{X}_{\mathscr{M}}^{>0} := \bigsqcup_{\Gamma:\mathscr{M}_\Gamma=\mathscr{M} } \mathcal{X}_\Gamma^{>0} \!\Big/\! \text{moves}.\]

Let \({\mathrm{wt}}\) be an edge weight. For \(I \in \binom{[n]}{k}\), let \[Z_I({\mathrm{wt}}):=\sum_{M: \partial M=I } \prod_{e \in M}{\mathrm{wt}}(e)\] denote the partition function for dimer covers with boundary \(I\). The map \[\operatorname{Meas}_\Gamma: \mathcal{X}_\Gamma^{>0} \rightarrow \mathbb{P}^{\binom{n}{k}-1}, \qquad [{\mathrm{wt}}] \mapsto (Z_I({\mathrm{wt}}))_{I \in \binom{[n]}{k}}\] is called boundary measurement. Here, \({\mathrm{wt}}\) denotes any edge weight representing the gauge equivalence class \([{\mathrm{wt}}]\). Different choices of representatives result in the same vector \((Z_I({\mathrm{wt}}))_{I \in \binom{[n]}{k}}\) modulo rescaling, and therefore the point \((Z_I({\mathrm{wt}}))_{I \in \binom{[n]}{k}} \in \mathbb{P}^{\binom{n}{k}-1}\) is the same.

Consider the following action of \(\mathbb{R}_{>0}^n\) on \(\mathcal{X}_\Gamma^{>0}\). For \(\boldsymbol{\lambda} = (\lambda_1,\dots,\lambda_n) \in \mathbb{R}_{>0}^n\), \[\boldsymbol{\lambda} \cdot {\mathrm{wt}}(e) := \begin{cases} \lambda_i {\mathrm{wt}}(e), &\text{if e is the boundary edge incident to u_i^\partial and a white internal vertex},\\ \lambda_i^{-1} {\mathrm{wt}}(e), &\text{if e is the boundary edge incident to u_i^\partial and a black internal vertex},\\ {\mathrm{wt}}(e),&\text{otherwise}. \end{cases}\] The action is compatible with moves, and therefore gives an action of \(\mathbb{R}_{>0}^n\) on \(\mathcal{X}_\mathscr{M}^{>0}\). Moreover, the definition is such that \(\operatorname{Meas}_\Gamma\) and \(\operatorname{Meas}_\mathscr{M}\) are equivariant.

2.4 \(A\)-networks↩︎

An \(A\)-network is a pair \((\Gamma,A)\) where \(\Gamma\) is a reduced plabic graph and \(A: F(\Gamma) \rightarrow \mathbb{R}_{>0}\) is a function defined modulo rescaling by \(\mathbb{R}_{>0}\). Let \[\mathcal{A}_\Gamma^{>0} \cong \mathbb{R}_{>0}^{|F(\Gamma)|-1}\] denote the space of \(A\)-networks with underlying plabic graph \(\Gamma\).

Each move \(\Gamma \to \Gamma'\) induces a bijection between the corresponding spaces of \(A\)-networks \[\mathcal{A}_\Gamma^{>0} \xrightarrow[]{\sim} \mathcal{A}_{\Gamma'}^{>0}\] as shown in 4. Given a positroid \(\mathscr{M}\), we define the space of \(A\)-networks with positroid \(\mathscr{M}\) to be \[\mathcal{A}_{\mathscr{M}}^{>0} := \bigsqcup_{\Gamma:\mathscr{M}_\Gamma=\mathscr{M} } \mathcal{A}_\Gamma^{>0} \!\Big/\! \text{moves}.\]

Recall that Plücker coordinates are defined modulo rescaling. For any reduced plabic graph \(\Gamma\) with \(\mathscr{M}_\Gamma = \mathscr{M}\), we define a map \[\boldsymbol{F}_\Gamma: \Pi^{>0}_{\mathscr{M}} \rightarrow \mathcal{A}_\Gamma^{>0}, \qquad V \mapsto (\Delta_{\boldsymbol{T}(f)}(V))_{f \in F(\Gamma)}\] assigning to each point in the positroid cell \(\Pi^{>0}_{\mathscr{M}}\) an \(A\)-network.

