Random Walks and Diffusions on Fractals
We investigate the asymptotic motion of a random walker, which at time n is at X(n), on certain “fractal lattices”. For the “Sierpinski lattice” in dimension d we show that as ℓ → ∞, the process Yℓ(t) ≡ X([(d+3)ℓ t])/2ℓ converges in distribution (so that, in particular, |X(n)| ~ nγ, where γ = (ln 2)/ln(d + 3)) to a diffusion on the Sierpinski gasket, a Cantor set of Lebesgue measure zero. The analysis is based on a simple “renormalization group” type argument, involving self-similarity and “decimation invariance”.
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