Abstract
We consider the continuous time, zero-temperature heat-bath dynamics for the nearest-neighbor Ising model on \(\mathbb Z^2\) with positive magnetic field. For a system of size \(L\in{\mathbb N}\), we start with initial condition σ such that \(\sigma_x=-1\) if \(x\in[-L,L]^2\) and \(\sigma_x=+1\) and investigate the scaling limit of the set of • spins when both time and space are rescaled by L. We compare the obtained result and its proof with the case of zero-magnetic fields, for which a scaling result was proved by Lacoin et al. (J Eur Math Soc, in press). In that case, the time-scaling is diffusive and the scaling limit is given by anisotropic motion by curvature.
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Acknowledgments
The author would like to thank the organizers of the 2012 PASI conference, where he had stimulating discussion with other participants, Milton Jara, for valuable bibliographic help concerning TASEP, and François Simenhaus and Fabio Toninelli for numerous enlightening discussions on the subject. He is also grateful to the anonymous referee for his detailed report. This work was partially written during the author’s stay at Instituto de Matematica Pura e Applicada, he acknowledges the kind hospitality and the support of CNPq.
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Lacoin, H. (2014). The Scaling Limit for Zero-Temperature Planar Ising Droplets: With and Without Magnetic Fields. In: Ramírez, A., Ben Arous, G., Ferrari, P., Newman, C., Sidoravicius, V., Vares, M. (eds) Topics in Percolative and Disordered Systems. Springer Proceedings in Mathematics & Statistics, vol 69. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0339-9_4
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DOI: https://doi.org/10.1007/978-1-4939-0339-9_4
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