The following is due to Scott [27] for the uniform positroid \(\mathscr{M}=\binom{[n]}{k}\) and Muller–Speyer [6] in general.

The target face labels change under moves as shown in 5. Therefore, the commutativity of 5 is trivial for the contraction-uncontraction move and is a consequence of the three-term Plücker relation \[\Delta_{Sik} \Delta_{Sjl} = \Delta_{Sij} \Delta_{Skl}+\Delta_{Sil}\Delta_{Sjk}\] for the spider move.

The group \(\mathbb{R}_{>0}^n\) acts on \(\mathcal{A}_\Gamma^{>0}\) by \[\boldsymbol{\lambda} \cdot A = \left( \left(\prod_{i \in \mathbf{T}(f)} \lambda_i \right) A_f \right)_{f \in F(\Gamma)}\] making the map \(\boldsymbol{F}_\Gamma\) equivariant. The action is compatible with moves, and therefore the same is true for \(\boldsymbol{F}_\mathscr{M}\).

2.5 The twist maps↩︎

Let \(\mathscr{M}\) be a positroid, let \(\boldsymbol{I}_{\mathscr{M}} = (I_1,\dots,I_n)\) be its Grassmann necklace, and let \(\Pi_{\mathscr{M}}^{>0}\) denote the corresponding positroid cell. The right twist \(\vec{\tau}_{\mathscr{M}}\) is the map \[\vec{\tau}_{\mathscr{M}}: \Pi_{\mathscr{M}}^{>0} \rightarrow \Pi_{\mathscr{M}}^{>0}\] defined as follows. Let \(M\) be a \(k \times n\) matrix representative of \(V \in \Pi_{\mathscr{M}}\), and denote by \(M_i\) the \(i\)-th column of \(M\). Define a new matrix \(\vec{\tau}_{\mathscr{M}}(M)\) by the conditions \[\langle \vec{\tau}_{\mathscr{M}}(M)_i, M_i \rangle = 1, \qquad \langle \vec{\tau}_{\mathscr{M}}(M)_i, M_j \rangle = 0 \quad \text{for } j \in I_i \setminus \{i\}. \label{eq:def95right95twist}\tag{6}\] Here \(\langle \cdot , \cdot \rangle\) is the standard inner product on \(\mathbb{C}^k\). Finally, set \[\vec{\tau}_{\mathscr{M}}(V) := \operatorname{rowspan}\!\big(\vec{\tau}_{\mathscr{M}}(M)\big).\] The left twist \(\reflectbox{\vec{\reflectbox{\tau}}}_{\mathscr{M}}\) is defined similarly but using the reverse Grassmann necklace instead in 6 .

Let \(\mathscr{M}\) be a positroid. There is a canonical map \[p_\mathscr{M}: \mathcal{A}_{\mathscr{M}}^{>0} \rightarrow \mathcal{X}_{\mathscr{M}}^{>0},\] called the cluster ensemble map, defined as follows. Let \(\Gamma\) be a reduced plabic graph such that \(\mathscr{M}_\Gamma = \mathscr{M}\). Let \(A \in \mathcal{A}_{\Gamma}^{>0}\), let \[{\mathrm{wt}}(e) :=\begin{cases} 1/{A_f A_g}, &\text{if e is internal and adjacent to faces f and g},\\ 1/{A_{f_{i-1,i}^\partial}}, &\text{if e is adjacent to u_i^\partial and a white internal vertex},\\ 1/{A_{f_{i,i+1}^\partial}}, &\text{if e is adjacent to u_i^\partial and a black internal vertex},\\ \end{cases}\] and define \(p_\Gamma(A):=[{\mathrm{wt}}]\). If \(\Gamma \to \Gamma'\) is a move, then the following diagram commutes: \[\begin{tikzcd} \mathcal{A}_{\Gamma}^{>0} \arrow[r, "\sim"] \arrow[d, "p_{\Gamma}"'] & \mathcal{A}_{\Gamma'}^{>0} \arrow[d, "p_{\Gamma'}"] \\ \mathcal{X}_{\Gamma}^{>0} \arrow[r, "\sim"] & \mathcal{X}_{\Gamma'}^{>0} \end{tikzcd}.\] Thus, the maps \((p_\Gamma)_{\Gamma: \mathscr{M}_\Gamma = \mathscr{M}}\) glue together to give \(p_{\mathscr{M}}\).

