1,471 research outputs found
Asymptotic aspects of Schreier graphs and Hanoi Towers groups
We present relations between growth, growth of diameters and the rate of
vanishing of the spectral gap in Schreier graphs of automaton groups. In
particular, we introduce a series of examples, called Hanoi Towers groups since
they model the well known Hanoi Towers Problem, that illustrate some of the
possible types of behavior.Comment: 5 page
Diameters, distortion and eigenvalues
We study the relation between the diameter, the first positive eigenvalue of
the discrete -Laplacian and the -distortion of a finite graph. We
prove an inequality relating these three quantities and apply it to families of
Cayley and Schreier graphs. We also show that the -distortion of Pascal
graphs, approximating the Sierpinski gasket, is bounded, which allows to obtain
estimates for the convergence to zero of the spectral gap as an application of
the main result.Comment: Final version, to appear in the European Journal of Combinatoric
Optimal Vertex Cover for the Small-World Hanoi Networks
The vertex-cover problem on the Hanoi networks HN3 and HN5 is analyzed with
an exact renormalization group and parallel-tempering Monte Carlo simulations.
The grand canonical partition function of the equivalent hard-core repulsive
lattice-gas problem is recast first as an Ising-like canonical partition
function, which allows for a closed set of renormalization group equations. The
flow of these equations is analyzed for the limit of infinite chemical
potential, at which the vertex-cover problem is attained. The relevant fixed
point and its neighborhood are analyzed, and non-trivial results are obtained
both, for the coverage as well as for the ground state entropy density, which
indicates the complex structure of the solution space. Using special
hierarchy-dependent operators in the renormalization group and Monte-Carlo
simulations, structural details of optimal configurations are revealed. These
studies indicate that the optimal coverages (or packings) are not related by a
simple symmetry. Using a clustering analysis of the solutions obtained in the
Monte Carlo simulations, a complex solution space structure is revealed for
each system size. Nevertheless, in the thermodynamic limit, the solution
landscape is dominated by one huge set of very similar solutions.Comment: RevTex, 24 pages; many corrections in text and figures; final
version; for related information, see
http://www.physics.emory.edu/faculty/boettcher
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