THE WIRED MINIMAL SPANNING FOREST ON THE POISSON-WEIGHTED INFINITE TREE

Asaf Nachmias, Pengfei Tang

Research output: Contribution to journalArticlepeer-review

Abstract

We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let M be the tree containing the root in the WMSF on the PWIT and (Yn)n≥0 be a simple random walk on M starting from the root. We show that almost surely M has P[Y2n = Y0] = n-3/4+o(1) and dist(Y0, Yn) = n1/4+o(1) with high probability. That is, the spectral dimension of M is 3/2 and its typical displacement exponent is 1/4, almost surely. These confirm Addario–Berry’s predictions (Addario-Berry (2013)).

Original languageEnglish
Pages (from-to)2415-2446
Number of pages32
JournalAnnals of Applied Probability
Volume34
Issue number2
DOIs
StatePublished - Apr 2024

Keywords

  • Minimal spanning tree
  • Poisson-weighted infinite tree
  • local limit
  • spectral dimension
  • wired minimal spanning forest

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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