Suggestions for papers to read and present in connection with the course Couplings: old and new ideas

• Coupling is extremely useful in percolation theory. See, e.g., Schonmann, R.H. (1999) Stability of infinite clusters in supercritical percolation, Probability Theory and Related Fields 113, 287-300; the preprint can be found here.

• For another application of coupling in percolation theory, see Häggström, O.: Uniqueness of the infinite entangled component in three-dimensional bond percolation, to appear in Annals of Probability; click here for the preprint.

• Disagreement percolation is a way to combine percolation and coupling ideas, in a way which is very useful for analysing Markov random fields, see Van den Berg, J. and Maes, C. (1994) Disagreement percolation in the study of Markov fields, Annals of Probability 22, 749--763.

• The Propp-Wilson algorithm can be used for simulating Markov random fields even in cases where the sandwiching trick used for the Ising model is not available. See, e.g., Häggström, O. and Steif, J. (2000) Propp-Wilson algorithms and finitary codings for high noise Markov random fields, Combinatorics, Probability and Computing 9, 425-439; click here for the preprint.

• A useful alternative to the Propp-Wilson algorithm is Fill's algorithm which was introduced in Fill, J. (1998) An interruptible algorithm for perfect sampling via Markov chains, Annals of Applied Probability 8, 131--162; the preprint is available here.

• There are many cases where stochastic domination holds even though the condition of Holley's inequality (discussed in Lecture 4) does not. Then, it is sometimes nevertheless possible to derive some variation of Holley's inequality, which does work to establish stochastic domination. One such example can be found in Häggström, O. and Turova, T.: A strict inequality for the random triangle model, to appear in Journal of Statistical Physics; preprint available here.

• The most exciting recent development in the general thoery of stochastic domination and coupling, is, without doubt, the paper by Fill, J. and Motoya, M.: Stochastic monotonicity and realizable monotonicity, to appear in Annals of Probability; preprint available here.

• In Chapter 1, Section 10 of Thorisson, H. (2000) Coupling, Stationarity and Regeneration (Springer), a very interesting connection between coupling and quantum physics is pointed out. See also the twin papers
  • Kummerer, B. and Maassen, H. (1998), Elements of Quantum Probability, Quantum Probability Communications 10, 73-100, and
  • Gill, R., Critique of `Elements of Quantum Probability', Quantum Probability Communications 10, 351-361
  to obtain a slightly broader picture; preprints of both papers are available here.

• An interesting problem on reconstruction in tree-shapred communications networks (which also has applications in genetics), where couplings are useful, is studied in Evans, W., Kenyon, C., Peres, Y. and Schulman, L. (2000) Broadcasting on trees and the Ising model, Annals of Applied Probability 10, 410--433; click here to find the preprint.

• One important area of application of coupling is Poisson approximation. See, e.g., Barbour, A., Topics in Poisson approximation, to appear in Stochastic Processes: Theory and Methods, Handbook of Statistics, Wiley; preprint available here.

• The modern way of proving Blackwell's renewal theorem is via couplings. After Lindvall's pioneering 1977 paper, the arguments have gradually been refined, see
  • Lindvall, T. (1977) A probabilistic proof of Blackwell's renewal theorem, Annals of Probability 5, 57-70,
  • Thorisson, H. (1987) A complete coupling proof of Blackwell's renewal theorem, Stochastic Processes and their Applications 26, 87-97, and
  • Lindvall, T. and Rogers, C. (1996) On coupling of random walks and renewal processes, Journal of Applied Probability 33, 122-126.

• The most recent Ph.D. thesis here at Chalmers and Göteborg University, dealing with couplings, is Svensson, D. (2000) A Class of Renewal Processes in Random Environment, which can be downloaded here.