3 The positive orthogonal Grassmannian and the Ising model↩︎

Let \(n \geq 1\) be a positive integer. We consider the following quadratic form on \(\mathbb{R}^{2n}\) \[q \colon \mathbb{R}^{2n} \xrightarrow[]{} \mathbb{R}, \quad \boldsymbol{x} = (x_1,\dots,x_{2n}) \mapsto \sum_{i = 1}^{2n} (-1)^{i-1} \;x_{i}^2.\] We denote by \(Q := {\rm diag}(1, -1, 1, \dots 1, -1) \in \mathbb{R}^{2n \times 2n}\) the matrix that represents \(q\) in the standard basis. The orthogonal Grassmannian \(\mathrm{OGr}_{n,2n}\) is the subvariety of the Grassmannian \(\mathrm{Gr}_{n,2n}\) that consists of vector spaces \(V\) such that \[q(\boldsymbol{x}) = 0 \quad \text{for all}~\boldsymbol{x} \in V.\] It is known that this variety has two isomorphic connected components that are irreducible and cut out in \(\mathrm{Gr}_{n,2n}\) by linear equations as follows \[\begin{align} \mathrm{OGr}_{n,2n}^{\pm} := \left\{ V \in \mathrm{Gr}_{n,2n} \colon \Delta_I(V) = \pm \Delta_{I^c}(V) \quad \text{for all } I \in \binom{[2n]}{n} \right\}. \end{align}\] We define \[\mathrm{OGr}_{n,2n}^{ \geq 0} := \mathrm{OGr}_{n,2n} \cap \mathrm{Gr}_{n,2n}^{\geq 0} \quad \text{and} \quad \mathrm{OGr}_{n,2n}^{ > 0} := \mathrm{OGr}_{n,2n} \cap \mathrm{Gr}_{n,2n}^{> 0} \;.\] Note that \(\mathrm{OGr}_{n,2n}^{ \geq 0} \subset \mathrm{OGr}_{n,2n}^{+}\). Intersecting \(\mathrm{OGr}_{n,2n}^{ \geq 0}\) with the positroid stratification 4 , we get an induced stratification \[\mathrm{OGr}_{n,2n}^{\geq 0} = \bigsqcup_{\mathscr{M}} \mathrm{OGr}^{\geq 0}_{n,2n} \cap \Pi_{\mathscr{M}}^{>0}.\]

Let us denote by \(P_n\) the set of fixed-point-free involutions on \([2n]\). Given \(\pi \in P_n\), viewing it as a decorated permutation, we have an associated positroid \(\mathscr{M}_\pi\) via the bijections in 2.

3.1 Cactus graphs↩︎

Let \(\mathbb{D}\) be a disk with boundary vertices labeled \(b_1^\partial, \dots, b_n^\partial\) in clockwise cyclic order. Let \(P\) be a noncrossing partition of \([n]\). A cactus with shape \(P\) is the topological space \(\mathbb{D}/P\) obtained by gluing together the points \(b_1^\partial, \dots, b_n^\partial\) whose indices are in the same part of \(P\). A cactus graph with shape \(P\) is a graph \(G\) embedded in a cactus \(\mathbb{D}/P\) with boundary vertices \(b_1^\partial, \dots, b_n^\partial\).

