186 research outputs found
Generalizations of the Kolmogorov-Barzdin embedding estimates
We consider several ways to measure the `geometric complexity' of an
embedding from a simplicial complex into Euclidean space. One of these is a
version of `thickness', based on a paper of Kolmogorov and Barzdin. We prove
inequalities relating the thickness and the number of simplices in the
simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved
for graphs. We also consider the distortion of knots. We give an alternate
proof of a theorem of Pardon that there are isotopy classes of knots requiring
arbitrarily large distortion. This proof is based on the expander-like
properties of arithmetic hyperbolic manifolds.Comment: 45 page
Isoperimetric inequalities and mixing time for a random walk on a random point process
We consider the random walk on a simple point process on ,
, whose jump rates decay exponentially in the -power of jump
length. The case corresponds to the phonon-induced variable-range
hopping in disordered solids in the regime of strong Anderson localization.
Under mild assumptions on the point process, we show, for ,
that the random walk confined to a cubic box of side has a.s. Cheeger
constant of order at least and mixing time of order . For the
Poisson point process, we prove that at , there is a transition from
diffusive to subdiffusive behavior of the mixing time.Comment: Published in at http://dx.doi.org/10.1214/07-AAP442 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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