On the double random current nesting field
 68 Downloads
Abstract
We relate the planar random current representation introduced by Griffiths, Hurst and Sherman to the dimer model. More precisely, we provide a measurepreserving map between double random currents (obtained as the sum of two independent random currents) on a planar graph and dimers on an associated bipartite graph. We also define a nesting field for the double random current, which, under this map, corresponds to the height function of the dimer model. As applications, we provide an alternative derivation of some of the bozonization rules obtained recently by Dubédat, and show that the spontaneous magnetization of the Ising model on a planar biperiodic graph vanishes at criticality.
Mathematics Subject Classification
82B20 60C051 Introduction
The goal of this paper is to present a new connection between the Ising model and dimers through double random currents, and to show some of its applications. The link between dimers and the Ising model has a long history that we will not describe in detail here (we refer the reader to the extensive literature for more information). The articles that we choose to mention in the introduction are the ones directly relevant to our new connection.
1.1 Random currents and dimers
The Ising model is a random configuration of \(\pm \, 1\) spins. In this article we think of the spins as living on the faces of a planar graph \(G=(V,E)\) with vertex set V and edge set E. In [21] Peierls used the socalled lowtemperature expansion of the model to show the existence of an orderdisorder phase transition in the Ising model on \({\mathbb {Z}}^2\). In this representation, configurations of spins assigned to the faces of G are mapped to contour configurations on G. More precisely, for \(B\subset V\), write \({\mathcal {E}}^B\) for the collection of sets of edges \(\omega \subseteq E\) such that the graph \((V,\omega )\) has odd degrees at B and even degrees everywhere else. A connected component of \(\omega \in {\mathcal {E}}^B\) is called a contour, and \(\omega \) itself is called a contour configuration. Each spin configuration on the faces of G is naturally associated with the collection \(\omega \) of edges bordering two faces with different spins. Clearly, \(\omega \) belongs to \({\mathcal {E}}^\emptyset \), and conversely, every element of \({\mathcal {E}}^\emptyset \) is associated with exactly two spin configurations, one with spin \(+\,1\) on the unbounded face, and one with spin \(\,1\).
Remark 1
Our definition of random currents is derived directly from the original one of Griffiths, Hurst and Sherman [13], where a current is a function assigning to each edge a natural number. It is left to the reader to check that our representation is obtained by forgetting the numerical value of the current but keeping the information about its parity and whether it is zero or not. More precisely, \(\omega _{\mathrm{odd}}\) is the set of edges with odd current, \(\omega _{\mathrm{even}}\) with strictly positive even current, and \(E{\setminus }\omega \) with zero current.
The random current model has been successful in several ways. In the original article [13], it was used to derive correlation inequalities. In 1982 it was used by Aizenman [1] to prove triviality of the Ising model in dimension \(d\ge 5\) and a few years later, Aizenman, Barsky and Fernandez proved that the phase transition is sharp [2] (see also [10] for an alternative proof). In recent years the representation has been the object of a revived interest. It was used to study the continuity of the phase transition (see below) and it was also related to other models. For instance, a new distributional relation between random currents, Bernoulli percolation and the FKIsing model was discovered by Lupu and Werner [20]. For a more exhaustive account of random currents, we refer the reader to [9].
The graph \(G^d\) is constructed from \({\vec {G}}\) as follows (the reader may look at Fig. 2 for an illustration). For a vertex z, let r(z) be the number of pairs of consecutive edges in \({\vec {E}}\) around z with the same orientation, and let \(\text {deg}(z)\) be the degree of z. Replace each z with a cycle of \(3\text {deg}(z)r(z)\) edges, called short edges. By construction, the length of the cycle is even, and hence its vertices can be colored black and white in an alternating way. Now, add long edges corresponding to the edges of \({\vec {G}}\). We do it in such a way that if (z, w) is a directed edge of \({\vec {G}}\), then the corresponding edge in \(G^d\) connects a white vertex in the cycle of z with a black vertex in the cycle of w, and moreover, the cyclic order of edges around each cycle in \(G^d\) matches the one in \({\vec {G}}\). The resulting graph \(G^d\) is therefore bipartite. We finish the construction by assigning weights. The long edges inherit their weights from their counterparts in \({\vec {G}}\), and short edges get weight 1.