Let \(G\) be a cactus graph with shape \(P\). The medial graph \(G^\times\) of \(G\) is the graph in \(\mathbb{D}\) obtained from \(G\) as follows. Place boundary vertices \(u_1^\partial,\dots,u_{2n}^\partial\) of \(G^\times\) around the boundary of the cactus \(\mathbb{D}/P\) so that \(b_i^\partial\) is between \(u_{2i-1}^\partial\) and \(u_{2i}^\partial\). Replace each edge of \(G\) as in 7. Finally, take the preimage of the resulting graph under the quotient map \(\mathbb{D}\rightarrow \mathbb{D}/P\). By construction, each internal vertex of \(G^\times\) has degree \(4\). A strand of \(G\) is a maximal (unoriented) path in \(G^\times\) that contains the opposite pair of edges at each internal vertex. The medial pairing of \(G\) is the fixed-point-free involution \(\pi_G : [2n] \rightarrow [2n]\) given by \[\pi_G(i) = j \quad \text{if there is a strand between \( u_i^\partial\) and \(u_j^\partial\)}.\] If \(\pi_G(i) = j\), then we write \(j = \bar i\). We denote the strand between \(i\) and \(\bar i\) by \(\gamma_{\{i,\bar i\}}.\)

We say that a cactus graph \(G\) is reduced if:

1. No strand is a closed loop.

2. No pair of strands intersects twice.

The Y-\(\Delta\) move is a local modification of cactus graphs shown in 8. We say that two cactus graphs \(G\) and \(G'\) are move-equivalent if they are connected by a sequence of Y-\(\Delta\) moves. Similarly to 2, we have:

In other words, move-equivalence classes of reduced cactus graphs with \(n\) boundary vertices are classified by fixed-point-free involutions of \([2n]\).

3.2 Ising models↩︎

An Ising model is a pair \((G,J)\) where \(G\) is a reduced cactus graph and \(J: E(G) \rightarrow \mathbb{R}_{>0}\) is a function called coupling constant. Let \[\mathcal{I}_G^{>0} \cong \mathbb{R}_{>0}^{E(G)}\] be the space of Ising models with underlying graph \(G\). Each Y-\(\Delta\) move \(G \to G'\) induces a bijection between the corresponding spaces of Ising models \[\mathcal{I}_G^{>0} \xrightarrow[]{\sim} \mathcal{I}_{G'}^{>0}\] as illustrated in 8. The edge parameters shown in the figure are \(\exp(2J_e)\) and the bijection is given in terms of these parameters by \[A=\sqrt{ \frac{(abc+1)(a+bc)}{(b+ac)(c+ab)}},\quad B=\sqrt{ \frac{(abc+1)(b+ac)}{(a+bc)(c+ab)}},\quad C=\sqrt{ \frac{(abc+1)(c+ab)}{(a+bc)(b+ac)}}.\]

Let \(\pi\) be a fixed-point-free involution on \([2n]\). The space of Ising models associated to \(\pi\) is defined as \[\mathcal{I}_{\pi}^{>0} := \bigsqcup_{G: \pi_G=\pi } \mathcal{I}_G^{>0} \!\Big/\! \text{Y-\Delta moves}.\]

A spin configuration is a function \(\sigma : V \rightarrow \{ -1,1\}\) assigning to each vertex \(v \in V\) a spin \(\sigma_v\). The probability of a spin configuration is defined as \[\mathbf{P}(\sigma):= \frac{1}{Z} \prod_{e = uv \in E} e^{J_e \sigma_u \sigma_v},\] where \(Z := \sum_{\sigma: V \rightarrow \{-1,1\} } \prod_{e = uv \in E} e^{J_e \sigma_u \sigma_v}\) is the partition function. Given \(i,j \in [n]\), we define the boundary correlation function by \[\langle \sigma_i \sigma_j \rangle : = \sum_{\sigma : V \rightarrow \{-1,1\}} \mathbf{P}(\sigma) \sigma_{b_i^\partial} \sigma_{b_j^\partial}.\] By definition, we have \(\langle \sigma_i \sigma_j \rangle =\langle \sigma_j \sigma_i \rangle\) and \(\langle \sigma_i \sigma_i \rangle =1\) for all \(i,j \in [n]\). Thus, the \(n \times n\)-matrix \[M(G,J):=(\langle \sigma_i \sigma_j \rangle)_{ i,j \in [n]}\] of boundary correlation functions is a symmetric real matrix with \(1\)s on the diagonal. We denote the space of such matrices by \(\operatorname{Mat}_n^{\text{sym}}(\mathbb{R},1)\). The map \[\operatorname{Corr}_G: \mathcal{I}_G^{>0} \rightarrow \operatorname{Mat}_n^{\text{sym}}(\mathbb{R},1), \quad J \mapsto M(G,J)\] is called the boundary correlation map.

