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Tunneling in Two Dimensions

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Abstract

Tunneling is studied here as a variational problem formulated in terms of a functional which approximates the rate function for large deviations in Ising systems with Glauber dynamics and Kac potentials, [9]. The spatial domain is a two-dimensional square of side L with reflecting boundary conditions. For L large enough the penalty for tunneling from the minus to the plus equilibrium states is determined. Minimizing sequences are fully characterized and shown to have approximately a planar symmetry at all times, thus departing from the Wulff shape in the initial and final stages of the tunneling. In a final section (Sect. 11), we extend the results to d = 3 but their validity in d > 3 is still open.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133, Roma, Italy

    G. Bellettini & E. Presutti

  2. INFN Laboratori Nazionali di Frascati, Frascati, Italy

    G. Bellettini

  3. Dipartimento di Matematica, Università di L’Aquila, via Vetoio, loc. Coppito, 67100, l’Aquila, Italy

    A. De Masi

  4. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103, Leipzig, Germany

    N. Dirr

Authors
  1. G. Bellettini
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  2. A. De Masi
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  3. N. Dirr
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  4. E. Presutti
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Corresponding author

Correspondence to E. Presutti.

Additional information

Communicated by J.L. Lebowitz

This research has been partially supported by MURST and NATO Grant PST.CLG.976552 and COFIN, Prin n.2004028108.

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Bellettini, G., Masi, A.D., Dirr, N. et al. Tunneling in Two Dimensions. Commun. Math. Phys. 269, 715–763 (2007). https://doi.org/10.1007/s00220-006-0143-9

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  • DOI: https://doi.org/10.1007/s00220-006-0143-9

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