35,576 research outputs found
Entropy and Hausdorff Dimension in Random Growing Trees
We investigate the limiting behavior 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 of attachment
The Einstein Relation on Metric Measure Spaces
This note is based on F. Burghart's master thesis at Stuttgart university
from July 2018, supervised by Prof. Freiberg.
We review the Einstein relation, which connects the Hausdorff, local walk and
spectral dimensions on a space, in the abstract setting of a metric measure
space equipped with a suitable operator. This requires some twists compared to
the usual definitions from fractal geometry. The main result establishes the
invariance of the three involved notions of fractal dimension under
bi-Lipschitz continuous isomorphisms between mm-spaces and explains, more
generally, how the transport of the analytic and stochastic structure behind
the Einstein relation works. While any homeomorphism suffices for this
transport of structure, non-Lipschitz maps distort the Hausdorff and the local
walk dimension in different ways. To illustrate this, we take a look at
H\"older regular transformations and how they influence the local walk
dimension and prove some partial results concerning the Einstein relation on
graphs of fractional Brownian motions. We conclude by giving a short list of
further questions that may help building a general theory of the Einstein
relation.Comment: 28 pages, 3 figure
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