论文标题
接近危险的二聚体和巨大的SLE
Near-critical dimers and massive SLE
论文作者
论文摘要
我们考虑具有双重周期性重量的正方形和六角形晶格上的二聚体模型。本文的目的是三倍：（a）我们与Makarov和Smirnov建造的大量SLE $ _2 $建立了严格的联系（最近由Chelkak和Wan重新审视）； （b）我们表明，收敛性发生在受到候选人边界条件的任意界面域中，并且缩放限度是普遍的。 （c）我们证明了缩放极限的共形协方差。为此，我们引入了一个不均匀的近临界二聚体模型，对应于基础随机步行的漂移，该步行是一个平稳变化的矢量场，或者与不均匀的质量分布。当向量场源自对数凸电势时，我们证明基于相应的循环随机步行具有通用缩放限制。我们的技术依赖于可能具有独立关注的三角形晶格上的确切离散的Girsanov身份。我们通过陈述在自由费点上与广义正弦模型建立连接的精确猜想来补充结果。
We consider the dimer model on the square and hexagonal lattices with doubly periodic weights. The purpose of this paper is threefold: (a) we establish a rigourous connection with the massive SLE$_2$ constructed by Makarov and Smirnov (and recently revisited by Chelkak and Wan); (b) we show that the convergence takes place in arbitrary bounded domains subject to Temperleyan boundary conditions, and that the scaling limit is universal; and (c) we prove conformal covariance of the scaling limit. For this we introduce an inhomogeneous near-critical dimer model, corresponding to a drift for the underlying random walk which is a smoothly varying vector field or alternatively to an inhomogeneous mass profile. When the vector field derives from a log-convex potential we prove that the corresponding loop-erased random walk has a universal scaling limit. Our techniques rely on an exact discrete Girsanov identity on the triangular lattice which may be of independent interest. We complement our results by stating precise conjectures making connections to a generalised Sine-Gordon model at the free fermion point.
