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Front evolution of the Fredrickson-Andersen one spin facilitated model. (English) Zbl 1406.60127

Summary: The Fredrickson-Andersen one spin facilitated model (FA-1f) on \(\mathbb Z\) belongs to the class of kinetically constrained spin models (KCM). Each site refreshes with rate one its occupation variable to empty (respectively occupied) with probability \(q\) (respectively \(p=1-q\)), provided at least one nearest neighbor is empty. Here, we study the non equilibrium dynamics of FA-1f started from a configuration entirely occupied on the left half-line and focus on the evolution of the front, namely the position of the leftmost zero. We prove, for \(q\) larger than a threshold \(\bar q<1\), a law of large numbers and a central limit theorem for the front, as well as the convergence to an invariant measure of the law of the process seen from the front.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J27 Continuous-time Markov processes on discrete state spaces
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[1] [AD02] D. Aldous and P. Diaconis. The asymmetric one-dimensional constrained ising model: rigorous results. J. Stat. Phys., 107(5-6):945?975, 2002. · Zbl 1006.60095 · doi:10.1023/A:1015170205728
[2] [BCM+13] O. Blondel, N. Cancrini, F. Martinelli, C. Roberto, and C. Toninelli. Fredrickson-Andersen one spin facilitated model out of equilibrium. Markov Process. Related Fields, 19(3):383-406, 2013. · Zbl 1321.82025
[3] [BD88] M.A. Burschka and R. Dickman. Nonequilibrium critical poisoning in a single-species model. Physics Letters A, 127(3):132-137, 1988.
[4] [BFM78] R.C. Brower, M.A. Furman, and M. Moshe. Critical exponents for the reggeon quantum spin model. Physics Letters, 76B:213-219, 1978.
[5] [Blo13] Oriane Blondel. Front progression in the East model. Stochastic Process. Appl., 123(9):3430-3465, 2013. · Zbl 1291.60199 · doi:10.1016/j.spa.2013.04.014
[6] [Bol82] E. Bolthausen. On the central limit theorem for stationary mixing random fields. Ann. Probab., 10(4):1047-1050, 1982. · Zbl 0496.60020 · doi:10.1214/aop/1176993726
[7] [DG83] Richard Durrett and David Griffeath. Supercritical contact processes on \(\bf Z\). Ann. Probab., 11(1):1-15, 1983. · Zbl 0508.60080 · doi:10.1214/aop/1176993655
[8] [Dur80] Richard Durrett. On the growth of one-dimensional contact processes. Ann. Probab., 8(5):890-907, 1980. · Zbl 0457.60082 · doi:10.1214/aop/1176994619
[9] [GK11] Olivier Garet and Aline Kurtzmann. De l’intégration aux probabilités. Références sciences. Ellipses Marketing, 2011. · Zbl 1327.28001
[10] [GLM15] S. Ganguly, E. Lubetzky, and F. Martinelli. Cutoff for the east process. Comm. Math. Phys., 335(3):1287-1322, 2015. · Zbl 1408.82007
[11] [Har74] T. E. Harris. Contact interactions on a lattice. Ann. Probability, 2:969-988, 1974. · Zbl 0334.60052
[12] [LPW09] David A. Levin, Yuval Peres, and Elizabeth L. Wilmer. Markov chains and mixing times. American Mathematical Society, Providence, RI, 2009. With a chapter by James G. Propp and David B. Wilson. · Zbl 1160.60001
[13] [MV18] T. Mountford and G. Valle. Exponential convergence for the fredrikson-andersen one spin facilitated model. ArXiv e-prints 1609.01364, 2018.
[14] [RS03] F. Ritort and P. Sollich. Glassy dynamics of kinetically constrained models. Advances in Physics, 52(4):219?342, 2003.
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