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Order-Invariant Measures on Fixed Causal Sets

Published online by Cambridge University Press:  19 January 2012

GRAHAM BRIGHTWELL  and
MALWINA LUCZAK
GRAHAM BRIGHTWELL
Affiliation:
Department of Mathematics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK (e-mail: [email protected])
MALWINA LUCZAK
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK (e-mail: [email protected])
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Abstract

A causal set is a countably infinite poset in which every element is above finitely many others; causal sets are exactly the posets that have a linear extension with the order-type of the natural numbers; we call such a linear extension a natural extension. We study probability measures on the set of natural extensions of a causal set, especially those measures having the property of order-invariance: if we condition on the set of the bottom k elements of the natural extension, each feasible ordering among these k elements is equally likely. We give sufficient conditions for the existence and uniqueness of an order-invariant measure on the set of natural extensions of a causal set.

Keywords

Primary 06A0760C05
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 3 , May 2012 , pp. 330 - 357
Copyright
Copyright © Cambridge University Press 2012

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References

[1]Aldous, D. and Diaconis, P. (1999) Longest increasing subsequences: From patience sorting to the Baik–Deift–Johansson theorem. Bull. Amer. Math. Soc. 36 413–432.CrossRefGoogle Scholar
[2]Billingsley, P. (1999) Convergence of Probability Measures, Wiley.CrossRefGoogle Scholar
[3]Bovier, A. (2006) Statistical Mechanics of Disordered Systems, Cambridge University Press.CrossRefGoogle Scholar
[4]Brightwell, G. R. (1988) Linear extensions of infinite posets. Discrete Math. 70 113–136.CrossRefGoogle Scholar
[5]Brightwell, G. R. (1989) Semiorders and the 1/3–2/3 conjecture. Order 5 369–380.CrossRefGoogle Scholar
[6]Brightwell, G. R. (1999) Balanced pairs in partial orders. Discrete Math. 201 25–52.CrossRefGoogle Scholar
[7]Brightwell, G. R. and Georgiou, N. (2010) Continuum limits for classical sequential growth models. Random Struct. Alg. 36 218–250.CrossRefGoogle Scholar
[8]Brightwell, G. and Luczak, M. (2011) Order-invariant measures on causal sets. Ann. Appl. Probab. 21 1493–1536.CrossRefGoogle Scholar
[9]Frame, J. S., Robinson, G. de B. and Thrall, R. M. (1954) The hook graphs of the symmetric group. Canad. J. Math. 6 316–324.CrossRefGoogle Scholar
[10]Georgii, H.-O. (1988) Gibbs Measures and Phase Transitions, Vol. 9 of De Gruyter Studies in Mathematics, de Gruyter.CrossRefGoogle Scholar
[11]Gnedin, A. and Kerov, S. (2000) The Plancherel measure of the Young–Fibonacci graph. Math. Proc. Cambridge Philos. Soc. 129 433–446.CrossRefGoogle Scholar
[12]Graham, R. L., Yao, A. C. and Yao, F. F. (1980) Some monotonicity properties of partial orders. SIAM J. Alg. Discrete Methods 1 251–258.CrossRefGoogle Scholar
[13]Grimmett, G. R. and Stirzaker, D. (2001) Probability and Random Processes, Oxford University Press.CrossRefGoogle Scholar
[14]Kerov, S. (1996) The boundary of Young lattice and random Young tableaux. In Formal Power Series and Algebraic Combinatorics: New Brunswick 1994, Vol. 24 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., AMS, pp. 33–158.Google Scholar
[15]Rideout, D. P. and Sorkin, R. D. (2000) Classical sequential growth dynamics for causal sets. Phys. Rev. D (3) 61 024002.CrossRefGoogle Scholar
[16]Rideout, D. P. and Sorkin, R. D. (2001) Evidence for a continuum limit in causal set dynamics. Phys. Rev. D (3) 63 104011.CrossRefGoogle Scholar
[17]Ross, S. M. (2007) Introduction to Probability Models, ninth edition, Academic Press.Google Scholar
[18]Vershik, A. M. and Kerov, S. K. (1977) Asymptotics of the Plancherel measure of the symmetric group and the limiting shape of Young tableaux. Soviet Math. Dokl. 18 527–531.Google Scholar
[19]Vershik, A. M. and Tsilevich, N. V. (2006) Markov measures on Young tableaux and induced representations of the infinite symmetric group (in Russian). Teor. Veroyatn. Primen. 51 47–63. Translation in Theory Probab. Appl. 51 (2007), 211–223.Google Scholar
[20]Williams, D. (2007) Probability with Martingales, Cambridge University Press.Google Scholar
[21]Winkler, P. (1986) Correlation and order. Contemporary Mathematics 57 151–174.CrossRefGoogle Scholar