Spectral asymptotics of the Laplacian on supercritical bond-percolation graphs. (English) Zbl 1127.60090
Summary: We investigate Laplacians on supercritical bond-percolation graphs with different boundary conditions at cluster borders. The integrated density of states of the Dirichlet Laplacian is found to exhibit a Lifshits tail at the lower spectral edge, while that of the Neumann Laplacian shows a van Hove asymptotics, which results from the percolating cluster. At the upper spectral edge, the behaviour is reversed.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 47B80 | Random linear operators |
| 82B43 | Percolation |
| 34B45 | Boundary value problems on graphs and networks for ordinary differential equations |
| 05C80 | Random graphs (graph-theoretic aspects) |
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