Scholarlink

<p>This note provides a sufficient condition for linear lower bounds on chemical distances (compared to the Euclidean distance) in general spatial random graphs. The condition is based on the scarceness of long edges in the graph and weak correlations at large distances and is valid for all translation invariant and locally finite graphs that fulfil these conditions. We apply the result to various examples, thereby confirming a conjecture on graph distances in the heavy-tailed Boolean model posed by Hirsch (Braz J Probab Stat 31(1):111–143, 2017). The proof is based on a renormalisation scheme introduced by Berger (<a href="http://arxiv.org/abs/math.PR/0409021">arXiv:math/0409021</a> [math.PR], 2004).</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

All Spatial Random Graphs with Weak Long-Range Effects have Chemical Distance Comparable to Euclidean Distance

  • 期刊论文

  • 发表日期：

    2025年12月1日

查看全文
  • Lukas Lüchtrath

摘要

This note provides a sufficient condition for linear lower bounds on chemical distances (compared to the Euclidean distance) in general spatial random graphs. The condition is based on the scarceness of long edges in the graph and weak correlations at large distances and is valid for all translation invariant and locally finite graphs that fulfil these conditions. We apply the result to various examples, thereby confirming a conjecture on graph distances in the heavy-tailed Boolean model posed by Hirsch (Braz J Probab Stat 31(1):111–143, 2017). The proof is based on a renormalisation scheme introduced by Berger (arXiv:math/0409021 [math.PR], 2004).