TY - JOUR

T1 - Spectral dimension of simple random walk on a long-range percolation cluster

AU - Can, V. H.

AU - Croydon, D. A.

AU - Kumagai, T.

N1 - Funding Information:
*This research was partially supported by JSPS KAKENHI, grant numbers 17F17319, 17H01093 and 19K03540, by the Singapore Ministry of Education Academic Research Fund Tier 2 grant number MOE2018-T2-2-076, by the Vietnam Academy of Science and Technology grant number CTTH00.02/22-23, and by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. †Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam. ‡Department of Statistics and Data Science, National University of Singapore, 6 Science Drive 2 Singapore 117546. E-mail: cvhao89@gmail.com §Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. E-mail: croydon@kurims.kyoto-u.ac.jp ¶Department of Mathematics, Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan. E-mail: t-kumagai@waseda.jp
Publisher Copyright:
© 2022, Institute of Mathematical Statistics. All rights reserved.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - heat kernel estimates

KW - long-range percolation

KW - random walk

KW - spectral dimension

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U2 - 10.1214/22-EJP783

DO - 10.1214/22-EJP783

M3 - Article

AN - SCOPUS:85134832462

VL - 27

SP - 1

EP - 37

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

ER -