scholarly journals Statistical Regularity of Apollonian Gaskets

Author(s):  
Xin Zhang

Abstract Apollonian gaskets are formed by repeatedly filling the gaps between three mutually tangent circles with further tangent circles. In this paper we give explicit formulas for the limiting pair correlation and the limiting nearest neighbor spacing of centers of circles from a fixed Apollonian gasket. These are corollaries of the convergence of moments that we prove. The input from ergodic theory is an extension of Mohammadi–Oh’s theorem on the equidisribution of expanding horospheres in the frame fundles of infinite volume hyperbolic spaces.

Author(s):  
Giovanni Antinucci ◽  
Alessandro Giuliani ◽  
Rafael L. Greenblatt

AbstractIn this paper, meant as a companion to Antinucci et al. (Energy correlations of non-integrable Ising models: the scaling limit in the cylinder, 2020. arXiv: 1701.05356), we consider a class of non-integrable 2D Ising models in cylindrical domains, and we discuss two key aspects of the multiscale construction of their scaling limit. In particular, we provide a detailed derivation of the Grassmann representation of the model, including a self-contained presentation of the exact solution of the nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic estimates of the fermionic Green’s function in the cylinder, required for the multiscale analysis of the model. We also review the multiscale construction of the effective potentials in the infinite volume limit, in a form suitable for the generalization to finite cylinders. Compared to previous works, we introduce a few important simplifications in the localization procedure and in the iterative bounds on the kernels of the effective potentials, which are crucial for the adaptation of the construction to domains with boundaries.


Author(s):  
Tuncer Kaya

In this work, an exact renormalization group treatment of honeycomb lattice leading to an exact relation between the coupling strengths of the honeycomb and the triangular lattices is presented. Using the honeycomb and the triangular duality relation, the critical coupling values of honeycomb and triangular lattices are calculated exactly by the simultaneous solution of the renormalized relation and the duality relation, without using the so-called star-triangular transformation. Apparently, the obtained coupling relation is unique. It not only takes place the role of the star triangular relation, but it is also the only exact relation obtained from renormalization group theory other than the 1D Ising chain. An exact pair correlation function expression relating the nearest neighbors and the next nearest neighbor correlation functions are also obtained for the honeycomb lattice. Utilizing this correlation relation, an exact expression of the correlation length of the honeycomb lattice is calculated analytically for the coupling constant values less than the critical value in the realm of the scaling theory. The critical exponents [Formula: see text] and [Formula: see text] are also calculated as [Formula: see text] and [Formula: see text].


2014 ◽  
Vol 70 (a1) ◽  
pp. C861-C861
Author(s):  
Hiroshi Arima ◽  
Toru Kawamata ◽  
Yoshihiko Yokoyama ◽  
Kazumasa Sugiyama ◽  
Toshiya Otomo

Transition metal-metalloid amorphous systems such as Fe-B and Ni-B are usually applied for the soft magnetic metallic glassy alloys and are counted as one of prominent categories in the field of amorphous alloy technology. Since glass forming ability of these systems correlates closely with the atomic level structure depending on chemical species of metal and metalloid, the local structure analysis for these glassy alloys is strongly required. In order to obtain the reliable structural model for these metal (Fe, Ni)-metalloid (B) amorphous samples, we determined the partial structural functions by combinational use of neutron diffraction (ND) and anomalous X-ray scattering (AXS). The amorphous ribbon samples were produced by the single-roller melt-spinning technique. The AXS measurements at Fe and Ni absorption edges were carried out at BL-7C of Photon Factory, KEK. The ND experiment was performed by using the time-of-flight technique and high intensity total diffractometer, NOVA at MLF, J-PARC. The figure shows the g(r)s for Fe80B20, Ni81B19, and Ni60B40calculated by the interference functions obtained by ND measurements. The dashed lines in the figure indicate the interatomic distances estimated from Goldschmidt atomic radii (Fe: 1.28 Å, Ni: 1.25 Å, B: 0.97 Å). At the nearest neighbor region up to about 3 Å, the first peak could be accounted for a harmony of metal (M)–B and B–B pair correlations and the second peak is mainly contributed by the M-M pair correlation. As for the three dimensional structural modeling of the amorphous samples, reverse Monte Carlo simulation has been performed starting from an initial model of 2,000 atoms with the b.c.c. structure. The present simulation results are found to reproduce the experimental interference functions obtained by the ND, ordinary X-ray diffraction, and AXS measurements. We will present the obtained structural model and local structural units around B including their arrangement.


1999 ◽  
Vol 13 (29n31) ◽  
pp. 3552-3554
Author(s):  
HAI QING LIN ◽  
JUN LI

We discuss frustration effects by studying the two-dimensional Hubbard model with nearest-neighbor(t) and next-nearest-neighbor(t′) hoppings. We present results obtained by the mean-field solution, the exact diagonalization technique, and the quantum Monte Carlo simulations. Questions to be addressed are: (1) magnetic phase diagram; (2) domain formation; and (3) superconducting pair correlation functions.


Author(s):  
J. M. Oblak ◽  
W. H. Rand

The energy of an a/2 <110> shear antiphase. boundary in the Ll2 expected to be at a minimum on {100} cube planes because here strue ture is there is no violation of nearest-neighbor order. The latter however does involve the disruption of second nearest neighbors. It has been suggested that cross slip of paired a/2 <110> dislocations from octahedral onto cube planes is an important dislocation trapping mechanism in Ni3Al; furthermore, slip traces consistent with cube slip are observed above 920°K.Due to the high energy of the {111} antiphase boundary (> 200 mJ/m2), paired a/2 <110> dislocations are tightly constricted on the octahedral plane and cannot be individually resolved.


Author(s):  
S. R. Herd ◽  
P. Chaudhari

Electron diffraction and direct transmission have been used extensively to study the local atomic arrangement in amorphous solids and in particular Ge. Nearest neighbor distances had been calculated from E.D. profiles and the results have been interpreted in terms of the microcrystalline or the random network models. Direct transmission electron microscopy appears the most direct and accurate method to resolve this issue since the spacial resolution of the better instruments are of the order of 3Å. In particular the tilted beam interference method is used regularly to show fringes corresponding to 1.5 to 3Å lattice planes in crystals as resolution tests.


Author(s):  
Karl E. Petersen
Keyword(s):  

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