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, 287300; the preprint can be found
here.
 For another application of coupling in percolation theory, see
Häggström,
O.: Uniqueness of the infinite entangled component in
threedimensional 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, 749763.
 The ProppWilson 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) ProppWilson algorithms and finitary codings for high noise
Markov random fields, Combinatorics, Probability
and Computing 9, 425439; click
here
for the preprint.
 A useful alternative to the ProppWilson 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, 131162;
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, 73100, and

Gill, R.,
Critique of `Elements of Quantum Probability', Quantum
Probability Communications 10, 351361
to obtain a slightly broader picture; preprints of both papers are
available here.
 An interesting problem on reconstruction in treeshapred
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, 410433; 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, 5770,
 Thorisson, H.
(1987) A complete coupling proof of Blackwell's renewal
theorem, Stochastic Processes and their Applications
26, 8797, and
 Lindvall,
T. and
Rogers,
C. (1996) On coupling of random walks
and renewal processes, Journal of Applied
Probability 33, 122126.
 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.
Last modified: Mon Feb 19 13:59:02 MET 2001