Theorem 1.1
For any finite simple planar graph G, we have \(\pi _* \mathbf {P}_{\text {dim}}^\emptyset =\mathbf {P}_{\text {dcurr}}^{\emptyset }\).
Remark 2
The theorem can be extended to graphs that are properly embedded in an orientable surface.
1.2 The nesting field of a double random current
One of the main applications of Theorem 1.1 is the study of the socalled nesting field. The idea behind introducing the nesting field is the interpretation of the contours of a current as level lines of a random surface whose discretization is an integervalued function defined on the faces of G. The change in height of the discretized surface when crossing a contour is either \(+\,1\) or \(\,1\), and for two contours belonging to different clusters, the respective height changes are independent.
For a current \(\omega \), a connected component of the graph \((V,\omega )\) will be called a cluster. In particular, each contour \(C\) of \(\omega _{\mathrm{odd}}\) (also called a contour of \(\omega \)) is contained in a unique cluster of \(\omega \), and each cluster \(\mathscr {C}\) of \(\omega \) gives rise to a contour configuration \(\mathscr {C}\cap \omega _{\mathrm{odd}}\). Call a cluster \(\mathscr {C}\)odd around a faceu if the spin configuration associated via the lowtemperature expansion with the contour configuration \(\mathscr {C}\cap \omega _{\mathrm{odd}}\) assigns spin \(\,1\) to u if the exterior face has spin \(+\,1\).
One of the main features of Theorem 1.1 is that it enables to connect the nesting field of a random current \(\omega \) drawn from the double random current measure to the height function associated with dimer covers of \(G^d\). While the latter notion is classical, we still take a moment to recall it here. In the whole article, a path is a sequence of neighboring faces.
To each dimer cover M on \(G^d\), we associate a 1form\(f_M\) (i.e. a function defined on directed edges which is antisymmetric under changing orientation) satisfying \(f_M((z,w))=f_M((w,z))=1\) if \(\{z,w\} \in M\) and z is white, and \(f_M((z,w))=0\) otherwise. From now on, we fix a reference 1form\(f_0\) given by \(f_0((z,w))=f_0((w,z))=1/2\) if \(\{z,w\}\) is a short edge and z is white, and \(f_0((z,w))=0\) otherwise.
 (i)
\(h(u_0)=0\) for the unbounded face \(u_0\),
 (ii)
for every other face u, choose a path \(\gamma \) connecting \(u_0\) and u, and define h(u) to be the total flux of \(f_Mf_{0}\) through \(\gamma \), i.e., the sum of values of \(f_Mf_{0}\) over the edges crossing \(\gamma \) from left to right.
Note that both the faces and vertices of G are embedded naturally in the faces of \(G^d\).
Theorem 1.2
The law of h under \( \mathbf {P}_{\text {dim}}^\emptyset \) restricted to the faces of G is the same as the law of the nesting field \({\mathcal {S}}\) under \(\mathbf {P}_{\text {dcurr}}^{\emptyset }\).
Remark 3
Again, the theorem can be extended to graphs G that are properly embedded in the torus. In this case, the total increment of the nesting field on G between two faces u and v, as it is the case for the dimer height function on \(G^d\), is defined only up to homotopy of the path \(\gamma \) connecting u and v along which the divergence free flows is summed up. We denote these increments by \({\mathcal {S}}_{\gamma }\) and \(h_{\gamma }\) respectively, and conclude that \({\mathcal {S}}_{\gamma }\) drawn according to \(\mathbf {P}^{\emptyset }_{\text {dcurr}, G}\) has the same distribution as \(h_{\gamma }\) drawn according to \(\mathbf {P}^{\emptyset }_{\dim , G^d}\). Also, after fixing \(\gamma \), the increment \({\mathcal {S}}_{\gamma }\) is equal to the sum of the \(\pm \, 1\) variables \(\xi _{\mathscr {C}}\) for the clusters \(\mathscr {C}\) that are odd with respect to \(\gamma \), meaning that the contour configuration \(\mathscr {C}\cap \omega _{\mathrm{odd}}\) crosses an odd number of edges of \(\gamma \).
