Statistical Geometry  

In General
* Idea: Includes statistical techniques for studying a geometry, usually Euclidean (random sampling/sprinkling), and the study of properties of stochastically distributed subsets of a geometry ("stochastic geometry").

Point Process > s.a. Poisson Distribution; random process [including departures from randomness]; Sprinkling.
* Random sprinkling: Can be defined if we have a volume element on a set X, as a random process; It corresponds to a random measure on the set.
* Stationary: The statistical properties of the point process do not depend on location (what we would call "homogeneous").
* Types: Homogeneous (the number variance in a given region grows like the number mean), super-homogeneous (the number variance grows slower than the number mean), hyper-uniform (variance growth saturates).
* Moments: Each moment corresponds to a measure on the set, like the process itself.
* Palm distribution: Gives the conditional probability of point process events, given that a point is observed at a specific location.
* Operations on point processes: Thinning (a type of "coarse-graining"), clustering (replacing points by clusters), superposition (union).
* Other constructions: From a distribution of points, can get a (Delaunay) triangulation, and a (Voronoi) cell complex.
@ General references: Macchi AAP(75); Ambartzumian 90; van Hameren & Kleiss NPB(98)mp, et al NPB(99) [quantum field theory methods]; Barndorff-Nielsen et al 98; van Lieshout 00 [Markov point processes]; Ramiche AAP(00) [of phase-type]; Daley & Vere-Jones 07; Gabrielli et al PRE(08)-a0711 [superhomogeneous]; Møller & Schoenberg AAP(10) [random thinning]; Kendall & Molchanov ed-10; Nehring JMP(13), et al JMP(13) [method of cluster expansion].
@ Poisson point process: Cowan et al AAP(03) [gamma-distributed domains]; Bhattacharyya & Chakrabarti EJP(08) [distance to nth neighbor]; Balister et al AAP(09) [k-nearest-neighbour model, critical constant]; Chatterjee et al AM(10) [with allocation of measure to points]; Davydov et al AAP(10) [peeling procedure]; Serinaldi & Kilsby PhyA(13) [the Allan factor as an estimator of homogeneity]; Sevilla a1310 [Poisson processes with pile-up]; Cristina a1603 [in Minkowski space, and Noldus limit]; > s.a. Wikipedia page.
@ Other point processes: Hahn et al AAP(03) [inhomogeneous]; in Vuletić IMRN-mp/07 [Pfaffian]; Kuna et al AAP(11)-a0910 [realizability]; Gupta & Iyer AAP(10) [with exponentially decaying density]; Caron et al AAP(11) [conditional distributions].
@ Correlations: Kerscher A&A(99)ap/98 [correlation estimators]; de Coninck et al PhyA(07) [correlation structure]; Lenz & Moody a0902 [correlations].
@ Related topics: Barbour & Månsson AAP(00) [clustering of points]; Soshnikov RMS-m.PR/00, AAP-m.PR/00 [determinantal random point fields]; Valdarnini ASP-ap/01 [analysis of point distributions]; Lytvynov RVMP(02)mp/01 [fermion and boson]; Chiu & Molchanov AAP(03) [nearest neighbors, degree]; Koyama & Shinomoto JPA(05) [Bayesian interpretation]; Cowan AAP(06) [complementary theorem for n-tuples]; Sangaletti et al JPA(07) [high-d, Cox probabilities]; Kuna et al JSP(07) [realizability of functions as correlation functions]; Bárány BAMS(08) [convex polytopes]; Majumdar et al JSP(10) [properties of convex hulls]; Møller & Berthelsen AAP(12) [superposition of spatial point processes]; Rohrmann & Zurbriggen PRE(12) [conditional pair distributions]; > s.a. Betti Numbers.

Other Processes and Applications > s.a. cover [coverage process]; e; random tiling.
* Buffon's needle: An experimental method for determining the value of π, used by Georges Leclerc, Count de Buffon, in 1777; Rule a series of equidistant lines on a sheet of paper, a distance d apart; Drop a needle of length l < d on the paper so it falls in a random position; The probability it will cross a line is 2l / πd.
@ Buffon's needle: in Gardner 81, 127-128 & ref [use (fudged)]; {> s.a. #Lazzarini}.
@ Related topics: Donetti & Destri JPA(04)cm/03 [scale-free random trees]; Vickers & Brown PRS(01) [projected area and perimeter of solids]; Roberts & Garboczi JMPS(02) [elastic properties of solids]; Arsuaga et al JPA(07) [uniform random polygons, linking].
> Online resources: Garboczi page on elastic properties of solids.

Results and Special Cases
* 2D flat: For any non-concave 2D figure, the average width over all orientations is exactly perimeter/π; For any non-concave solid, the average projected area on a plane over all orientations is (surface area)/4.
* 2D curved:
* 3D flat:
* 3D curved:
@ On spheres: Tu & Fischbach mp/00/JMP, JPA(02)mp [n dimensions, distances between random points]; comp.graphics.algorithms page(06).
@ On other manifolds: Parry & Fischbach JMP(00) [distances on an ellipsoid].

References > s.a. ergodic theory.
@ General: Meijering PRR(53); Smith & Guttman JoM(53); Gilbert AMS(62); Miles MB(70)-mr; Harding & Kendall ed-74 [see intro]; Matheron 75; Santaló 76; Solomon 78; Stoyan et al 95; Beneš & Rataj 04.
@ Related topics: Schindler CG(94) [and equivariant mappings]; Grimmett a1110 [three theorems].

Computational Geometry > s.a. geometry.
* Simulating binomial point processes:
* Simulating Poisson point processes:
@ General references: Preparata & Shamos 85; Ripley 87, in Stoyan et al 87 [statistical simulations]; O'Rourke 98 [in C]; Goodman & O'Rourke 04; Boucetta & Morvan ed-05; de Berg et al 08; Devadoss & O'Rourke 11; Joswig & Theobald 13 [polyhedral and algebraic methods].
@ Graphs: Di Battista et al CG(94) [drawing algorithms].
@ Voronoi diagrams: Bespamyatnikh & Snoeyink CG(00) [queries with segments].
@ Delaunay triangulations: Su & Drysdale CG(97) [algorithm comparison]; Mücke et al CG(99) [point location, 2D and 3D]; Lemaire & Moreau CG(00); Hjelle & Dæhlen 06.
@ Greedy triangulations: Dickerson et al CG(97) [algorithms]; Levcopoulos & Krznaric CG(99).
@ Related topics: Mehlhorn et al CG(98) [higher-dimensional].

And Physics > s.a. causal sets; lattice field theory; semiclassical quantum gravity.
@ References: David et al ed-96 [fluctuating geometry and statistical mechanics]; Requardt & Roy CQG(01) [fuzzy lumps].


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