Michael Aizenman and Almut Burchard
H\"older Regularity and Dimension Bounds for Random Curves
(204K, Latex (2e) with EPSF figures)
ABSTRACT. Random systems of curves exhibiting fluctuating features on arbitrarily small
scales ($\delta$) are often encountered in critical models. For such systems it
is shown that scale-invariant bounds on the probabilities of crossing events
imply that typically all the realized curves admit H\"older continuous
parametrizations with a common exponent and a common random pre\-factor, which
in the scaling limit ($\delta\to 0$) remains stochastically bounded. The
regularity is used for the construction of scaling limits, formulated in terms
of probability measures on the space of closed sets of curves. Under the
hypotheses presented here the limiting measures are supported on sets of curves
which are H\"older continuous but not rectifiable, and have Hausdorff
dimensions strictly greater than one. The hypotheses are known to be satisfied
in certain two dimensional percolation models. Other potential applications are
also mentioned.