Let \[\phi_n: \operatorname{Mat}_n^{\text{sym}}(\mathbb{R},1) \rightarrow \mathrm{Gr}_{n,2n}\] be the doubling map defined as follows: given \(M = (m_{i,j}) \in \operatorname{Mat}_n^{\text{sym}}(\mathbb{R},1)\), let \(\phi_n(M) = \mathrm{rowspan}(\tilde{M})\) where \(\tilde{M} = (\tilde{M}_{i,j})\) is the \(n \times 2n\)-matrix defined as follows. For \(i=j\), let \[\tilde{M}_{i,2i-1} = \tilde{M}_{i,2i} = m_{i,i} =1,\] and for \(i \neq j\), let \[\tilde{M}_{i,2j-1} = -\tilde{M}_{i,2j} = (-1)^{i+j+ \mathbb{1}(i<j)} m_{i,j},\] where \(\mathbb{1}(i<j)\) denotes \(1\) if \(i<j\) and \(0\) otherwise. Similarly to 3, we have:

3.3 Embedding Ising models in dimer models↩︎

Let \(G\) be a reduced cactus graph. We orient the edges of the medial graph \(G^\times\) so that:

1. The boundary vertex \(u_i^\partial\) is a source if and only if \(i\) is odd.

2. Each internal vertex of \(G^\times\) is incident to two incoming and two outgoing edges which alternate in orientation around the vertex.

The existence of this ordering shows that \(\pi_G\) pairs even numbers with odd numbers and vice versa. We construct a reduced plabic graph \(G^\square\) by replacing each vertex of \(G^\times\) as shown on the left hand side of 10 with the bipartite graph on the right hand side. We refer to the degree-\(4\) faces of \(G^\square\) corresponding to edges of \(G\) as square faces. The remaining faces of \(G^\square\) are in bijection with the vertices and faces of \(G\). Note that there is a \(2:1\) correspondence between strands in \(G^\square\) and strands in \(G\) as follows. The strand in \(G\) with source \(u_i^\partial\) corresponds to the pair of strands in \(G^\square\) with sources \(u_i^\partial,u_{\bar{i}}^\partial\). Moreover, we have \(\pi_{G^\square}=\pi_G\).

Given an Ising model \((G,J)\), we define the two functions \(s, c : E(G) \rightarrow (0,1)\) by \[s_e := \operatorname{sech}(2J_e),\qquad c_e:= \operatorname{tanh}(2J_e)\] for every edge \(e \in E(G)\) and replace \(e\) with the dimer model \((G^\square,[{\mathrm{wt}}^\square])\) shown in 10. The map \[\mathcal{I}_{G}^{>0} \hookrightarrow \mathcal{X}_{G^\square}^{>0}, \quad (G,J) \mapsto (G^\square,[{\mathrm{wt}}^\square]).\] is injective (see the proof of Theorem 5.17 in [2]).

We give another characterization of the image \[\mathcal{I}_{G}^{>0} \hookrightarrow \mathcal{X}_{G^\square}^{>0}\] found in [32] for Ising models on a torus. The proof given there however works for arbitrary surfaces. Let \(\bar{G^\square}\) be the plabic graph obtained from \(G^\square\) by changing the colors of all the vertices. We define two operations:

1. Color-change: We replace \(G^\square\) with \(\overline{G^\square}\), i.e., change colors of all vertices, but keep all the edge weights the same.

