On the resistance of the Sierpiński carpet. (English) Zbl 0729.60108

Barlow, M. T.; Bass, R. F.

Proc. R. Soc. Lond., Ser. A 431, No. 1882, 345-360 (1990).

Summary: Let \(F_ n\) be the n-th stage in the construction of the Sierpiński carpet. Let \(R_ n\) be the electrical resistance of \(F_ n\) when the left and right sides are each short-circuited, and a voltage is applied between them. We prove that there exists a constant \(\rho\) such that \(\rho^ n/4\leq R_ n\leq 4\rho^ n\). The motivation for this result came from the problem of establishing (a) the existence and (b) the value of the ‘spectral dimension’ of the Sierpiński carpet. In this and a subsequent paper, we settle (a) and give bounds for (b).

MSC:

60K35

Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

construction of the Sierpiński carpet; electrical resistance; spectral dimension