Journal
ELECTRONIC JOURNAL OF PROBABILITY
Volume 27, Issue -, Pages -Publisher
INST MATHEMATICAL STATISTICS-IMS
DOI: 10.1214/22-EJP783
Keywords
long-range percolation; random walk; heat kernel estimates; spectral dimension
Categories
Funding
- JSPS KAKENHI [17F17319, 17H01093, 19K03540]
- Singapore Ministry of Education Academic Research Fund Tier 2 grant [MOE2018T2-2-076]
- Vietnam Academy of Science and Technology [CTTH00.02/22-23]
- Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University
- Grants-in-Aid for Scientific Research [19K03540, 17F17319, 17H01093] Funding Source: KAKEN
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This article considers the long-range percolation model on the integer lattice and determines the spectral dimensions of the associated simple random walk. By applying a general technique, new on-diagonal heat kernel bounds are derived and the main result is partially extended beyond the nearest-neighbour setting and within a range of parameters.
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.
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