Rudas Anna (BME)
"Entropy and Hausdorff dimension in random growing trees"
We investigate the limiting behaviour of random tree growth in preferential
attachment models. The tree stems from a root, and we add vertices to the
system one-by-one at random, according to a rule which depends on the
degree distribution of the already existing tree. The so-called `weight
function`, in terms of which the rule of attachment is formulated, is such
that each vertex in the tree can have at most K children.
We define the concept of a certain random measure $\mu$ on the leaves of
the limiting tree, which captures a global property of the tree growth in a
natural way. We prove that the Hausdorff and the packing dimension of this
limiting measure is equal and constant with probability one. Moreover, the
local dimension of $\mu$ equals the Hausdorff dimension at $\mu$-almost
every point. We give an explicit formula for the dimension, given the rule