13,596 research outputs found
Restoration of Isotropy in the Ising Model on the Sierpinski Gasket
We study the ferromagnetic Ising model on the Sierpinski gasket (SG), where
the spin-spin interactions depends on the direction. Using the renormalization
group method, we show that the ratios of the correlation lengths restore the
isotropy of the lattice as the temperature approaches zero. This restoration is
either partial or perfect, depending on the interactions. In case of partial
restoration, we also evaluate the leading-order singular behavior of the
correlation lengths.Comment: 17 pages, 10 figures. References added in v.2 and 3. Small
improvements in v.4, 5. This version will appear in Prog. Theor. Phy
The delta invariant and the various GIT-stability notions of toric Fano varieties
In this article, we give combinatorial proofs of the following two theorems:
(1) If a Gorenstein toric Fano variety is asymptotically Chow semistable then
it is Ding polystable. (2) For a smooth toric Fano manifold , the delta
invariant defined by Fujita and Odaka coincides with the greatest
Ricci lower curvature . In the proof, neither toric test configuration
nor toric Minimal Model Program (MMP) we use. We also verify the reductivity of
automorphism group of toric Fano -folds by computing Demazure's roots for
each. All the results are listed in Table with the value of and
.Comment: 19 pages, 2 figures, 1 table. Fixed an error in Proposition 4.3.
Section 5 in the previous version removed. The appendix added. The title
changed from the first versio
Facets of secondary polytopes and Chow stability of toric varieties
Chow stability is one notion of Mumford's Geometric Invariant Theory for
studying the moduli space of polarized varieties. Kapranov, Sturmfels and
Zelevinsky detected that Chow stability of polarized toric varieties is
determined by its inherent {\it secondary polytope}, which is a polytope whose
vertices correspond to regular triangulations of the associated polytope
\cite{KSZ}. In this paper, we give a purely convex-geometrical proof that the
Chow form of a projective toric variety is -semistable if and only if it is
-polystable with respect to the standard complex torus action . This
\emph{essentially} means that Chow semistability is equivalent to Chow
polystability for any (not-necessaliry-smooth) projective toric varieties.Comment: 13 pages, to appear in Osaka Journal of Mathematics Vol. 53, No. 3,
(2016
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