The Annals of Applied Probability

Order-invariant measures on causal sets

Graham Brightwell and Malwina Luczak

Full-text: Open access


A causal set is a partially ordered set on a countably infinite ground-set such that each element is above finitely many others. A natural extension of a causal set is an enumeration of its elements which respects the order.

We bring together two different classes of random processes. In one class, we are given a fixed causal set, and we consider random natural extensions of this causal set: we think of the random enumeration as being generated one point at a time. In the other class of processes, we generate a random causal set, working from the bottom up, adding one new maximal element at each stage.

Processes of both types can exhibit a property called order-invariance: if we stop the process after some fixed number of steps, then, conditioned on the structure of the causal set, every possible order of generation of its elements is equally likely.

We develop a framework for the study of order-invariance which includes both types of example: order-invariance is then a property of probability measures on a certain space. Our main result is a description of the extremal order-invariant measures.

Article information

Ann. Appl. Probab., Volume 21, Number 4 (2011), 1493-1536.

First available in Project Euclid: 8 August 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 06A07: Combinatorics of partially ordered sets 60F99: None of the above, but in this section

Causal sets infinite posets random linear extensions invariant measures


Brightwell, Graham; Luczak, Malwina. Order-invariant measures on causal sets. Ann. Appl. Probab. 21 (2011), no. 4, 1493--1536. doi:10.1214/10-AAP736.

Export citation


  • [1] Albert, M. H. and Frieze, A. M. (1989). Random graph orders. Order 6 19–30.
  • [2] Alon, N., Bollobás, B., Brightwell, G. and Janson, S. (1994). Linear extensions of a random partial order. Ann. Appl. Probab. 4 108–123.
  • [3] Berti, P. and Rigo, P. (2008). A conditional 0–1 law for the symmetric σ-field. J. Theoret. Probab. 21 517–526.
  • [4] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [5] Bollobás, B. and Brightwell, G. (1995). The width of random graph orders. Math. Sci. 20 69–90.
  • [6] Bovier, A. (2006). Statistical Mechanics of Disordered Systems: A Mathematical Perspective. Cambridge Univ. Press, Cambridge.
  • [7] Brightwell, G. and Georgiou, N. (2010). Continuum limits for classical sequential growth models. Random Structures Algorithms 36 218–250.
  • [8] Brightwell, G. and Luczak, M. Order-invariant measures on fixed causal sets. Unpublished manuscript. Available at arXiv:0901.0242.
  • [9] Brightwell, G. R. (1988). Linear extensions of infinite posets. Discrete Math. 70 113–136.
  • [10] Brightwell, G. R. (1989). Semiorders and the ⅓–⅔ conjecture. Order 5 369–380.
  • [11] Brightwell, G. R., Felsner, S. and Trotter, W. T. (1995). Balancing pairs and the cross product conjecture. Order 12 327–349.
  • [12] Dowker, F. and Surya, S. (2006). Observables in extended percolation models of causal set cosmology. Classical Quantum Gravity 23 1381–1390.
  • [13] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [14] Fishburn, P. C. (1984). A correlational inequality for linear extensions of a poset. Order 1 127–137.
  • [15] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics 9. de Gruyter, Berlin.
  • [16] Georgiou, N. (2005). The random binary growth model. Random Structures Algorithms 27 520–552.
  • [17] Grimmett, G. R. and Stirzaker, D. R. (2001). Probability and Random Processes, 3rd ed. Oxford Univ. Press, New York.
  • [18] Hewitt, E. and Savage, L. J. (1955). Symmetric measures on Cartesian products. Trans. Amer. Math. Soc. 80 470–501.
  • [19] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [20] Kerov, S. (1996). The boundary of Young lattice and random Young tableaux. In Formal Power Series and Algebraic Combinatorics (L. J. Billera, C. Greene, R. Simion and R. P. Stanley, eds.). DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 24 133–158. Amer. Math. Soc., Providence, RI.
  • [21] Luczak, M. and Winkler, P. (2004). Building uniformly random subtrees. Random Structures Algorithms 24 420–443.
  • [22] Maitra, A. (1977). Integral representations of invariant measures. Trans. Amer. Math. Soc. 229 209–225.
  • [23] Pittel, B. and Tungol, R. (2001). A phase transition phenomenon in a random directed acyclic graph. Random Structures Algorithms 18 164–184.
  • [24] Rideout, D. P. and Sorkin, R. D. (2000). Classical sequential growth dynamics for causal sets. Phys. Rev. D (3) 61 024002, 16.
  • [25] Rideout, D. P. and Sorkin, R. D. (2001). Evidence for a continuum limit in causal set dynamics. Phys. Rev. D (3) 63 104011, 15.
  • [26] Stanley, R. P. (1981). Two combinatorial applications of the Aleksandrov–Fenchel inequalities. J. Combin. Theory Ser. A 31 56–65.
  • [27] Varadarajan, M. and Rideout, D. (2006). General solution for classical sequential growth dynamics of causal sets. Phys. Rev. D (3) 73 104021, 10.
  • [28] Williams, D. (2007). Probability with Martingales. Cambridge Univ. Press, Cambridge.