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<p>In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that are visited only once) of a simple symmetric random walk on ℤ. Let <i>α</i>(<i>n</i>) be the number of rarely visited edges up to time <i>n</i>. First, we evaluate <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb{E}(\alpha(n))\)</EquationSource> </InlineEquation>, show that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n\to \mathbb{E}(\alpha(n))\)</EquationSource> </InlineEquation> is non-decreasing in <i>n</i> and that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\lim\limits_{n\to \infty}\mathbb{E}(\alpha(n))=2\)</EquationSource> </InlineEquation>. Then, we study the asymptotic behavior of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb{P} (\alpha(n)&gt;a(\log n)^2)\)</EquationSource> </InlineEquation> for any <i>a</i> &gt; 0 and use it to show that there exists a constant <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(C\in (\frac{1}{32}, \frac{1}{2}]\)</EquationSource> </InlineEquation> such that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\limsup\limits_{n\to \infty}\frac{\alpha(n)}{(\log n)^2}=C\)</EquationSource> </InlineEquation> almost surely.</p>

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The asymptotic behavior of rarely visited edges of the simple random walk

  • 期刊论文

  • 发表日期：

    2026年8月5日

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  • Zechun Hu,
  • Xue Peng,
  • Renming Song,
  • Yuan Tan

摘要

In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that are visited only once) of a simple symmetric random walk on ℤ. Let α(n) be the number of rarely visited edges up to time n. First, we evaluate \(\mathbb{E}(\alpha(n))\) , show that \(n\to \mathbb{E}(\alpha(n))\) is non-decreasing in n and that \(\lim\limits_{n\to \infty}\mathbb{E}(\alpha(n))=2\) . Then, we study the asymptotic behavior of \(\mathbb{P} (\alpha(n)>a(\log n)^2)\) for any a > 0 and use it to show that there exists a constant \(C\in (\frac{1}{32}, \frac{1}{2}]\) such that \(\limsup\limits_{n\to \infty}\frac{\alpha(n)}{(\log n)^2}=C\) almost surely.