On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets

Abstract

We consider natural Laplace operators on random recursive affine nested fractals based on the Sierpinski gasket and prove an analogue of Weyl’s classical result on their eigenvalue asymptotics. The eigenvalue counting function N(λ) is shown to be of order λ ds/2 as λ→∞ where we can explicitly compute the spectral dimension d s . Moreover the limit N(λ) λ −ds/2 will typically exist and can be expressed as a deterministic constant multiplied by a random variable. This random variable is a power of the limiting random variable in a suitable general branching process and has an interpretation as the volume of the fractal.