Consider an infinite biperiodic (i.e. invariant under the action of a \({\mathbb {Z}}^2\)isomorphic lattice) planar graph \(\mathbb G\). The graph \({\mathbb {G}}\) is assumed to be nondegenerate, in the sense that the complement of the edges is the union of topological disks (in other words, the faces are topological disks). Then, the dimer graph \({\mathbb {G}}^d\) constructed as in the finite case, is biperiodic and bipartite. The height function of dimers on biperiodic bipartite graphs has been studied in detail, for instance in [15]. Kenyon, Okounkov and Sheffield identified three possible behaviors depending on the phase: gaseous, liquid or frozen, in which the associated dimer model lies. In particular, the height function of dimers in the liquid phase, which is specified by the property that the characteristic polynomial has zeroes on the torus \({\mathbb {T}}^2\), has unbounded fluctuations. Let \({\mathbb {G}}_n = {\mathbb {G}}/(n{\mathbb {Z}}\oplus n {\mathbb {Z}})\). The relation between the nesting field and the height function of dimers can be hence combined with Theorem 4.5 of [15] to give the following.
Corollary 1.3
We will not use the specific form of \(\phi \), but let us say that it is expressed in terms of the characteristic polynomial.
1.3 Application 1: bozonization rules for the Ising model
By construction, the Ising model is related to the double random current on G with parameters \(x_e=\tanh (\beta J_e)\) and hence, Theorem 1.1 gives a connection between the Ising model and dimers on a bipartite graph. It is known since [11] that the Ising model on a graph G is related to a dimer model on a modified graph, called the Fisher graph of G. This connection enables to express the partition function of the former model in terms of the partition function of the later, which is more amenable to computations. The Fisher graph of G is not bipartite, a fact which renders the study of the dimer model on it more difficult.
Recently, Dubédat [7] (see also [5]) proved that the Ising model can be related to a dimer model on a bipartite graph \(C_G\) where each edge of G is replaced by a quadrilateral and each vertex of degree d by a 2dgon face (see Fig. 2). The dimer model defined in this article on \(G^d\) can in fact be mapped to the dimer model on \(C_G\) with weights as in Fig. 2 via an explicit sequence of vertex splittings and urban renewals (operations which partially preserve the distribution of dimers, and in particular, the height function, see Remark 4). This means that Dubédat’s mapping and our mapping are two facets of the same relation.
Theorem 1.4
Note that for a vertex x and a face u, \(\sin (\pi h_x)=(1)^{h_x1/2}\) and \(\cos (\pi h_u)=(1)^{h_u}\). Also, the fact that the disorder lines are starting on the unbounded face \(u_0\) is a convenient convention to state the result elegantly in terms of the notation introduced in the previous section. The theorem can be extended to graphs G properly embedded in the torus with appropriate modifications.
1.4 Application 2: continuity of the phase transition for the Ising model on biperiodic planar graphs
In [6], the critical parameter \(\beta _c\) of the Ising model was proved to correspond to the only value of \(\beta \) for which the dimer model introduced in [7] on \(C_{{\mathbb {G}}}\) (and therefore the one defined here on \({\mathbb {G}}^d\)) is in the liquid phase. Here, we combined this result with the information above to prove the following statement.
Theorem 1.5
For the square lattice, the result goes back to the exact computation of Yang [28]. In higher dimension, the fact that \(\mu _{{\mathbb {G}},\beta _c}^+[\sigma _u]=0\) is known for the nearest neighbor Ising model on \({\mathbb {G}}={\mathbb {Z}}^d\) [3, 4]. On trees, the result was proved in [14]. Recently, Raoufi [25] showed that amenable groups with exponential growth undergo a continuous phase transition. To the best of our knowledge, a proof which is valid for any infinite biperiodic planar graph was not available until now.
A byproduct of the proof is the following result about nonpercolation of spins.
Corollary 1.6
Let \({\mathbb {G}}\) be a nondegenerate infinite biperiodic planar graph, then the \(\mu _{{\mathbb {G}},\beta _c}^+\)probability that there exists an infinite cluster of pluses or minuses is zero.
