Conformal invariance of double random currents I: identification of the limit

Abstract

This is the first of two papers devoted to the proof of conformal invariance of the critical double random current model on the square lattice. More precisely, we show the convergence of loop ensembles obtained by taking the cluster boundaries in the sum of two independent currents with free and wired boundary conditions. The strategy is first to prove convergence of the associated height function to the continuum Gaussian free field, and then to characterize the scaling limit of the loop ensembles as certain local sets of this Gaussian Free Field. In this paper, we identify uniquely the possible subsequential limits of the loop ensembles. Combined with the second paper, this completes the proof of conformal invariance.

Figures (18)

• Figure 1.1: Left: We depict the outermost clusters in a double random current with free boundary conditions. The outer boundaries of these clusters are in red (they form a CLE$_4$). The inner boundaries of the clusters are in black. Right: We depict the unique outermost cluster in a double random current with wired boundary conditions. The inner boundaries of this cluster are in black. For both: In each domain encircled by an inner boundary loop, one has (the scaling limit of) an independent double random current with free boundary conditions. This allows us to iteratively sample the nested interfaces.
• Figure 3.1: An illustration of the coupling from Theorem \ref{['thm:mastercoupling']}. A piece of the (rotated) primal square lattice with white vertices, and its dual square lattice with black vertices is shown. The primal and dual double random current clusters are drawn in blue and red respectively. The odd parts of the current are marked with solid lines, whereas the nonzero even parts are marked with dashed lines. Each vertex (primal black vertex) and a face (dual white vertex) carries both a $\pm 1$ spin ($\tau$ and $\tau^\dagger$ respectively) and the value of the height function $H$. The height function takes integer values in $\mathbb Z$ on the black vertices and in $\tfrac{1}{2}+\mathbb Z$ on the white vertices as implied by property \ref{['itm:cp6']} of the master coupling. Property \ref{['itm:cp6']} and the fact that the spins $\tau$ and $\tau^\dagger$ are constant on the primal and dual clusters respectively imply that the height function is also constant on both the primal and dual clusters. This is why in the figure we marked the values of the spins and height only at the rightmost vertices of the clusters (including isolated vertices).
• Figure 3.2: A configuration of primal (red) and dual (blue) double random currents ${\mathbf n}$ and ${\mathbf n}^\dagger$. The outermost blue circuit is part of a cluster of the boundary in ${\mathbf n}^\dagger$ whose remainder is not shown here. The green edges denote the inner boundary loop $\ell$ of this cluster (i.e. a loop in $Q_0$ as defined in Sec. \ref{['sec:mainres']}). The primal vertices on this loop are identified with each other in the exploration process described in property \ref{['itm:bc']} of the master coupling from Theorem \ref{['thm:mastercoupling']}. After this identification the primal current ${\mathbf n}$ has wired boundary conditions. The clusters of the modified current ${\mathbf n}_\ell$ defined in Sec. \ref{['sec:mainres']} are given by the union of the green loop and the red clusters surrounded by it. Finally, $Q_{1}(\ell)$ is defined as the collection of loops in the inner boundary of the external most cluster (touching $\ell$) of this modified current ${\mathbf n}^\ell$. These loops come in two types, the yellow loops that are part of $A_0(\ell)$, and the orange loops are in $Q_1(\ell) \setminus A_0(\ell)$. Each orange loop traces the red clusters from the outside and/or the green loop from the inside. This property is used in Lemma \ref{['lem:QH1']} to obtain precompactness of the the orange loops given precompactness of the red and green loops. Inside each yellow loop of the inner boundary of the primal clusters, the procedure is repeated and now the primal clusters surrounded by each such loop have wired boundary conditions.
• Figure 3.3: One can construct the graphs $\vec{G}$, $G^d$ and $C_G$ ($\vec{G}$ is formally a multigraph) locally around each vertex of $G$. The weights satisfy $y=\tfrac{2x}{1-x^2}$, $w=\tfrac{2x}{1+x^2}$, $z=\tfrac{1-x^2}{1+x^2}$. Here $x=x_e$ is the high-temperature weight equal to $\tanh( \beta J_e$). The edges carrying weight $1$ in $G^d$ (resp. in $C_G$) are called short (resp. roads), and the remaining edges are called long (resp. streets).
• Figure 3.4: Left: A configuration $({\mathbf n}_{\rm odd},{\mathbf n}_{\rm even})$ on a piece of the hexagonal lattice $G$. The blue edges represent ${\mathbf n}_{\rm odd}$ and the red edges represent ${\mathbf n}_{\rm even}$. The blue and red edges together form one cluster $\mathscr{C}$. Middle: Two alternating flow configurations on $\vec{G}$ mapped to $({\mathbf n}_{\rm odd},{\mathbf n}_{\rm even})$ under $\theta$. The two clusters have opposite orientations of the outer boundary. Depending on this orientation the height function either increases or decreases by one when going from the outside to the inside of the lower hexagon. This corresponds to two different outcomes for the label $\epsilon_\mathscr{C}$ in the definition of the nesting field \ref{['def:nestingfield']}. Right: Two dimer configurations on $G^d$ that map to the corresponding alternating flows under $\pi$. Note that the parity of the height function on $G^d$ restricted to the vertices of $\mathscr{C}$ and shifted by $1/2$ changes whenever the sign of $\epsilon_\mathscr{C}$ changes. This can be seen from the placement of the dimers on the short edges. This property is used in the proof of Theorem \ref{['thm:mastercoupling']}. On the other hand the parity of the height function on the faces of $G$ is independent of $\epsilon_\mathscr{C}$. We also note that both $\pi$ and $\theta$ are many-to-one maps.

Theorems & Definitions (95)

• Theorem 1.1: Convergence of double random current clusters with free boundary conditions
• Theorem 1.2: Convergence of double random current clusters with wired boundary conditions
• Remark 1.3
• Theorem 1.4: Convergence of the nesting field
• Theorem 2.1: Aizenman--Burchard criterion for the double random current model
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4: Connection probabilities close to the boundary for double random current
• Theorem 3.1: Master coupling
• Corollary 3.2