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Largest-loop-first loop-erased random walk on $\mathbb{Z}^{4}$

Daisuke Shiraishi, Satomi Watanabe

Abstract

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.

Largest-loop-first loop-erased random walk on $\mathbb{Z}^{4}$

Abstract

Let be a simple random walk on started at the origin. We study a loop-erasing procedure of that differs from Lawler's chronological loop-erasure. Specifically, we remove loops from in decreasing order of their lengths. The resulting random simple path is called the largest-loop-first (LLF) LERW. For , we prove that the expected length of LLF LERW is of the order . 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.

Paper Structure

This paper contains 4 sections, 10 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Proof of Theorem \ref{['main']}
  4. Convergence to Brownian motion

Key Result

Theorem 1.1

Let $d=4$. As $n \to \infty$, the expectation of $L_{n}$ satisfies $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 3.1
  • Lemma 3.2: Lemma 4.1.1 in exact
  • Remark 3.3
  • ...and 19 more