1.5 Extension to Dobrushin boundary conditions
Much of what has been described above can be extended to cover the case of the Ising model with Dobrushin boundary conditions. Consider two vertices a and b on the exterior face of G. Configurations in \({\mathcal {E}}^{\{a,b\}}\) correspond to (the socalled Dobrushin) spin configurations where the external face is split into two faces of opposite spins by adding an additional edge joining a and b. In particular, this construction implies that \(\omega \in {\mathcal {E}}^{\{a,b\}} \) necessarily contains a contour connecting a and b.
The definition of the nesting field for a current with \(\{a,b\}\)boundary conditions is almost the same with the exception that the variable \(\xi _{\mathscr {C}_0}\) corresponding to the cluster \(\mathscr {C}_0\) connecting a and b is set to 1. Moreover, the cluster \(\mathscr {C}_0\) is called odd around u if its contours assign spin \(\,1\) to u in the model with Dobrushin boundary conditions with \(+\,1\) spin on the (external) face adjacent to the clockwise boundary arc from a to b, and \(\,1\) spin on the face adjacent to the arc from b to a.
Consider an augmented graph \({\vec {G}}_{(a,b)}\) where an additional edge \(e_{(a,b)}\) directed from b to a is added in the external face of \({\vec {G}}\) in such a way that the clockwise boundary arc of \({\vec {G}}\) from a to b is bordering the unbounded face of \({\vec {G}}_{(a,b)}\). We define the graph \(G^d_{(a,b)}\) out of \({\vec {G}}_{(a,b)}\) exactly as we defined \(G^d\) out of \({\vec {G}}\). Let \(\mathcal {M}_{(a,b)}\) be the set of dimer covers of \(G^d_{(a,b)}\) containing the edge (b, a). Also, introduce the height function h of M in \(\mathcal {M}_{(a,b)}\) by choosing a reference 1form corresponding to a matching that represents a current composed only of a path of odd edges that form the clockwise arc from b to a on the boundary of G.
Then, we have the following extension of Theorems 1.1 and 1.2.
Theorem 1.7
 (i)
\(\pi _* \mathbf {P}_{\text {dim}}^{\{a,b\}}=\mathbf {P}_{\text {dcurr}}^{\{a,b\}}\),
 (ii)
the law of h under \( \mathbf {P}_{\text {dim}}^{(a,b)}\) restricted to the faces of G is the same as the law of the nesting field \({\mathcal {S}}\) under \(\mathbf {P}_{\text {dcurr}}^{\{a,b\}}\).
2 Proofs of Theorems 1.1, 1.2 and 1.7
There will be no difference in working with \(B=\emptyset \) or \(B=\{a,b\}\). For this reason, we simply refer to B as being the set of sources. The proofs rely on the notion of alternating flows and their height function. For this reason, we define a probability measure on flows which will be later naturally related to the double random current measure and its nesting field. We should mention that the proofs of the theorems can be obtained by hand, meaning without using alternating flows. Nonetheless, we believe that alternative flows offer an elegant way of deriving the connection between dimers and double random currents.
A sourceless alternating flowF is a set of directed edges of \({\vec {G}}\) such that for each vertex v, the edges in F around v alternate between being oriented towards and away from v when going around v (see Fig. 4). In particular, the same number of edges enters and leaves v. For two vertices a and b on the outer face of \({\vec {G}}\), an alternating flow with source a and sink b is a sourceless alternating flow on \({\vec {G}}_{(a,b)}\) containing \(e_{(a,b)}\) (note that, here, (a, b) is an oriented edge and should not be confused with \(\{a,b\}\)). Denote the set of sourceless alternating flows on \({\vec {G}}\) by \(\mathcal {F}^\emptyset \), and the set of alternating flows with source a and sink b by \(\mathcal {F}^{(a,b)}\).
Define a map \(\theta : \mathcal {F}^B \rightarrow \Omega ^B\) as follows. For every \(F \in \mathcal {F}^B\) and every \(e \in E\), consider the number of corresponding directed edges \({\vec {e}}_m\), \({\vec {e}}_{s1}\), \({\vec {e}}_{s2}\) present in F. Let \(\omega _{\mathrm{odd}}\subset E\) be the set with one or three such present edges, and \(\omega _{\mathrm{even}} \subset E\) the set with exactly two such edges. Then, set \(\theta (F)=\omega \). It follows from the definition of alternating flows that \(\omega =\omega _{\mathrm{odd}} \cup \omega _{\mathrm{even}}\) is a current with sources B. Denote by \(\theta _* \mathbf {P}_{\text {aflow}}^B\) the pushforward measure on \(\Omega ^B\). The following result was previously obtained in [19].
