The determination of the Hausdorff dimension of the scaling limit of loop-erased random walk is closely related to the study of the one-point function of loop-erased random walk, i.e., the probability a loop-erased random walk passes through a given vertex. Recent work in the theoretical physics literature has investigated the Hausdorff dimension of loop-erased random walk in three dimensions by applying field theory techniques to study spin systems that heuristically encode the one-point function of loop-erased random walk. Inspired by this, we introduce two different spin systems whose correlation functions can be rigorously shown to encode the one-point function of loop-erased random walk.

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循环擦除随机游走作为可观察的自旋系统

循环擦除随机游走标度极限Hausdorff维数的确定与循环擦除随机游走单点函数的研究密切相关，即循环擦除随机游走通过给定顶点的概率. 理论物理文献中的最新工作通过应用场论技术来研究对循环擦除随机游走的单点函数进行启发式编码的自旋系统，在三个维度上研究了循环擦除随机游走的 Hausdorff 维。受此启发，我们引入了两种不同的自旋系统，它们的相关函数可以严格显示来编码循环擦除随机游走的单点函数。