Causal Set Kinematics

In General > s.a. Alexandrov Sets; causal structure and spacetime; spin-foam models.
* Hauptvermutung: (Original version) If a causal set can be faithfully embedded in two Lorentzian manifolds (M, g) and (M', g'), then those two manifolds are close down to scales of the order of the embedding density.
* Coarse graining: A random coarse-graining procedure consists in starting with a causal set C and removing each point with probability p.
* Observables: All covariant statements one can make about causal sets are statements about stems.
* Feature: Causal sets can implement the notion that spacetime topology may be scale-dependent; No known continuum approach can do this.
@ And posets: Low JMP(00); Droste JMP(05)gq [universal past-finite causal set]; Vatandoost & Bahrampour JMP(11) [and sphere orders].
@ Set of causal sets: Yazdi & Kempf CQG(17)-a1612 [spectral distance, from the Feynman propagator]; > s.a. spectral geometry.
@ Related topics: Eichhorn CQG(18)-a1709 [coarse graining]; Zapatrin a1801 [timelets].

Geometry and Relationship with the Continuum > s.a. emergence; Global Hyperbolicity.
* Past / future of an element: The layer-k past/future $$L^\mp_k(x)$$ of an element x is the set of elements y such that the (inclusive) interval [y, x] has k +1 elements; The rank-k past/future $$R^\mp_k(x)$$ of an element x is the set of elements y that are k links to its past.
* Preferred past structure: The choice of a rank-2 past element for each causal set element.
* Definitions of curvature: One definition of curvature at an element x of a causal set with a minimal element 0 is Gudder's, K(x) = number of $$\cal L$$-geodesics between 0 and x, where $${\cal L}(x):= \big[\sum_i(l(a_{i+1})-l(a_i))^2\big]{}^{1/2}$$.
@ General references: Bombelli & Meyer PLA(89); Daughton CQG(98) [symmetric case]; Brightwell & Gregory PRL(91); Filk CQG(01)gq [time]; Requardt JMP(03)gq/01 [renormalization group]; Henson CQG(06)gq [manifoldlike causal sets]; Surya TCS(08)-a0712 [topology]; Krugly a1006 [unfaithful embeddings and matter]; Glaser & Surya PRD(13)-a1309 [proposed definition of locality]; Saravani & Aslanbeigi CQG(14)-a1403; Bolognesi & Lamb a1407; Gudder a1502 [and the emergence of 4D]; Gudder a1507 [c-causets and curvature]; Fewster et al PRD(21)-a2011 [sprinkled causal sets, preferred pasts].
@ Dimension: Meyer PhD(88), Ord(93); Reid PRD(03)gq/02; Eichhorn & Mizera CQG(14)-a1311 [spectral dimension]; Carlip CQG(15)-a1506, Belenchia et al PRD(16)-a1507, Abajian & Carlip PRD(18)-a1710 [reduction to spectral dimension = 2].
@ Curvature: Benincasa & Dowker PRL(10)-a1001 [scalar curvature]; Roy et al PRD(13)-a1212, Kambor & Nomaan CQG(20)-a2007 [using chains, and dimension].
@ Timelike distances and geodesics: Ilie et al CQG(06)gq/05.
@ Spacelike distances and hypersurfaces: Rideout & Wallden CQG(09)-a0810, JPCS(09)-a0811 [and timelike]; Eichhorn et al CQG(19)-a1809; Eichhorn et al CQG(19)-a1905 [spectral dimension].
@ 2D causal sets: Brightwell et al CQG(08)-a0706, JPCS(09); Surya CQG(12)-a1110 [phase transition]; Glaser et al CQG(18)-a1706 [scaling and transition between non-continuum and continuum phases]; Glaser a2011 [coupled to Ising model, phase transitions].
@ Properties of manifoldlike causal sets: Eichhorn et al CQG(17)-a1703 & CQG+ [manifestations of asymptotic silence].

Hypersurfaces, Foliations
@ Thickened spatial hypersurfaces: Major et al CQG(06)gq/05; Major et al JMP(07)gq/06; Major et al CQG(09)-a0902 [stable homology and manifoldlikeness]; Jubb CQG(17)-a1611.
@ Related topics: Bleybel & Zaiour FP(18)-a1508 [results on temporal foliations]; Cunningham CQG(18)-a1710 [extrinsic geometry of boundaries]; Barton et al PRD(19)-a1909 [horizon molecules].

Special Type of Metrics
@ Specific types of metrics: He & Rideout CQG(09)-a0811 [Schwarzschild]; D'Ariano & Tosini a1008, SHPMP(13)-a1109, Cristina CQG(16)-a1603 [Minkowski].