Theorem 2.1
[19] For any finite simple planar graph G, we have that \(\theta _* \mathbf {P}_{\text {aflow}}^B=\mathbf {P}_{\text {dcurr}}^{B}\).
Proof
Since the theorem is a special case of [19, Thm 4.1], we only outline the proof here for completeness.
The result follows from the fact that the outer boundary of each nontrivial cluster can be oriented in two possible ways, hence the weight \(2^{k(\omega )V^c(F)}\). \(\square \)
We now describe a straightforward measure preserving mapping from the dimer model to alternating flows. To each matching \(M\in \mathcal {M}^B\), we associate a flow \(\eta (M)\in \mathcal {F}^B\) by replacing each long edge in \(M\) by the corresponding directed edge in \({\vec {G}}\). One can see that this always produces an alternating flow. Indeed, assuming otherwise, there would be two consecutive edges in \(\eta (M)\) of the same orientation, and therefore, the path of odd length connecting them in a cycle would have a dimer cover, which is a contradiction. Let \(\eta _*\mathbf {P}_{\text {dim}}^B\) be the pushforward measure on \(\mathcal {F}^B\) under the map \(\eta \).
Theorem 2.2
For any finite simple planar graph G, we have that \(\eta _* \mathbf {P}_{\text {dim}}^B=\mathbf {P}_{\text {aflow}}^{B}\).
Proof
Comparing (2.1) with (1.2), and knowing that the long edges of \(G^d\) have the same weights as in \({\vec {G}}\), we only need to account for the factor \(2^{V^c(F)}\) from the definition of the alternating flow measure. To this end, note that the only freedom in the dimer covers in \(\eta ^{1}(F)\) is the way they match the short edges in the cycles corresponding to the isolated vertices of (V, F). Each such cycle has two matchings, and the matchings of different cycles are independent. This completes the proof. \(\square \)
Proof of Theorems 1.1 and 1.7 (i)
We define \(\pi =\theta \circ \eta : \mathcal {M}^B \rightarrow \Omega ^B\) to be the manytoone map projecting dimer covers to currents (note that it is the mapping defined in the introduction). Let \(\pi _*\mathbf {P}_{\text {dim}}^B\) be the pushforward measure on \(\Omega ^B\). Combining the two previous theorems yields the corresponding statements of the introduction. \(\square \)
 (i)
\(h(u_0)=0\) for the unbounded face \(u_0\),
 (ii)
for every other face u, choose a path \(\gamma \) connecting \(u_0\) and u, and define h(u) to be total flux of F through \(\gamma \), i.e., the number of edges in F crossing \(\gamma \) from left to right minus the number of edges crossing \(\gamma \) from right to left.
Proof of Theorem 1.2 and 1.7(ii)
It is clear that \(h_F\) is equal to the height function of the dimer cover \(M=\eta (F)\). We therefore relate \(h_F\) to \(\mathcal S(\omega )\), where \(\omega =\theta (F)\).
Recall from the proof of Theorem 2.1 that for each cluster of a double random current, there are two opposite orientations of the boundary of the corresponding connected component of the associated alternating flows in \(\theta ^{1}(\omega )\). Set \(\xi _{{\mathcal {C}}}(F)=+1\) if F is oriented counterclockwise around the boundary of the cluster \({\mathcal {C}}\) of \(\omega \), and \(\xi _{{\mathcal {C}}}(F)=1\) otherwise. By the proof of Theorem 2.1, \((\omega ,\xi (F))\) is in direct correspondence with F. Furthermore, by construction, \(h_F\) is equal to the nesting field \({\mathcal {S}}(\omega )\) obtained from the \(\xi (F)\). The fact that for each cluster \({\mathcal {C}}\), the two opposite orientations carry the same weight implies that under the law of alternating flows, conditionally on \(\omega \), \(\xi (F)\) is a iid family of random variables which are equal to \(+\,1\) or \(\,1\) with probability 1 / 2. This concludes the proof. \(\square \)
3 Proof of Theorem 1.4
Remark 4
The relations obtained in Theorem 1.4 are the same as in Lemma 3 of [7]. Indeed, the dimer model on \(G^d\) is associated with the dimer model of [5, 7] as follows. Given an edge of G, select a quadrilateral face in \(G^d\) corresponding to the edge and (if necessary) split each vertex that the chosen quadrilateral shares with a quadrilateral corresponding to a different edge of G. In this way we find ourselves in the situation from the upper left panel in Fig. 5. After performing urban renewal (i.e. the transformation from Fig. 5) and collapsing the doubled edge, we are left with one quadrilateral as desired. One can check that the weights that we obtain match those from Fig. 2. We then repeat the procedure for every edge of G and the resulting graph is \(C_G\).
