The mathematics of causal sets

The causal set approach to quantum gravity is based on the hypothesis that the underlying structure of spacetime is that of a random partial order. We survey some of the interesting mathematics that has arisen in connection with the causal set hypothesis, and describe how the mathematical theory can be translated to the application area. We highlight a number of open problems of interest to those working in causal set theory.

[1]  Observables in extended percolation models of causal set cosmology , 2005, gr-qc/0504069.

[2]  R. Sorkin,et al.  Classical sequential growth dynamics for causal sets , 1999, gr-qc/9904062.

[3]  David A. Meyer,et al.  The origin of Lorentzian geometry , 1989 .

[4]  Graham R. Brightwell,et al.  Order-Invariant Measures on Fixed Causal Sets , 2009, Combinatorics, Probability and Computing.

[5]  Gregory,et al.  Structure of random discrete spacetime. , 1990, Physical review letters.

[6]  Andrew King,et al.  A new topology for curved space–time which incorporates the causal, differential, and conformal structures , 1976 .

[7]  Graham R. Brightwell.  Linear extensions of infinite posets , 1988, Discret. Math..

[8]  Boris G. Pittel,et al.  A phase transition phenomenon in a random directed acyclic graph , 2001, Random Struct. Algorithms.

[9]  Graham Brightwell,et al.  A 2D model of causal set quantum gravity: the emergence of the continuum , 2008 .

[10]  P. Diaconis,et al.  Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem , 1999 .

[11]  Sergei V. Kerov,et al.  The Boundary of Young Lattice and Random Young Tableaux , 1994, Formal Power Series and Algebraic Combinatorics.

[12]  M. El-Zahar,et al.  Asymptotic enumeration of two-dimensional posets , 1988 .

[13]  Rafael D. Sorkin.  Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School) , 2003 .

[14]  B. Bollobás,et al.  The longest chain among random points in Euclidean space , 1988 .

[15]  Anusch Taraz,et al.  Counting Partial Orders With A Fixed Number Of Comparable Pairs , 2001, Comb. Probab. Comput..

[16]  Klaus Simon,et al.  On the Distribution of the Transitive Closure in a Random Acyclic Digraph , 1993, ESA.

[17]  Noga Alon,et al.  LINEAR EXTENSIONS OF A RANDOM PARTIAL ORDER , 1994 .

[18]  Graham R. Brightwell,et al.  Continuum limits for classical sequential growth models , 2010, Random Struct. Algorithms.

[19]  G. Brightwell.  Models of random partial orders , 1993 .

[20]  Rafael D. Sorkin,et al.  ``Observables'' in causal set cosmology , 2003 .

[21]  J. Baik,et al.  On the distribution of the length of the longest increasing subsequence of random permutations , 1998, math/9810105.

[22]  Joe Henson,et al.  QUANTUM GRAVITY PHENOMENOLOGY, LORENTZ INVARIANCE AND DISCRETENESS , 2003, gr-qc/0311055.

[23]  Anusch Taraz,et al.  Phase Transitions in the Evolution of Partial Orders , 2001, J. Comb. Theory, Ser. A.

[24]  Béla Bollobás,et al.  The Structure of Random Graph Orders , 1997, SIAM J. Discret. Math..

[25]  B. Rothschild,et al.  Asymptotic enumeration of partial orders on a finite set , 1975 .

[26]  Bombelli,et al.  Space-time as a causal set. , 1987, Physical review letters.

[27]  B. Bollobás,et al.  Box-spaces and random partial orders , 1991 .

[28]  Sumati Surya,et al.  Directions in Causal Set Quantum Gravity , 2011, 1103.6272.

[29]  Boris G. Pittel,et al.  On Tail Distribution of Interpost Distance , 2000, J. Comb. Theory, Ser. B.

[30]  D. Meyer.  Spherical containment and the Minkowski dimension of partial orders , 1993 .

[31]  Oleg Pikhurko,et al.  Poset limits can be totally ordered , 2012, 1211.2473.

[32]  Statistical Lorentzian geometry and the closeness of Lorentzian manifolds , 2000, gr-qc/0002053.

[33]  R. Sorkin,et al.  Spacetime as a Causal Set , 1989 .

[34]  D. Malament.  The class of continuous timelike curves determines the topology of spacetime , 1977 .

[35]  Petros Wallden,et al.  Spacelike distance from discrete causal order , 2008, 0810.1768.

[36]  David Rideout,et al.  General solution for classical sequential growth dynamics of causal sets , 2006 .

[37]  Stefan Felsner,et al.  Finite three dimensional partial orders which are not sphere orders , 1999, Discret. Math..

[38]  Noga Alon,et al.  Degrees of freedom versus dimension for containment orders , 1988 .

[39]  Svante Janson,et al.  Poset limits and exchangeable random posets , 2009, Comb..

[40]  Paul Erdös,et al.  On the Maximal Number of Strongly Independent Vertices in a Random Acyclic Directed Graph , 1984 .

[41]  The random binary growth model , 2005 .

[42]  Peter Winkler.  Random orders of dimension 2 , 1990 .

[43]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .

[44]  Fay Dowker,et al.  Introduction to causal sets and their phenomenology , 2013 .

[45]  Fay Dowker,et al.  The Causal Set Approach to Quantum Gravity , 2013 .

[46]  Graham R. Brightwell,et al.  The Average Number of Linear Extensions of a Partial Order , 1996, J. Comb. Theory, Ser. A.

[47]  G. Brightwell,et al.  Order-invariant measures on causal sets , 2009, 0901.0240.

[48]  R. Sorkin,et al.  Evidence for a continuum limit in causal set dynamics , 2000, gr-qc/0003117.

[49]  Nicholas Georgiou,et al.  Continuum limits for classical sequential growth models , 2010 .

[50]  L. J. Savage,et al.  Symmetric measures on Cartesian products , 1955 .

[51]  Béla Bollobás,et al.  The Dimension of Random Graph Orders , 2013, The Mathematics of Paul Erdős II.

[52]  Michael H. Albert,et al.  Random graph orders , 1989 .