2. Spider moves at all square faces: We perform a spider move at each square face of \(G^\square\).

Under boundary measurement, color-change is the map \(\Delta_I \mapsto \Delta_{[2n]\setminus I}\) while doing spider moves at all square faces does not change the boundary measurement. Since \(\mathrm{OGr}^+_{n,2n}\) is defined by \(\Delta_I = \Delta_{[2n]\setminus I}\) for all \(I \in \binom{[2n]}{n}\), this gives an alternative proof that the composition \[\mathcal{I}_\pi^{>0} \hookrightarrow \mathcal{X}^{>0}_{\mathscr{M}_\pi} \xrightarrow[]{\operatorname{Meas}_{\mathscr{M}_\pi}} \Pi^{>0}_{\mathscr{M}_\pi}\] lands in \(\mathrm{OGr}_{n,2n}^{\geq 0}\). We will see that an identical characterization exists on the cluster \(\mathcal{A}\) side (2). However, the cluster ensemble and twist maps are not compatible with color-change and will need to be modified (see 5).

4 \(C\)-networks and locally Lagrangian positroid cells↩︎

4.1 \(C\)-networks↩︎

A \(C\)-network is a pair \((G,C)\) where \(G\) is a reduced cactus graph and \[C: V(G) \sqcup F(G) \rightarrow \mathbb{R}_{>0}\] is a function defined modulo rescaling by \(\mathbb{R}_{>0}\). Let \[\mathcal{C}_G^{>0} \cong \mathbb{R}_{>0}^{|V(G)| + |F(G)|-1}\] be the space of \(C\)-networks with underlying cactus graph \(G\).

A Y-\(\Delta\) move \(G \to G'\) induces a map \[\mathcal{C}_G^{>0} \xrightarrow[]{\sim} \mathcal{C}_{G'}^{>0}\] given in the notation of 13 by the positive Kashaev equation \[\begin{align} C_{111} &= \frac{1}{C_{000}^2}\Bigg( 2C_{100}C_{010}C_{001} + C_{000}\bigl( C_{100}C_{011} + C_{010}C_{101} + C_{001}C_{110} \bigr)\\ & + 2 \sqrt{ (C_{000}C_{011}+C_{010}C_{001}) (C_{000}C_{101}+C_{100}C_{001}) (C_{000}C_{110}+C_{100}C_{010}) }\Bigg) \end{align}\]

By construction, we have a bijection \[V(G) \sqcup E(G) \sqcup F(G) \xrightarrow[]{\sim} F(G^\square).\] Note that the edges of \(G\) correspond to the square faces of \(G^\square\). We define an embedding \[\mathcal{C}_G^{>0} \hookrightarrow \mathcal{A}_{G^\square}^{>0}\] as follows. We let \(A_x := C_x\) for \(x \in V(G) \sqcup F(G)\) and for an edge \(e = uv\in E(G)\) incident to faces \(f,g\), we let \[\label{eq:A95in95terms95of95C} A_e := \sqrt{C_u C_v + C_f C_g}.\tag{10}\]

We now define the subset of \(\Pi_{\mathscr{M}_\pi}\) parameterized by \(C\)-networks. We denote by \(\mathcal{S}_G^\square\) the set of all \(I \in \binom{[2n]}{n}\) which appear as the target label of a square face of \(G^\square\), where \(G\) is a reduced cactus graph with \(\pi_G=\pi\), and let \(\mathcal{S}_\pi^\square := \bigcup_{G:\pi_G=\pi} \mathcal{S}_{G}^\square\).

Any such \(I\) is of the form \(S i\bar i\) where:

1. \(\{\gamma_{\{ i, \bar i\}}\), \(\gamma_{\{j,\bar{j} \}} \}\) is a crossing, i.e., a pair of strands \(\{\gamma_{\{ i, \bar i\}}\), \(\gamma_{\{j,\bar{j} \}} \}\) with \(i<j<\bar{i}<\bar{j}\) in cyclic order around the boundary of \(\mathbb{D}\).

2. \(S\) is an \((n-2)\)-element subset of \([2n]\) that contains exactly one element from each pair \(\{k,\bar k\}\) for all strands \(\gamma_{\{k,\bar k\}} \neq \gamma_{\{i,\bar i\}},\gamma_{\{j,\bar j\}}\).