Note that the height function on faces is not modified by vertex splitting and urban renewal. Nonetheless, there is indeed loss of information between the dimer model on \(G^d\) and the one on \(C_G\), and we the former is more suitable for understanding double random currents.
4 Proof of Theorem 1.5
We will in fact work with the Ising model on the dual graph \(\mathbb G^*\) obtained by putting a vertex in each face of \({\mathbb {G}}\), and edges between vertices corresponding to neighboring faces. As such, the Ising model below will be seen as a random assignment of spins to the faces of \({\mathbb {G}}\). While we use the notation \({\mathbb {G}}\) as in the introduction, the outcome of the proof will be Theorem 1.5 for \({\mathbb {G}}^*\). Since the dual graph of a nondegenerated biperiodic graph is itself nondegenerated and biperiodic, this is sufficient. The reason for working with the Ising model on \({\mathbb {G}}^*\) is that we will use the connection with the dimer on \({\mathbb {G}}\), and that this makes the study more coherent with other sections of the article.
 Step 0

We introduce relevant auxiliary infinite volume measures.
 Step 1

We show that \(\mu ^+_{{\mathbb {G}}^*}[\mathbf C_u(\sigma )]=+\infty \).
 Step 2

We prove that \(\mu ^+_{{\mathbb {G}}^*}[\mathbf C_u(\sigma )=0]=0\).
 Step 3

We deduce that \(\mu ^+_{\mathbb G^*}[\sigma _u]=0\).
Remark 5
Note that Step 2 can be restated as follows: there is no infinite cluster of pluses or minuses \(\mu ^+_{{\mathbb {G}}^*}\)almost surely. As a byproduct, we obtain Corollary 1.6.
Step 1 is the major novelty of the proof. It relies on Theorem 1.2 and Corollary 1.3. Step 3 is directly extracted from [8, Prop. 4.1]. We refer to [12] for classical facts on the Ising model.
Let \(\Lambda \approx {\mathbb {Z}}\oplus {\mathbb {Z}}\) be a group acting transitively on \({\mathbb {G}}\). Let \({\mathbb {G}}_n={\mathbb {G}}/(n\mathbb Z\oplus n{\mathbb {Z}})\) be the toroidal graph of size \(n\in {\mathbb {N}}\), and let \({\mathbb {G}}^d_n = {\mathbb {G}}^d/(n{\mathbb {Z}}\oplus n{\mathbb {Z}})\) be the bipartite toroidal dimer graph corresponding to \(\mathbb G_n\). Below, we consider the random current, double random current and dimer models on \({\mathbb {G}}_n\) and \({\mathbb {G}}^d_n\) with n tending to infinity, and where the weights \(x_e\) on \({\mathbb {G}}_n\) are defined as follows: if e is the edge between the faces u and v, then \(x_e:=\exp [2\beta ({\mathbb {G}}^*) J_{\{u,v\}}]\). In what follows, we add subscripts to the already introduced notation to mark the dependency of the probability measures on the underlying graph.
Step 0 Note that for topological reasons, some current configurations on \({\mathbb {G}}_n\) do not correspond to spin configurations on the faces of \({\mathbb {G}}_n\). To overcome this obstacle, we will resort to the construction of infinite volume measures for the different models, where planarity is recovered in the limit as n tends to infinity. There are several ways to proceed and we simply explain here the shortest one (this is not the most selfcontained one).
