Sigurdur Orn Stefansson, Stefan Zohren
Using generating functions techniques we develop a relation between the
Hausdorff and spectral dimension of trees with a unique infinite spine.
Furthermore, it is shown that if the outgrowths along the spine are independent
and identically distributed, then both the Hausdorff and spectral dimension can
easily be determined from the probability generating function of the random
variable describing the size of the outgrowths at a given vertex, provided that
the probability of the height of the outgrowths exceeding n falls off as the
inverse of n. We apply this new method to both critical non-generic trees and
the attachment and grafting model, which is a special case of the vertex
splitting model, resulting in a simplified proof for the values of the
Hausdorff and spectral dimension for the former and novel results for the
latter.
View original:
http://arxiv.org/abs/1202.5388
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