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 $X$, the delta invariant $\delta(X)$ defined by Fujita and Odaka coincides with the greatest Ricci lower curvature $R(X)$. 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 $3$-folds by computing Demazure's roots for each. All the results are listed in Table $1$ with the value of $\delta(X)$ and $R(X)$.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