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Anderson model on the fractal lattice

The fractal lattice Γ is a skeleton, i.e. the discrete approximations of the nested fractal, say, the infinite Sierpinski gasket.

The dimension d (Γ) of such lattice (Hausdorff’s dimension or spectral dimension) can be different but in all cases it has values on the interval (1,2).

The Anderson Hamiltonian has the standard definition : Δ is the lattice Laplacian, (X_i) are i.i.d. random variables and σ is a coupling constant.

We will discuss the following recent results :

a) The spectrum of H has no a.c. component P-a.s. for any non-degenerated potential.

b) If the random variables are heavy tailed, then the spectrum of H is p.p. P-a.s. This fact must be true for the arbitrary random variable, but it has not been proved.

Several results for the deterministic potential will be also presented.

CMAP UMR 7641 École Polytechnique CNRS, Route de Saclay, 91128 Palaiseau Cedex France, Tél: +33 1 69 33 46 23 Fax: +33 1 69 33 46 46