Random Tilings and Triangulations |
In General > s.a. statistical
geometry; tilings [space of tilings].
* Examples: The Voronoi
or Delaunay complexes defined by a uniformly random set of points in
(M, qab),
or by a continuous nucleation (Mehl-Johnson) model.
* Aboav-Weaire law:
A correlation between the number of faces of a cell and that of
its neighbors; In 2D, m(n) = 6 – a
+ (6a+σ(n))/n, where
m(n) is the average number of sides of cells with
n-sided neighbors, and σ(n) the
variance of the number of edges per cell.
@ General references: Miles MB(70)-mr;
Santaló 76;
Brilliantov et al JPA(94) [continuous nucleation];
Richard JPA(99)cm;
Matzutt a0712.
@ Aboav-Weaire law: Weaire Met(74);
Aboav Met(80);
Lambert & Weaire Met(81);
Peshkin et al PRL(91);
Lauritsen et al JPI(93)cm;
Fortes JPA(95);
Mason et al JPA(12) [geometric formulation]; > s.a. networks.
@ Coloring: Di Francesco et al NPB(98)cm/97;
Bouttier et al NPB(02).
@ Related topics: Lauritsen et al IJMPC(94)cm/93 [Monte Carlo];
Richard et al JPA(98)cm/97 [entropy];
Baake & Höffe JSP(00)mp/99 [diffraction];
Veerman et al CMP(00) [Brillouin zones, constant curvature];
Kenyon AIHP(97)m.CO/01 [random domino, measure];
Desoutter & Destainville JPA(05)cm/04 [3D rhombus tilings, flip dynamics];
Mecke et al AAP(08) [iteration];
Colomo & Pronko PRE(13)-a1306 [third-order phase transition].
@ Johnson-Mehl models:
Chiu AAP(95) [limit theorems];
Garcia AAP(95);
Bollobás & Riordan PTRF(07) [2D, percolation].
@ Other generalized types: Lautensack & Zuyev AAP(08) [random Laguerre tessellations];
Cowan AAP(10) [from iterative cell division].
2D Riemannian Manifolds
> s.a. Percolation; spin models.
@ Euclidean plane: Mecke MOS(84);
Joseph & Baake JPA(96) [entropy];
Di Francesco et al NPB(98)cm/97 [coloring];
Kostov PLB(02)ht/00 [3-color problem];
Hayen & Quine AAP(02) [moments of area distribution];
Calka AAP(02) [sizes of circles containing or contained in cells],
AAP(03) [principal geometric characteristics],
AAP(03) [distribution of the number of sides];
Destainville et al JSP(05),
Widom et al JSP(05) [high symmetry];
Pinchasi et al JCTA(06) [empty convex polygons];
Böröczky et al JGP(06) [Weaire sum rule].
@ 2D sphere: Miles Sankhya(71).
@ 2D torus: Higuchi NPB(99) [number of Hamiltonian cycles].
Higher-Dimensional Riemannian Manifolds
@ In E3: Meijering Philips(53);
Gilbert AMS(62);
Miles SAAP(72);
Mecke MOS(84);
Hug et al AAP(04) [shape of large cells].
@ In En: Zähle AnnProb(88);
Møller AAP(89) [convex cells, mean-value relations];
Mecke & Stoyan AAP(01) [connectivity number];
Chatterjee et al AM(10) [allocation rule of Lebesgue measure with subpolynomial decay of the tail];
Xu et al TMP(12)
[Poisson-distributed vertices and randomly assigned edges].
@ Other manifolds:
Escudero JGP(08) [spherical manifolds].
Of a Lorentzian Manifold
@ Triangulations:
Di Francesco et al NPB(01) [1+1 model].
And Physics > s.a. lattice field theory;
tilings; voronoi tilings.
* Applications: Random
tilings are used as models of nucleation in crystals, or random lattices for
gauge theory and quantum gravity.
@ General references: Ziman 79; Lee in(85).
@ Polycrystals and foams: Aboav Met(83),
Met(84).
@ Examples and effects: Davison & Sherrington JPA(00) [glassy behavior];
Charbonnier et al a1701
[large-N limit and and topological 2D gravity];
Stéphan a2003-ln [effect of boundary conditions on bulk properties].
@ Thermodynamics: Leuzzi & Parisi JPA(00) [with Wang tiles].
@ In field theory: Ciucu MAMS-mp/03 [2D electromagnetism].
@ In cosmology: de Laix & Vachaspati PRD(99)hp/98;
Schaap & van de Weygaert A&A(00)ap,
ESO(01)ap/00,
ap/01-in [Delaunay].
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