Statistical Mechanics on Isoradial Graphs

Boutillier, Cédric; Tilière, Béatrice de

doi:10.1007/978-3-642-23811-6_20

Cited by 18 publications

(25 citation statements)

References 50 publications

(38 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…That work is, in a sense, combinatorial in nature, and it provides connections between percolation on a graph embedded in R 2 and on a type of dual graph obtained via a generalized star-triangle transformation. In contrast, the work reported here is closely connected to the property of isoradiality (see [3,8]), and is thus more geometric in nature. It permits the proof of relations between a variety of two-dimensional graphs.…”

mentioning

confidence: 66%

“…and (3,4,6,4). The second may be transformed into the hexagonal lattice by one sequence of star-triangle transformations as marked.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…of star-triangle transformations, as well as to their dual graphs. This includes a number of tessellations (see [7]) and, in particular, two further Archimedean lattices, namely those denoted (3 3 , 4 2 ) and (3,4,6,4) and illustrated in Figure 2. The measures on these two lattices are as follows.…”

Section: Introduction and Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Universality for bond percolation in two dimensions

Grimmett¹,

Manolescu²

2013

Ann. Probab.

View full text Add to dashboard Cite

All (in)homogeneous bond percolation models on the square, triangular, and hexagonal lattices belong to the same universality class, in the sense that they have identical critical exponents at the critical point (assuming the exponents exist). This is proved using the star-triangle transformation and the box-crossing property. The exponents in question are the one-arm exponent $\rho$, the 2j-alternating-arms exponents $\rho_{2j}$ for $j\ge1$, the volume exponent $\delta$, and the connectivity exponent $\eta$. By earlier results of Kesten, this implies universality also for the near-critical exponents $\beta$, $\gamma$, $\nu$, $\Delta$ (assuming these exist) for any of these models that satisfy a certain additional hypothesis, such as the homogeneous bond percolation models on these three lattices.Comment: Published in at http://dx.doi.org/10.1214/11-AOP740 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

show abstract

mentioning

confidence: 66%

“…and (3,4,6,4). The second may be transformed into the hexagonal lattice by one sequence of star-triangle transformations as marked.…”

Section: Introduction and Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Universality for bond percolation in two dimensions

Grimmett¹,

Manolescu²

2013

Ann. Probab.

View full text Add to dashboard Cite

show abstract

“…Furthermore, it is important to observe that our procedure allows many distinct types of tessellations as long as they meet the bounded-cell condition (18). Besides Voronoi tessellations and sphere tessellations studied in this work, examples include disordered isoradial graphs [76], Delaunay triangulations, "Delaunay-centroidal" tessellations [44,54], dissected tessellations [77], and various generalizations of Voronoi tessellations, such as Laguerre tessellations [74,75] and tessellations in Manhattan distances [103]. Since constructing progenitor packings is the most time-consuming step in our numerical implementations, one would increase sample sizes by employing other tessellations that can be efficiently generated.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Methodology to construct large realizations of perfectly hyperuniform disordered packings

Kim

Torquato

2019

Phys. Rev. E

View full text Add to dashboard Cite

Disordered hyperuniform packings (or dispersions) are unusual amorphous two-phase materials that are endowed with exotic physical properties. Such hyperuniform systems are characterized by an anomalous suppression of volume-fraction fluctuations at infinitely long-wavelengths, compared to ordinary disordered materials. While there has been growing interest in such singular states of amorphous matter, a major obstacle has been an inability to produce large samples that are perfectly hyperuniform due to practical limitations of conventional numerical and experimental methods. To overcome these limitations, we introduce a general theoretical methodology to construct perfectly hyperuniform packings in d-dimensional Euclidean space R d . Specifically, beginning with an initial general tessellation of space by disjoint cells that meets a "bounded-cell" condition, hard particles are placed inside each cell such that the local-cell particle packing fractions are identical to the global packing fraction. We prove that the constructed packings with a polydispersity in size in R d are perfectly hyperuniform in the infinite-sample-size limit and the hyperuniformity of such packings is independent of particle shapes, positions, and numbers per cell. We use this theoretical formulation to devise an efficient and tunable algorithm to generate extremely large realizations of such packings. We employ two distinct initial tessellations: Voronoi as well as sphere tessellations. Beginning with Voronoi tessellations, we show that our algorithm can remarkably convert extremely large nonhyperuniform packings into hyperuniform ones in R 2 and R 3 . Implementing our theoretical methodology on sphere tessellations, we establish the hyperuniformity of the classical Hashin-Shtrikman multiscale coated-spheres structures, which are known to be two-phase media microstructures that possess optimal effective transport and elastic properties. A consequence of our work is a rigorous demonstration that packings that have identical tessellations can either be nonhyperuniform or hyperuniform by simply tuning local characteristics. It is noteworthy that our computationally designed hyperuniform two-phase systems can easily be fabricated via state-of-theart methods, such as 2D photolithographic and 3D printing technologies. In addition, the tunability of our methodology offers a route for the discovery of novel disordered hyperuniform two-phase materials.

show abstract

“…It has been studied in fields like statistical me-105 chanics [19] where the rigidity of planar isoradial graphs is well understood, but has not been extended to R 3 . It is indeed possible to transform a planar Tchebycheff net into an isoradial mesh (cf Figure 2).…”

mentioning

confidence: 99%