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Causet Dynamics withDq.DgItai SeggevDepartment of MathematicsKnox College17th Midwest Relativity MeetingNovember 2, 2007

IntroductionA New CodeResultsCausetsClassical Sequential GrowthMotivationReminderPoints p and q in a (causal) spacetime will obey one of threerelations: p is to the future of q, to the past of q, or spacelikerelated to q.Theorem (Malament)The points of a manifold (M, g ) together with their causalstructure specify the pair (M, g ) up to conformal equivalence.In 4d, this gives topology and 9 of 10 metric components.Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsCausetsClassical Sequential GrowthGoing Discrete“Missing” metric component is local volume elementIf discretizing spacetime, natural to assume one point perPlank volumeGiven two points, only finitely many points in causal diamond.Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsCausetsClassical Sequential GrowthGoing Discrete“Missing” metric component is local volume elementIf discretizing spacetime, natural to assume one point perPlank volumeGiven two points, only finitely many points in causal diamond.Causal sets naturally encode geometry with a minimalphysical hypothesis.Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsCausetsClassical Sequential GrowthCausetsDefinition (Causal Set)A causal set, or causet, is a point set C together with a locallyfinite partial order.“Locally finite” means {z x z y } x, y C .Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsCausetsClassical Sequential GrowthClassical Dynamics?Need to implement Einstein’s equations somehowEventually lead to quantum dynamics using sum over historiesImplemented as a stochastic growth model of n-elementcauset to n k element causet.Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsCausetsClassical Sequential GrowthClassical Sequential GrowthRideout and Sorkin classified all models which obey12general covarianceBell causalityTwo basic varieties:12transitive percolationeverything elseItai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsCausetsClassical Sequential GrowthClassical Sequential GrowthRideout and Sorkin classified all models which obey12general covarianceBell causalityTwo basic varieties:12transitive percolation simple, long-studied, one parameter: peverything elseItai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsCausetsClassical Sequential GrowthClassical Sequential GrowthRideout and Sorkin classified all models which obey12general covarianceBell causalityTwo basic varieties:12transitive percolationgeneralized percolation more challengingItai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsCausetsClassical Sequential GrowthGeneralized PercolationGiven parameters nonnegative {t1 , t2 , t3 , . . .} nChoose 0 k n with relative probability tkkRandomly choose “proto-precursor” of k elements withuniform probability. 123Enforce implied relations using transitive closure.Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsPeopleGoalsImplementationWho We AreDqDefinition ( Dg)Discrete Quantum Dynamics Group.Luca Bombelli (University of Mississippi)Julio Tafoya (University of Mississippi)Itai Seggev (Knox College)Sam Watson (University of Mississippi)Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsPeopleGoalsImplementationPhysics Goals1Perform a wide variety of analyses on a CausetMyrheim Meyer dimensionmidpoint dimensionordering fractionnumber and location of postsheight, width (future)2Compute quantities locally as well as globally.3Explore some parameter space of generalized percolation.4Possibly add other dynamics.Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsPeopleGoalsImplementationProgramming Goals1Easy to add percolation “types” (functions for generating tn ).2Minimal external dependencies.3Portable.4Elegant.5Reasonably efficient, but not at the cost of the above.Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsPeopleGoalsImplementationBasic ChoicesLanguage: C Use GSL for random number generator and Γ(z).Implement our own “large double” type.Representation of causet hidden inside of class all analysis functions access the causet using class methodsAt present time, represent causet as an adjacency matriximplemented as a vector vector bool .Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsPeopleGoalsImplementationThe Generalized Percolation SolutionAchtung!This is the computer geek slide. Our regularly scheduled physicsprogram will resume on the next slide.1Define a class coupling t which has a “pure virtualfunction” called nthconstantclass coupling t {public:virtual large double t nthconstant(int n) 0;};Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsPeopleGoalsImplementationThe Generalized Percolation SolutionAchtung!This is the computer geek slide. Our regularly scheduled physicsprogram will resume on the next slide.1Define a class coupling t which has a “pure virtualfunction” called nthconstant2To define a new coupling type, define a class which inheritsfrom coupling tclass allones t : public coupling t {public:allones t() {one double to ldt(1.0);}large double t nthconstant(int n) {return one;}private:large double t one;};Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsPeopleGoalsImplementationThe Generalized Percolation SolutionAchtung!This is the computer geek slide. Our regularly scheduled physicsprogram will resume on the next slide.1Define a class coupling t which has a “pure virtualfunction” called nthconstant2To define a new coupling type, define a class which inheritsfrom coupling t3The general percolation method takes as an argument acoupling t*, so the coupling type is determined at run time.Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsPeopleGoalsImplementationThe Generalized Percolation SolutionAchtung!This is the computer geek slide. Our regularly scheduled physicsprogram will resume on the next slide.class coupling t {public:virtual large double t nthconstant(int n) 0;};class allones t : public coupling t {public:large double t nthconstant(int n) {return one;}.};class causet t {public:void seed causet gen perc(coupling t *couplingConsnats);.};Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsAccomplishmentsStatus and Future DirectionsA New ProofAlon, Bollobas, et al. proved that for transitive percolationand any p 0, infinitely many posts occur with probability 1.As part of our preparation for this project, we came up with asignificantly simplified “by-hand” proof.Forthcoming paper.Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsAccomplishmentsStatus and Future DirectionsSample OutputThe program ./csg simulation was compiled from an unmodified copy ofrevision 80 and was run with arguments -N 5000 --transitivePercolation--probability 0.3 --p ostsAll --myrheimMeyerDimension--midpointDimension --orderingFractionThe number of posts is 9.The posts are 877, 3114, 3115, 4209, 4210, 4211, 4354, 4355, and 000001.1050731.0000001.000000Ordering 9429881.0000001.000000The midpoint scaling dimension is 1.002894.The 2-chain abundance Myrheim-Meyer dimension is 1.003446.The ordering fraction is 0.998086.Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsAccomplishmentsStatus and Future DirectionsSample DataNumber of posts versus poset size, p"0.35Average ! of posts4321100200Itai Seggev300400Causet Dynamics with Dq/Dg .500n

IntroductionA New CodeResultsAccomplishmentsStatus and Future DirectionsWhat WorksTransitive percolation (nothing new)Proof-of-concept generalized percolation with tn simulatingtransitive percolation or tn 1 n.Non-sequential Minkowski “sprinkling”.Finding postsDimension estimation, both locally and globally.Itai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsAccomplishmentsStatus and Future DirectionsCurrent and Planned Work1Analyzing transitive percolation using local estimates(ongoing)2Bug-squashing (ongoing)3Analyzing more interesting general percolation models4Height and width calculationItai SeggevCauset Dynamics with Dq/Dg .

IntroductionA New CodeResultsAccomplishmentsStatus and Future DirectionsCurrent and Planned Work1Analyzing transitive percolation using local estimates(ongoing)2Bug-squashing (ongoing)3Analyzing more interesting general percolation models4Height and width calculation5Suggestions and questions?Itai SeggevCauset Dynamics with Dq/Dg .

program will resume on the next slide. 1 Define a class coupling t which has a “pure virtual function” called nthconstant 2 To define a new coupling type, define a class which inherits from coupling t 3 The general percolation method takes as an argument a coupling t*,

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