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, q_ab), 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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