Consider the condition \[\label{eq:cond95plucker95equal} \Delta_{S i \bar i }(V) = \Delta_{S j \bar j}(V).\tag{11}\]

When \(\pi\) is the matching \(i \mapsto i+n\) (modulo \(2n\)), \(\Lambda_\pi^{>0}\) is the intersection of the Lagrangian Grassmannian with \(\mathrm{Gr}_{n,2n}^{>0}\) or equivalently the space of positive-definite Hermitian matrices [13]. In general, 11 can be viewed as a local Lagrangian condition, but we do not know a global description of these spaces.

Note that the condition 11 implies 10 and therefore the following diagram commutes: \[\label{eq:C95A} \begin{tikzcd} \mathcal{C}_{G}^{>0} \arrow[r, hook] & \mathcal{A}_{G^\square}^{>0} \\ \Lambda_\pi^{>0} \arrow[r,hook]\arrow[u,"\boldsymbol{H}_G" ] & \Pi_{\mathscr{M}_\pi}^{>0} \arrow[u,"\boldsymbol{F}_{G^\square}"'] \end{tikzcd}.\tag{13}\]

Gluing the maps \((\boldsymbol{H}_G)_{G: \pi_G=\pi}\), we obtain a bijection \(\boldsymbol{H}_\pi: \Lambda_\pi^{>0} \rightarrow \mathcal{C}_\pi^{>0}\) such that the following diagram commutes: \[\begin{tikzcd} \mathcal{C}_{\pi}^{>0} \arrow[r, hook] & \mathcal{A}_{\mathscr{M}_\pi}^{>0} \\ \Lambda_\pi^{>0} \arrow[r,hook]\arrow[u,"\boldsymbol{H}_\pi" ] & \Pi_{\mathscr{M}_\pi}^{>0} \arrow[u,"\boldsymbol{F}_{\mathscr{M}_\pi}"'] \end{tikzcd}.\]

5 The Ising cluster ensemble and twist maps↩︎

5.1 Ising cluster ensemble↩︎

We now return to the Ising model and formulate the chamber ansatz from the introduction in cluster-ensemble language.

The following is a straightforward verification.

Gluing the maps \(q_G\) as \(G\) varies over all reduced graphs with \(\pi_G=\pi\), we get \[q_\pi: \mathcal{C}_\pi^{>0} \rightarrow \mathcal{I}_\pi^{>0}.\]

5.2 Ising twist map↩︎

Let \(\mathscr{M}\) be a positroid and let \(\Gamma\) be a reduced plabic graph with \(\mathscr{M}_\Gamma=\mathscr{M}\). We also define a modification \[\hat{p}_{\mathscr{M}}: \mathcal{A}_{\mathscr{M}}^{>0} \rightarrow \mathcal{X}_{\mathscr{M}}^{>0}.\] of the cluster ensemble map as follows. Let \(A \in \mathcal{A}_{\Gamma}^{>0}\) and for an edge \(e=bw\) with incident faces \(f\) and \(g\), let \[{\mathrm{wt}}(e) := \begin{cases}1/{A_{f} A_{g}}, &\text{if e is an internal edge},\\ 1/{\sqrt{A_{f} A_{g}}}, &\text{if e is a boundary edge}, \end{cases}\] and define \(\hat{p}_\Gamma(A):=[{\mathrm{wt}}]\). If \(\Gamma \to \Gamma'\) is a move, then the following diagram commutes: \[\begin{tikzcd} \mathcal{A}_{\Gamma}^{>0} \arrow[r, "\sim"] \arrow[d, "\hat{p}_{\Gamma}"'] & \mathcal{A}_{\Gamma'}^{>0} \arrow[d, "\hat{p}_{\Gamma'}"'] \\ \mathcal{X}_{\Gamma}^{>0} \arrow[r, "\sim"] & \mathcal{X}_{\Gamma'}^{>0} \end{tikzcd}.\] Thus, the maps \((\hat{p}_\Gamma)_{\Gamma: \mathscr{M}_\Gamma = \mathscr{M}}\) glue together to give \(\hat{p}_{\mathscr{M}}\).