By [15], \(\mathbf {P}^{\emptyset }_{\text {dim},{\mathbb {G}}^d_n}\) converges weakly to a \(\Lambda \)invariant measure \(\mathbf {P}^{\emptyset }_{\text {dim},{\mathbb {G}}^d}\) on dimer covers of \(\mathbb G^d\). Since the sourceless double random current on \({\mathbb {G}}_n\) is a local function of the dimer model on \({\mathbb {G}}^d_n\), we get that \(\mathbf {P}^{\emptyset }_{\text {dcurr},{\mathbb {G}}_n}\) converges weakly to an infinite volume measure \(\mathbf {P}^{\emptyset }_{\text {dcurr},{\mathbb {G}}}\) on sourceless currents on \({\mathbb {G}}\).
The measures \(\mathbf {P}^{\emptyset }_{\text {curr},{\mathbb {G}}_n}\) also converge weakly to a measure \(\mathbf {P}^{\emptyset }_{\text {curr},{\mathbb {G}}}\) on sourceless currents on \({\mathbb {G}}\). In order to see this, we go back to the original definition of single and double currents in terms of integervalued functions. Since the integer value of the double current at an edge is obtained from the parity independently for any edge, the integervalued double random current also converges. With this definition, the integervalued double random current is simply the sum of two iid integervalued single random currents, and therefore for any finite set D of edges, the characteristic function of the latter when restricted to D is the squareroot of the characteristic function of the former. In particular, it converges for any fixed D. This implies the convergence of the single random current.
We now define a probability measure \(\mu _{{\mathbb {G}}^*}\) on the space of \(\pm \, 1\) spin configurations on the faces of \({\mathbb {G}}\) by tossing a symmetric coin to decide the spin at a fixed face, and then using the odd part of a current \(\omega \) drawn according to \(\mathbf {P}^{\emptyset }_{\text {curr},{\mathbb {G}}}\) to define the interfaces between \(+\,1\) and \(\,1\) spins. This is well defined since \({\mathbb {G}}\) is planar and the degree of \(\omega _{\mathrm{odd}}\) at every vertex of \({\mathbb {G}}\) is even almost surely. Note that \(\mu _{{\mathbb {G}}^*}\) is \(\Lambda \)invariant since the infinitevolume version of the single random currents inherits the invariance under the action of \(\Lambda \) from the dimer measure.
Using the domain Markov property of \(\omega _{\mathrm{odd}}\) under \(\mathbf {P}^{\emptyset }_{\text {curr},{\mathbb {G}}_n}\), and the fact that a spin configuration under \(\mu _{{\mathbb {G}}^*}\) carries the same information (up to a spin flip) as \(\omega _{\mathrm{odd}}\), one can check that \(\mu _{{\mathbb {G}}^*}\) satisfies the Dobrushin–Lanford–Ruelle conditions for an infinite volume Gibbs state of the Ising model with parameters \(\beta \) and \((J_{e})_{e\in E}\).
Below, we will use the following notation. For a set of faces F, \(\partial F\) denotes the set of faces u such that there exists a neighboring face v which is not in F.
Step 3 It suffices to show that \(\mu ^+_{\mathbb G^*}[\sigma _u]\le 0\) since we already know by the first Griffiths inequality that \(\mu ^+_{{\mathbb {G}}^*}[\sigma _u]\ge 0\).
Notes
Acknowledgements
Open access funding provided by University of Vienna. M.L. is grateful to Sanjay Ramassamy for an inspiring remark, and the authors thank Aran Raoufi and Gourab Ray for many useful discussions. This article was finished during a stay of M.L. at IHES. M.L. thanks IHES for the hospitality. The research of M.L. was funded by EPSRC Grants EP/I03372X/1 and EP/L018896/1 and was conducted when M.L. was at the University of Cambridge. The research of H.D.C. was funded by a IDEX Chair from Paris Saclay, the ERC CriBLaM, and by the NCCR SwissMap from the Swiss NSF.
