Multiple SLE type scaling limits: from local to global
Multiple SLE type scaling limits: from local to global
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多种 SLE 类型扩展限制：从局部到全局
DOI：
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发表时间：
2019
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通讯作者：
Alex Karrila
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文献类型：
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作者：
Alex Karrila
当细网格图近似于简单连通域时，我们考虑从平面图上的临界晶格模型获得的 $N$ 弦随机曲线的集合。我们根据通过迭代 Loewner 方程生成的随机曲线的共形不变集合来定义和研究此类限制的候选者。这些曲线是早期引入的局部多 SLE 初始段到全局多 SLE 曲线的自然“域马尔可夫扩展”。为了将它们实现为缩放极限，我们提供了两个先验结果来保证离散随机曲线的预紧性，并允许将离散域马尔可夫属性提升到缩放极限。这些结果本质上仅将某些交叉条件作为输入，与 Kemppainen 和 Smirnov 引入的结果非常相似，并且它们允许通过经典 SLE 收敛证明的鞅策略来识别缩放限制。这些结果的使用通过各种格模型中的收敛证明进行了例证。
We consider collections of $N$ chordal random curves obtained from a critical lattice model on a planar graph, in the limit when a fine-mesh graph approximates a simply-connected domain. We define and study candidates for such limits in terms of conformally invariant collections of random curves, generated via iterated Loewner equations. These curves are a natural ``domain Markov extension' of the earlier introduced local multiple SLE initial segments to global multiple SLE curves. For realizing them as scaling limits, we provide two a priori results to guarantee the precompactness of the discrete random curves and to allow promoting a discrete domain Markov property to the scaling limit. These results essentially only take as input certain crossing conditions, very similar to those introduced by Kemppainen and Smirnov, and they allow the identification of scaling limits via the martingale strategy of classical SLE convergence proofs. The use of these results is exemplified with convergence proofs in various lattice models.