Since we have modified \(p_\mathscr{M}\), we also need to modify \(\vec{\tau}_{\mathscr{M}}\) to have a commutative diagram as in 5.

Now we return to the Ising model and relate the Ising twist map with the Ising cluster ensemble map.

5.3 Group action and inverse Ising twist↩︎

We have \(\dim \mathcal{I}_G^{>0}=|E(G)|\) and \(\dim \mathcal{C}_G^{>0}=|V(G)|+|F(G)|-1\). Applying Euler’s formula to the cell decomposition of \(\mathbb{D}\) given by \(G\) together with the \(n\) intervals that \((b_i^\partial)_{i=1}^n\) divide \(\partial \mathbb{D}\) into, we have \(|V(G)| - (|E(G)|+n) + |F(G)|=1,\) so \[\dim \mathcal{C}_G^{>0}-\dim \mathcal{I}_G^{>0}=n.\] Hence, the map \(\hat{\tau}_{\mathscr{M}_\pi}\) cannot be a bijection. We will see that there is an action of an \(n\)-dimensional group and \(\hat{\tau}_{\mathscr{M}_\pi}\) becomes a bijection upon taking the quotient.

Given \(\pi \in P_n\), consider the subgroup \[T^{>0}_\pi := \Big\{ \boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_{2n}): \lambda_{\bar{i}} = \lambda_i^{-1} \text{ for all i \in [2n]} \Big\} \cong \mathbb{R}_{>0}^n\] of \(\mathbb{R}_{>0}^{2n}\) with the induced action.

By 3 and 8, the induced map \(q_G: \mathcal{C}_G^{>0}/T_\pi^{>0} \rightarrow \mathcal{I}_G^{>0}\) sits in the commutative diagram: \[\begin{tikzcd} \mathcal{C}_{G}^{>0} /T_{\pi}^{>0} \arrow[r, "q_G"] & \mathcal{I}_{G}^{>0} \arrow[d, "\phi_n \circ \operatorname{Corr}_{G}"] \\ \Lambda_{\pi}^{>0}/T_\pi^{>0} \arrow[u, "\boldsymbol{H}_{G}"] \arrow[r, swap, "{\hat{\tau}_{{\mathscr{M}}_\pi}}"] & \mathrm{OGr}^{\ge 0}_{n,2n} \cap \Pi_{\mathscr{M}_\pi}^{>0} \end{tikzcd}\] and similarly, by gluing, for the induced map \({q}_\pi: \mathcal{C}_\pi^{>0}/T^{>0}_{\pi} \rightarrow \mathcal{I}_\pi^{>0}\). The goal of this section is to prove that the maps \(q_G, q_\pi, \hat{\tau}_{\mathscr{M}_\pi}\) are bijections. We do this by constructing an inverse map \(\hat{\tau}^{-1}_{\mathscr{M}_\pi}\) as a modification of the left twist.

Given \([{\mathrm{wt}}] \in \mathcal{X}_{G^\square}\), let \(m_i([{\mathrm{wt}}])\) denote the alternating product of edge weights along the strand with target \(u_i^\partial\) where the edge weight \({\mathrm{wt}}(bw)\) appears in the numerator if \(bw\) is traversed by the strand from \(b\) to \(w\) and in the denominator otherwise.

Different choices of \(\boldsymbol{\lambda}\) are related by the action of \(T^{>0}_\pi\) and therefore the definition is independent of the choice.

By 7, we have a well-defined map \[\hat{\tau}^{-1}_{\mathscr{M}_\pi}: \mathrm{OGr}^{\ge 0}_{n,2n} \cap \Pi_{\mathscr{M}_\pi}^{>0} \rightarrow \Lambda_\pi^{>0}/T_\pi^{>0}.\]

5.4 Proof of the main theorem↩︎

We are now ready to prove our main result.

5.5 Examples↩︎

We now compute some examples to illustrate the result of 9. The computations were carried out using the computer algebra system Macaulay2 [33]. The code used for our computations is available in [34].

References↩︎