References
 1.Aizenman, M.: Geometric analysis of \(\varphi ^{4}\) fields and Ising models. I, II. Commun. Math. Phys. 86(1), 1–48 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
 2.Aizenman, M., Barsky, D.J., Fernández, R.: The phase transition in a general class of Isingtype models is sharp. J. Stat. Phys. 47(3–4), 343–374 (1987)MathSciNetCrossRefGoogle Scholar
 3.Aizenman, M., DuminilCopin, H., Sidoravicius, V.: Random currents and continuity of Ising model’s spontaneous magnetization. Commun. Math. Phys. 334, 719–742 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
 4.Aizenman, M., Fernández, R.: On the critical behavior of the magnetization in highdimensional Ising models. J. Stat. Phys. 44(3–4), 393–454 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
 5.Boutillier, C., de Tilière, B.: Height representation of XORIsing loops via bipartite dimers. Electron. J. Probab. 19(80), 33 (2014)MathSciNetzbMATHGoogle Scholar
 6.Cimasoni, D., DuminilCopin, H.: The critical temperature for the Ising model on planar doubly periodic graphs. Electron. J. Probab. 18(44), 1–18 (2013)MathSciNetzbMATHGoogle Scholar
 7.Dubédat, J.: Exact bosonization of the Ising model (2011). arXiv:1112.4399
 8.DuminilCopin, H.: Lectures on the Ising and Potts models on the hypercubic lattice. arXiv:1707.00520
 9.DuminilCopin, H.: Random currents expansion of the Ising model (2016). arXiv:1607.06933
 10.DuminilCopin, H., Tassion, V.: A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Commun. Math. Phys. 343(2), 725–745 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
 11.Fisher, M.: On the dimer solution of planar Ising models. J. Math. Phys. 7(10), 1776–1781 (1966)CrossRefGoogle Scholar
 12.Friedli, S., Velenik, Y.: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction. Cambridge University Press, Cambridge (2017)CrossRefzbMATHGoogle Scholar
 13.Griffiths, R.B., Hurst, C.A., Sherman, S.: Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys. 11(3), 790–795 (1970)MathSciNetCrossRefGoogle Scholar
 14.Häggström, O.: The randomcluster model on a homogeneous tree. Probab. Theory Relat. Fields 104(2), 231–253 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
 15.Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. Math. (2) 163(3), 1019–1056 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
 16.Kramers, H.A., Wannier, G.H.: Statistics of the twodimensional ferromagnet I. Phys. Rev. (2) 60, 252–262 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
 17.Lam, T.: Dimers, webs, and positroids. J. Lond. Math. Soc. (2) 92(3), 633–656 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
 18.Lam, T.: Totally nonnegative Grassmannian and Grassmann polytopes (2015). arXiv:1506.00603
 19.Lis, M.: The planar Ising model and total positivity. J. Stat. Phys. 166(1), 72–89 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
 20.Lupu, T., Werner, W.: A note on Ising random currents, IsingFK, loopsoups and the Gaussian free field. Electron. Commun. Probab. 21(13) (2016). https://doi.org/10.1214/16ECP4733
 21.Peierls, R.: On Ising’s model of ferromagnetism. Math. Proc. Camb. Philos. Soc. 32, 477–481 (1936)CrossRefzbMATHGoogle Scholar
 22.Postnikov, A.: Total positivity, Grassmannians, and networks (2006). arXiv:math/0609764
 23.Postnikov, A., Speyer, D., Williams, L.: Matching polytopes, toric geometry, and the totally nonnegative grassmannian. J. Algebr. Comb. 30(2), 173–191 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
 24.Raoufi, A.: Translation invariant Ising Gibbs states, general setting (2017). arXiv:1710.07608
 25.Raoufi, A.: A note on continuity of magnetization at criticality for the ferromagnetic Ising model on amenable quasitransitive graphs with exponential growth (2016). arXiv:1606.03763
 26.Talaska, K.: A formula for Plücker coordinates associated with a planar network. Int. Math. Res. Not. IMRN, Art. ID rnn 081, 19 (2008)Google Scholar
 27.van der Waerden, B.L.: Die lange Reichweite der regelmassigen Atomanordnung in Mischkristallen. Z. Physik 118, 473–488 (1941)CrossRefzbMATHGoogle Scholar
 28.Yang, C.N.: The spontaneous magnetization of a twodimensional Ising model. Phys. Rev. (2) 85, 808–816 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
Copyright information
OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.