Spectral dimension of simple random walk on a long-range percolation cluster

V. H. Can, D. A. Croydon, T. Kumagai

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


Consider the long-range percolation model on the integer lattice Zd in which all nearest-neighbour edges are present and otherwise x and y are connected with probability qx,y:= 1 − exp(−|x − y|−s ), independently of the state of other edges. Throughout the regime where the model yields a locally-finite graph, (i.e. for s > d,) we determine the spectral dimension of the associated simple random walk, apart from at the exceptional value d = 1, s = 2, where the spectral dimension is discontinuous. Towards this end, we present various on-diagonal heat kernel bounds, a number of which are new. In particular, the lower bounds are derived through the application of a general technique that utilises the translation invariance of the model. We highlight that, applying this general technique, we are able to partially extend our main result beyond the nearest-neighbour setting, and establish lower heat kernel bounds over the range of parameters s ∈ (d, 2d). We further note that our approach is applicable to short-range models as well.

Original languageEnglish
Pages (from-to)1-37
Number of pages37
JournalElectronic Journal of Probability
Publication statusPublished - 2022


  • heat kernel estimates
  • long-range percolation
  • random walk
  • spectral dimension

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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