Piecewise Temperleyan dimers and a multiple SLE$_8$

TL;DR

This paper studies the scaling limits of dimer models on piecewise Temperleyan domains, showing convergence to multiple SLE$_8$ and Gaussian free field, extending previous results to more general graphs and boundary conditions.

Contribution

It generalizes the scaling limit results of dimer models to piecewise Temperleyan graphs, describing the limit as multiple SLE$_8$ and connecting height functions to the Gaussian free field.

Findings

Scaling limit of spanning trees described by multiple SLE$_8$

Interface between trees given by SLE$_2(-1,...,-1)

Height function converges to Gaussian free field with boundary jumps

Abstract

We consider the dimer model on piecewise Temperleyan, simply connected domains, on families of graphs which include the square lattice as well as superposition graphs. We focus on the spanning tree $\mathcal{T}_\delta$ associated to this model via Temperley's bijection, which turns out to be a Uniform Spanning Tree with singular alternating boundary conditions. Generalising the work of the second author with Peltola and Wu \cite{LiuPeltolaWuUST} we obtain a scaling limit result for $\mathcal{T}_\delta$. For instance, in the simplest nontrivial case, the limit of $\mathcal{T}_\delta$ is described by a pair of trees whose Peano curves are shown to converge jointly to a multiple SLE$_8$ pair. The interface between the trees is shown to be given by an SLE$_2(-1, \ldots, -1)$ curve. More generally we provide an equivalent description of the scaling limit in terms of imaginary geometry. This allows us to make use of the results developed by the first author and Laslier and Ray \cite{BLRdimers}. We deduce that, universally across these classes of graphs, the corresponding height function converges to a multiple of the Gaussian free field with boundary conditions that jump at each non-Temperleyan corner. After centering, this generalises a result of Russkikh \cite{RusskikhDimers} who proved it in the case of the square lattice. Along the way, we obtain results of independent interest on chordal hypergeometric SLE$_8$; for instance we show its law is equal to that of an SLE$_8 (\bar \rho)$ for a certain vector of force points, conditional on its hitting distribution on a specified boundary arc.