We give a simplified proof for the equivalence of loop-erased random walks to a lattice model containing two complex fermions, and one complex boson. This equivalence works on an arbitrary directed graph. Specifying to the $d$-dimensional hypercubic lattice, at large scales this theory reduces to a scalar $\phi^4$-type theory with two complex fermions, and one complex boson. While the path integral for the fermions is the Berezin integral, for the bosonic field we can either use a complex field $\phi(x)\in \mathbb C$ (standard formulation) or a nilpotent one satisfying $\phi(x)^2 =0$. We discuss basic properties of the latter formulation, which has distinct advantages in the lattice model.

中文翻译：

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循环擦除随机游走与具有两个费米子和一个玻色子的相互作用场理论之间的精确映射

我们为包含两个复费米子和一个复玻色子的晶格模型的循环擦除随机游走的等效性提供了简化的证明。这种等效作用于任意有向图。以$ d $维超立方晶格为例，该理论大规模地简化为具有两个复费米子和一个复玻色子的标量$ \ phi ^ 4 $型理论。尽管费米子的路径积分是Berezin积分，但对于波色子场，我们可以使用\ mathbb C $（标准公式）中的复数场$ \ phi（x）\，也可以使用满足$ \ phi（x）的幂等场^ 2 = 0 $。我们讨论后一种配方的基本特性，这在晶格模型中具有明显的优势。