Largest-loop-first loop-erased random walk on $\mathbb{Z}^{4}$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915909071798272 |
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| author | Shiraishi, Daisuke Watanabe, Satomi |
| author_facet | Shiraishi, Daisuke Watanabe, Satomi |
| contents | Let $S = (S(n))$ be a simple random walk on $\mathbb{Z}^{d}$ started at the origin. We study a loop-erasing procedure of $S[0,n]$ that differs from Lawler's chronological loop-erasure. Specifically, we remove loops from $S[0,n]$ in decreasing order of their lengths. The resulting random simple path is called the largest-loop-first (LLF) LERW. For $d=4$, we prove that the expected length of LLF LERW is of the order $n (\log n)^{-1/2 + o(1)}$. In particular, this suggests that chronological LERW and LLF LERW belong to different universality classes. Furthermore, we also prove the convergence of LLF LERW to Brownian motion in four dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01748 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Largest-loop-first loop-erased random walk on $\mathbb{Z}^{4}$ Shiraishi, Daisuke Watanabe, Satomi Probability 05C81, 60G50, 82B41 Let $S = (S(n))$ be a simple random walk on $\mathbb{Z}^{d}$ started at the origin. We study a loop-erasing procedure of $S[0,n]$ that differs from Lawler's chronological loop-erasure. Specifically, we remove loops from $S[0,n]$ in decreasing order of their lengths. The resulting random simple path is called the largest-loop-first (LLF) LERW. For $d=4$, we prove that the expected length of LLF LERW is of the order $n (\log n)^{-1/2 + o(1)}$. In particular, this suggests that chronological LERW and LLF LERW belong to different universality classes. Furthermore, we also prove the convergence of LLF LERW to Brownian motion in four dimensions. |
| title | Largest-loop-first loop-erased random walk on $\mathbb{Z}^{4}$ |
| topic | Probability 05C81, 60G50, 82B41 |
| url | https://arxiv.org/abs/2604.01748 |