1,297 research outputs found
Elliptic dimers on minimal graphs and genus 1 Harnack curves
This paper provides a comprehensive study of the dimer model on infinite
minimal graphs with Fock's elliptic weights [arXiv:1503.00289]. Specific
instances of such models were studied in [arXiv:052711, arXiv:1612.09082,
arXiv1801.00207]; we now handle the general genus 1 case, thus proving a
non-trivial extension of the genus 0 results of [arXiv:math-ph/0202018,
arXiv:math/0311062] on isoradial critical models. We give an explicit local
expression for a two-parameter family of inverses of the Kasteleyn operator
with no periodicity assumption on the underlying graph. When the minimal graph
satisfies a natural condition, we construct a family of dimer Gibbs measures
from these inverses, and describe the phase diagram of the model by deriving
asymptotics of correlations in each phase. In the -periodic case,
this gives an alternative description of the full set of ergodic Gibbs measures
constructed in [arXiv:math-ph/0311005] by Kenyon, Okounkov and Sheffield. We
also establish a correspondence between elliptic dimer models on periodic
minimal graphs and Harnack curves of genus 1. Finally, we show that a bipartite
dimer model is invariant under the shrinking/expanding of 2-valent vertices and
spider moves if and only if the associated Kasteleyn coefficients are
antisymmetric and satisfy Fay's trisecant identity.Comment: 71 pages, 16 figure
The combinatorics of plane curve singularities. How Newton polygons blossom into lotuses
This survey may be seen as an introduction to the use of toric and tropical
geometry in the analysis of plane curve singularities, which are germs
of complex analytic curves contained in a smooth complex analytic surface .
The embedded topological type of such a pair is usually defined to be
that of the oriented link obtained by intersecting with a sufficiently
small oriented Euclidean sphere centered at the point , defined once a
system of local coordinates was chosen on the germ . If one
works more generally over an arbitrary algebraically closed field of
characteristic zero, one speaks instead of the combinatorial type of .
One may define it by looking either at the Newton-Puiseux series associated to
relative to a generic local coordinate system , or at the set of
infinitely near points which have to be blown up in order to get the minimal
embedded resolution of the germ or, thirdly, at the preimage of this
germ by the resolution. Each point of view leads to a different encoding of the
combinatorial type by a decorated tree: an Eggers-Wall tree, an Enriques
diagram, or a weighted dual graph. The three trees contain the same
information, which in the complex setting is equivalent to the knowledge of the
embedded topological type. There are known algorithms for transforming one tree
into another. In this paper we explain how a special type of two-dimensional
simplicial complex called a lotus allows to think geometrically about the
relations between the three types of trees. Namely, all of them embed in a
natural lotus, their numerical decorations appearing as invariants of it. This
lotus is constructed from the finite set of Newton polygons created during any
process of resolution of by successive toric modifications.Comment: 104 pages, 58 figures. Compared to the previous version, section 2 is
new. The historical information, contained before in subsection 6.2, is
distributed now throughout the paper in the subsections called "Historical
comments''. More details are also added at various places of the paper. To
appear in the Handbook of Geometry and Topology of Singularities I, Springer,
202
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
インターネット問題のモデル化法と効率的算法の研究
平成16-17度科学研究費補助金(基盤研究(C))研究成果報告書 課題番号:16500010 研究代表者:伊藤大雄 (京都大学大学院情報学研究科
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