8,284 research outputs found
Pointwise Convergence in Probability of General Smoothing Splines
Establishing the convergence of splines can be cast as a variational problem
which is amenable to a -convergence approach. We consider the case in
which the regularization coefficient scales with the number of observations,
, as . Using standard theorems from the
-convergence literature, we prove that the general spline model is
consistent in that estimators converge in a sense slightly weaker than weak
convergence in probability for . Without further assumptions
we show this rate is sharp. This differs from rates for strong convergence
using Hilbert scales where one can often choose
Hierarchical models of rigidity percolation
We introduce models of generic rigidity percolation in two dimensions on
hierarchical networks, and solve them exactly by means of a renormalization
transformation. We then study how the possibility for the network to self
organize in order to avoid stressed bonds may change the phase diagram. In
contrast to what happens on random graphs and in some recent numerical studies
at zero temperature, we do not find a true intermediate phase separating the
usual rigid and floppy ones.Comment: 20 pages, 8 figures. Figures improved, references added, small
modifications. Accepted in Phys. Rev.
Self-organization with equilibration: a model for the intermediate phase in rigidity percolation
Recent experimental results for covalent glasses suggest the existence of an
intermediate phase attributed to the self-organization of the glass network
resulting from the tendency to minimize its internal stress. However, the exact
nature of this experimentally measured phase remains unclear. We modify a
previously proposed model of self-organization by generating a uniform sampling
of stress-free networks. In our model, studied on a diluted triangular lattice,
an unusual intermediate phase appears, in which both rigid and floppy networks
have a chance to occur, a result also observed in a related model on a Bethe
lattice by Barre et al. [Phys. Rev. Lett. 94, 208701 (2005)]. Our results for
the bond-configurational entropy of self-organized networks, which turns out to
be only about 2% lower than that of random networks, suggest that a
self-organized intermediate phase could be common in systems near the rigidity
percolation threshold.Comment: 9 pages, 6 figure
Self-organized criticality in the intermediate phase of rigidity percolation
Experimental results for covalent glasses have highlighted the existence of a
new self-organized phase due to the tendency of glass networks to minimize
internal stress. Recently, we have shown that an equilibrated self-organized
two-dimensional lattice-based model also possesses an intermediate phase in
which a percolating rigid cluster exists with a probability between zero and
one, depending on the average coordination of the network. In this paper, we
study the properties of this intermediate phase in more detail. We find that
microscopic perturbations, such as the addition or removal of a single bond,
can affect the rigidity of macroscopic regions of the network, in particular,
creating or destroying percolation. This, together with a power-law
distribution of rigid cluster sizes, suggests that the system is maintained in
a critical state on the rigid/floppy boundary throughout the intermediate
phase, a behavior similar to self-organized criticality, but, remarkably, in a
thermodynamically equilibrated state. The distinction between percolating and
non-percolating networks appears physically meaningless, even though the
percolating cluster, when it exists, takes up a finite fraction of the network.
We point out both similarities and differences between the intermediate phase
and the critical point of ordinary percolation models without
self-organization. Our results are consistent with an interpretation of recent
experiments on the pressure dependence of Raman frequencies in chalcogenide
glasses in terms of network homogeneity.Comment: 20 pages, 18 figure
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