4,596 research outputs found
Rigidity percolation on aperiodic lattices
We studied the rigidity percolation (RP) model for aperiodic (quasi-crystal)
lattices. The RP thresholds (for bond dilution) were obtained for several
aperiodic lattices via computer simulation using the "pebble game" algorithm.
It was found that the (two rhombi) Penrose lattice is always floppy in view of
the RP model. The same was found for the Ammann's octagonal tiling and the
Socolar's dodecagonal tiling. In order to impose the percolation transition we
used so c. "ferro" modification of these aperiodic tilings. We studied as well
the "pinwheel" tiling which has "infinitely-fold" orientational symmetry. The
obtained estimates for the modified Penrose, Ammann and Socolar lattices are
respectively: , , . The bond RP threshold of the pinwheel tiling was estimated to
. It was found that these results are very close to the
Maxwell (the mean-field like) approximation for them.Comment: 9 LaTeX pages, 3 PostScript figures included via epsf.st
Floppy modes and the free energy: Rigidity and connectivity percolation on Bethe Lattices
We show that negative of the number of floppy modes behaves as a free energy
for both connectivity and rigidity percolation, and we illustrate this result
using Bethe lattices. The rigidity transition on Bethe lattices is found to be
first order at a bond concentration close to that predicted by Maxwell
constraint counting. We calculate the probability of a bond being on the
infinite cluster and also on the overconstrained part of the infinite cluster,
and show how a specific heat can be defined as the second derivative of the
free energy. We demonstrate that the Bethe lattice solution is equivalent to
that of the random bond model, where points are joined randomly (with equal
probability at all length scales) to have a given coordination, and then
subsequently bonds are randomly removed.Comment: RevTeX 11 pages + epsfig embedded figures. Submitted to Phys. Rev.
The Betti poset in monomial resolutions
Let be a finite partially ordered set with unique minimal element
. We study the Betti poset of , created by deleting elements for which the open interval is acyclic. Using basic
simplicial topology, we demonstrate an isomorphism in homology between open
intervals of the form and corresponding open intervals
in the Betti poset. Our motivating application is that the Betti poset of a
monomial ideal's lcm-lattice encodes both its -graded Betti
numbers and the structure of its minimal free resolution. In the case of rigid
monomial ideals, we use the data of the Betti poset to explicitly construct the
minimal free resolution. Subsequently, we introduce the notion of rigid
deformation, a generalization of Bayer, Peeva, and Sturmfels' generic
deformation
Infinite-cluster geometry in central-force networks
We show that the infinite percolating cluster (with density P_inf) of
central-force networks is composed of: a fractal stress-bearing backbone (Pb)
and; rigid but unstressed ``dangling ends'' which occupy a finite
volume-fraction of the lattice (Pd). Near the rigidity threshold pc, there is
then a first-order transition in P_inf = Pd + Pb, while Pb is second-order with
exponent Beta'. A new mean field theory shows Beta'(mf)=1/2, while simulations
of triangular lattices give Beta'_tr = 0.255 +/- 0.03.Comment: 6 pages, 4 figures, uses epsfig. Accepted for publication in Physical
Review Letter
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