1,464 research outputs found
An Upper Bound for the Number of Planar Lattice Triangulations
We prove an exponential upper bound for the number of all maximal
triangulations of the grid: In particular,
this improves a result of S. Yu. Orevkov (1999).Comment: 4 pages, 3 figure
Moduli of Tropical Plane Curves
We study the moduli space of metric graphs that arise from tropical plane
curves. There are far fewer such graphs than tropicalizations of classical
plane curves. For fixed genus , our moduli space is a stacky fan whose cones
are indexed by regular unimodular triangulations of Newton polygons with
interior lattice points. It has dimension unless or .
We compute these spaces explicitly for .Comment: 31 pages, 25 figure
Topological Phases: An Expedition off Lattice
Motivated by the goal to give the simplest possible microscopic foundation
for a broad class of topological phases, we study quantum mechanical lattice
models where the topology of the lattice is one of the dynamical variables.
However, a fluctuating geometry can remove the separation between the system
size and the range of local interactions, which is important for topological
protection and ultimately the stability of a topological phase. In particular,
it can open the door to a pathology, which has been studied in the context of
quantum gravity and goes by the name of `baby universe', Here we discuss three
distinct approaches to suppressing these pathological fluctuations. We
complement this discussion by applying Cheeger's theory relating the geometry
of manifolds to their vibrational modes to study the spectra of Hamiltonians.
In particular, we present a detailed study of the statistical properties of
loop gas and string net models on fluctuating lattices, both analytically and
numerically.Comment: 38 pages, 22 figure
Random lattice triangulations: Structure and algorithms
The paper concerns lattice triangulations, that is, triangulations of the
integer points in a polygon in whose vertices are also integer
points. Lattice triangulations have been studied extensively both as geometric
objects in their own right and by virtue of applications in algebraic geometry.
Our focus is on random triangulations in which a triangulation has
weight , where is a positive real parameter, and
is the total length of the edges in . Empirically, this
model exhibits a "phase transition" at (corresponding to the
uniform distribution): for distant edges behave essentially
independently, while for very large regions of aligned edges
appear. We substantiate this picture as follows. For sufficiently
small, we show that correlations between edges decay exponentially with
distance (suitably defined), and also that the Glauber dynamics (a local Markov
chain based on flipping edges) is rapidly mixing (in time polynomial in the
number of edges in the triangulation). This dynamics has been proposed by
several authors as an algorithm for generating random triangulations. By
contrast, for we show that the mixing time is exponential. These
are apparently the first rigorous quantitative results on the structure and
dynamics of random lattice triangulations.Comment: Published at http://dx.doi.org/10.1214/14-AAP1033 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs
We study Markov chains for -orientations of plane graphs, these are
orientations where the outdegree of each vertex is prescribed by the value of a
given function . The set of -orientations of a plane graph has
a natural distributive lattice structure. The moves of the up-down Markov chain
on this distributive lattice corresponds to reversals of directed facial cycles
in the -orientation. We have a positive and several negative results
regarding the mixing time of such Markov chains.
A 2-orientation of a plane quadrangulation is an orientation where every
inner vertex has outdegree 2. We show that there is a class of plane
quadrangulations such that the up-down Markov chain on the 2-orientations of
these quadrangulations is slowly mixing. On the other hand the chain is rapidly
mixing on 2-orientations of quadrangulations with maximum degree at most 4.
Regarding examples for slow mixing we also revisit the case of 3-orientations
of triangulations which has been studied before by Miracle et al.. Our examples
for slow mixing are simpler and have a smaller maximum degree, Finally we
present the first example of a function and a class of plane
triangulations of constant maximum degree such that the up-down Markov chain on
the -orientations of these graphs is slowly mixing
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