AI 中文总结
本文证明四维和五维随机游走轨迹的图距离与有效电阻波动收敛于稳定律，结合前人六维及更高维的高斯收敛结果揭示五、六维间的相变，且所提长程渗流耦合技术有望应用于相关模型。
AI 中文摘要
本文证明，四维和五维中随机游走轨迹上的图距离与有效电阻的波动，依分布收敛于稳定律。此前工作中，第一作者与第二作者已证明，六维及更高维中对应波动收敛于高斯分布。综合这些结果，揭示了五维和六维之间存在相变。本文的证明发展了一种与长程渗流的新颖耦合，预计该技术将在广泛的相关模型中得到应用。
英文摘要
In this paper, we prove that the fluctuations of the graph distance and the effective resistance on the trace of a random walk in four and five dimensions converge in distribution to a stable law. In previous work, the first and second authors proved that the corresponding fluctuations converge to a Gaussian distribution in dimensions six and higher. Taken together, these results reveal a phase transition between dimensions five and six. Our proof develops a novel coupling with long range percolation, and we expect this technique to find applications in a broad class of related models.
