5,191 research outputs found
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
Convergence to the Tracy-Widom distribution for longest paths in a directed random graph
We consider a directed graph on the 2-dimensional integer lattice, placing a
directed edge from vertex to , whenever ,
, with probability , independently for each such pair of
vertices. Let denote the maximum length of all paths contained in an
rectangle. We show that there is a positive exponent , such
that, if , as , then a properly centered/rescaled
version of converges weakly to the Tracy-Widom distribution. A
generalization to graphs with non-constant probabilities is also discussed.Comment: 20 pages, 2 figure
The Tensor Track, III
We provide an informal up-to-date review of the tensor track approach to
quantum gravity. In a long introduction we describe in simple terms the
motivations for this approach. Then the many recent advances are summarized,
with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion,
Osterwalder-Schrader positivity...) which, while important for the tensor track
program, are not detailed in the usual quantum gravity literature. We list open
questions in the conclusion and provide a rather extended bibliography.Comment: 53 pages, 6 figure
Nonlinear Diffusion Through Large Complex Networks Containing Regular Subgraphs
Transport through generalized trees is considered. Trees contain the simple
nodes and supernodes, either well-structured regular subgraphs or those with
many triangles. We observe a superdiffusion for the highly connected nodes
while it is Brownian for the rest of the nodes. Transport within a supernode is
affected by the finite size effects vanishing as For the even
dimensions of space, , the finite size effects break down the
perturbation theory at small scales and can be regularized by using the
heat-kernel expansion.Comment: 21 pages, 2 figures include
Attributing a probability to the shape of a probability density
We discuss properties of two methods for ascribing probabilities to the shape
of a probability distribution. One is based on the idea of counting the number
of modes of a bootstrap version of a standard kernel density estimator. We
argue that the simplest form of that method suffers from the same difficulties
that inhibit level accuracy of Silverman's bandwidth-based test for modality:
the conditional distribution of the bootstrap form of a density estimator is
not a good approximation to the actual distribution of the estimator. This
difficulty is less pronounced if the density estimator is oversmoothed, but the
problem of selecting the extent of oversmoothing is inherently difficult. It is
shown that the optimal bandwidth, in the sense of producing optimally high
sensitivity, depends on the widths of putative bumps in the unknown density and
is exactly as difficult to determine as those bumps are to detect. We also
develop a second approach to ascribing a probability to shape, using Muller and
Sawitzki's notion of excess mass. In contrast to the context just discussed, it
is shown that the bootstrap distribution of empirical excess mass is a
relatively good approximation to its true distribution. This leads to empirical
approximations to the likelihoods of different levels of ``modal sharpness,''
or ``delineation,'' of modes of a density. The technique is illustrated
numerically.Comment: Published at http://dx.doi.org/10.1214/009053604000000607 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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