Random walk statistics on fractal structures. (English) Zbl 0587.60061
We consider some statistical properties of simple random walks on fractal structures viewed as networks of sites and bonds: range, renewal theory, mean first passage time, etc. Asymptotic behaviors are shown to be controlled by the fractal \((\bar d)\) and spectral \((\tilde d)\) dimensionalities of the considered structure.
A simple decimation procedure giving the value of \(\tilde d\) is outlined and illustrated in the case of the Sierpinski gaskets. Recent results for the trapping problem, the self-avoiding walk, and the true-self-avoiding walk are briefly reviewed. New numerical results for diffusion on percolation clusters are also presented.
A simple decimation procedure giving the value of \(\tilde d\) is outlined and illustrated in the case of the Sierpinski gaskets. Recent results for the trapping problem, the self-avoiding walk, and the true-self-avoiding walk are briefly reviewed. New numerical results for diffusion on percolation clusters are also presented.
MSC:
| 60G50 | Sums of independent random variables; random walks |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 60K05 | Renewal theory |
| 60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |
Keywords:
random walks on fractal structures; renewal theory; Asymptotic behaviors; diffusion on percolation clustersReferences:
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