117,141 research outputs found
Combinatorial models of rigidity and renormalization
We first introduce the percolation problems associated with the graph
theoretical concepts of -sparsity, and make contact with the physical
concepts of ordinary and rigidity percolation. We then devise a renormalization
transformation for -percolation problems, and investigate its domain of
validity. In particular, we show that it allows an exact solution of
-percolation problems on hierarchical graphs, for . We
introduce and solve by renormalization such a model, which has the interesting
feature of showing both ordinary percolation and rigidity percolation phase
transitions, depending on the values of the parameters.Comment: 22 pages, 6 figure
Conducting-angle-based percolation in the XY model
We define a percolation problem on the basis of spin configurations of the
two dimensional XY model. Neighboring spins belong to the same percolation
cluster if their orientations differ less than a certain threshold called the
conducting angle. The percolation properties of this model are studied by means
of Monte Carlo simulations and a finite-size scaling analysis. Our simulations
show the existence of percolation transitions when the conducting angle is
varied, and we determine the transition point for several values of the XY
coupling. It appears that the critical behavior of this percolation model can
be well described by the standard percolation theory. The critical exponents of
the percolation transitions, as determined by finite-size scaling, agree with
the universality class of the two-dimensional percolation model on a uniform
substrate. This holds over the whole temperature range, even in the
low-temperature phase where the XY substrate is critical in the sense that it
displays algebraic decay of correlations.Comment: 16 pages, 14 figure
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