9,157 research outputs found
Avalanche size distribution in a random walk model
We introduce a simple model for the size distribution of avalanches based on
the idea that the front of an avalanche can be described by a directed random
walk. The model captures some of the qualitative features of earthquakes,
avalanches and other self-organized critical phenomena in one dimension. We
find scaling laws relating the frequency, size and width of avalanches and an
exponent in the size distribution law.Comment: 16 pages Latex, macros included, 3 postscript figure
The spectral dimension of the branched polymers phase of two-dimensional quantum gravity
The metric of two-dimensional quantum gravity interacting with conformal
matter is believed to collapse to a branched polymer metric when the central
charge c>1. We show analytically that the spectral dimension of such a branched
polymer phase is four thirds. This is in good agreement with numerical
simulations for large c.Comment: 29 pages plain LateX2e, 7 eps figures included using eps
The phase diagram of an Ising model on a polymerized random surface
We construct a random surface model with a string susceptibility exponent one
quarter by taking an Ising model on a random surface and introducing an
additional degree of freedom which amounts to allowing certain outgrowths on
the surfaces. Fine tuning the Ising temperature and the weight factor for
outgrowths we find a triple point where the susceptibility exponent is one
quarter. At this point magnetized and nonmagnetized gravity phases meet a
branched polymer phase.Comment: Latex file, 10 pages, macros included. Two EPS figure
Cathodoluminescence read-out of the structural phase of gallium nanoparticles
We report on a method of phase identification of gallium nanoparticles via their cathodoluminescence when excited by a scanning electron beam. This feature can be used for high-density phase change memory element
A Numerical Analyst Looks at the "Cutoff Phenomenon" in Card Shuffling and Other Markov Chains
Diaconis and others have shown that certain Markov chains exhibit a "cutoff phenomenon" in which, after an initial period of seemingly little progress, convergence to the steady state occurs suddenly. Since Markov chains are just powers of matrices, how can such effects be explained in the language of applied linear algebra? We attempt to do this, focusing on two examples: random walk on a hypercube, which is essentially the same as the problem of Ehrenfest urns, and the celebrated case of riffle shuffling of a deck of cards. As is typical with transient phenomena in matrix processes, the reason for the cutoff is not readily apparent from an examination of eigenvalues or eigenvectors, but it is reflected strongly in pseudosprectra - provided they are measured in the 1-norm, not the 2-norm. We illustrate and explain the cutoff phenomenon with Matlab computations based in part on a new explicit formula for the entries of the "riffle shuffle matrix", and note that while the normwise cutoff may occur at one point, such as for the riffle shuffle, weak convergence may occur at an equally precise earlier point such as
On the spectral dimension of causal triangulations
We introduce an ensemble of infinite causal triangulations, called the
uniform infinite causal triangulation, and show that it is equivalent to an
ensemble of infinite trees, the uniform infinite planar tree. It is proved that
in both cases the Hausdorff dimension almost surely equals 2. The infinite
causal triangulations are shown to be almost surely recurrent or, equivalently,
their spectral dimension is almost surely less than or equal to 2. We also
establish that for certain reduced versions of the infinite causal
triangulations the spectral dimension equals 2 both for the ensemble average
and almost surely. The triangulation ensemble we consider is equivalent to the
causal dynamical triangulation model of two-dimensional quantum gravity and
therefore our results apply to that model.Comment: 22 pages, 6 figures; typos fixed, one extra figure, references
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