1,127 research outputs found
Asymptotic enumeration and limit laws for graphs of fixed genus
It is shown that the number of labelled graphs with n vertices that can be
embedded in the orientable surface S_g of genus g grows asymptotically like
where , and is the exponential growth rate of planar graphs. This generalizes the
result for the planar case g=0, obtained by Gimenez and Noy.
An analogous result for non-orientable surfaces is obtained. In addition, it
is proved that several parameters of interest behave asymptotically as in the
planar case. It follows, in particular, that a random graph embeddable in S_g
has a unique 2-connected component of linear size with high probability
Random geometric complexes
We study the expected topological properties of Cech and Vietoris-Rips
complexes built on i.i.d. random points in R^d. We find higher dimensional
analogues of known results for connectivity and component counts for random
geometric graphs. However, higher homology H_k is not monotone when k > 0. In
particular for every k > 0 we exhibit two thresholds, one where homology passes
from vanishing to nonvanishing, and another where it passes back to vanishing.
We give asymptotic formulas for the expectation of the Betti numbers in the
sparser regimes, and bounds in the denser regimes. The main technical
contribution of the article is in the application of discrete Morse theory in
geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete &
Computational Geometr
Non-contractible loops in the dense O(n) loop model on the cylinder
A lattice model of critical dense polymers is considered for the
finite cylinder geometry. Due to the presence of non-contractible loops with a
fixed fugacity , the model is a generalization of the critical dense
polymers solved by Pearce, Rasmussen and Villani. We found the free energy for
any height and circumference of the cylinder. The density of
non-contractible loops is found for and large . The
results are compared with those obtained for the anisotropic quantum chain with
twisted boundary conditions. Using the latter method we obtained for any
model and an arbitrary fugacity.Comment: arXiv admin note: text overlap with arXiv:0810.223
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