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<p>This paper focuses on last passage percolation on the complete graph <i>G</i><sub><i>n</i></sub> = ([<i>n</i>], <i>E</i><sub><i>n</i></sub>). Let {<i>X</i><sub><i>e</i></sub>: <i>e</i> ∈ <i>E</i><sub><i>n</i></sub>} be the non-negative passage times of edges and be i.i.d with tail probability <i>H</i>(·), let <i>W</i><sub><i>n</i></sub> be the largest passage time among all self-avoiding paths between vertices 1 and <i>n</i>. In this paper, when <i>H</i>(·) follows a Weibull or Pareto distribution, we have established the strong law of large numbers (SLLN) for <i>W</i><sub><i>n</i></sub>. Furthermore, the <i>suplinear</i> increasing function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb E}(W_{n})\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>W</mi> <mrow> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> is precisely characterised. The main methods we use are the greedy algorithm, variance estimation of <i>W</i><sub><i>n</i></sub>, and the Borel-Cantelli lemma.</p>

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A Strong Law of Large Numbers for Last Passage Percolation on the Complete Graph

  • 期刊论文

  • 发表日期：

    2026年7月7日

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  • Ya-meng Guo,
  • Feng Wang,
  • Xian-yuan Wu

摘要

This paper focuses on last passage percolation on the complete graph Gn = ([n], En). Let {Xe: e ∈ En} be the non-negative passage times of edges and be i.i.d with tail probability H(·), let Wn be the largest passage time among all self-avoiding paths between vertices 1 and n. In this paper, when H(·) follows a Weibull or Pareto distribution, we have established the strong law of large numbers (SLLN) for Wn. Furthermore, the suplinear increasing function \({\mathbb E}(W_{n})\) E ( W n ) is precisely characterised. The main methods we use are the greedy algorithm, variance estimation of Wn, and the Borel-Cantelli lemma.