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Dynamical coupling between Ising and FK percolation. (English) Zbl 1471.60146

Summary: We investigate the problem of constructing a dynamics on edge-spin configurations which realizes a coupling between a Glauber dynamics of the Ising model and a dynamical evolution of the percolation configurations. We dreamed of constructing a Markov process on edge-spin configurations which is reversible with respect to the Ising-FK coupling measure, and such that the marginal on the spins is a Glauber dynamics, while the marginal on the edges is a Markovian evolution. We present two local dynamics, one which fulfills only the first condition and one which fulfills the first two conditions. We show next that our dream process is not feasible in general. We present a third dynamics, which is non local and fulfills the first and the third conditions. We finally present a localized version of this third dynamics, which can be seen as a contraction of the first dynamics.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J27 Continuous-time Markov processes on discrete state spaces
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B43 Percolation
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References:

[1] F. Ball and G. F. Yeo.Lumpability and marginalisability for continuous-time Markov chains.J. Appl. Probab.30(3), 518-528 (1993) · Zbl 0781.60057
[2] R. G. Edwards and A. D. Sokal. Generalization of the Fortuin-Kasteleyn-SwendsenWang representation and Monte Carlo algorithm.Phys. Rev. D (3)38(6), 2009-2012 (1988)
[3] G. Grimmett. The stochastic random-cluster process and the uniqueness of randomcluster measures.Ann. Probab.23(4), 1461-1510 (1995) · Zbl 0852.60105
[4] G. Grimmett.The random-cluster model, volume 333 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin (2006). ISBN 978-3-540-32890-2; 3-540-32890-4 · Zbl 1122.60087
[5] O. Häggström. Dynamical percolation: early results and open problems. InMicrosurveys in discrete probability (Princeton, NJ, 1997), volume 41 ofDIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 59-74. Amer. Math. Soc., Providence, RI (1998) · Zbl 0906.60081
[6] T. M. Liggett.Interacting particle systems. Classics in Mathematics. SpringerVerlag, Berlin (2005). ISBN 3-540-22617-6 · Zbl 1103.82016
[7] E. Olivieri and M. E. Vares.Large deviations and metastability, volume 100 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2005). ISBN 0-521-59163-5 · Zbl 1075.60002
[8] R. H. Schonmann. Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region.Comm. Math. Phys.161(1), 1-49 (1994) · Zbl 0796.60103
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