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Phase coexistence of gradient Gibbs states
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  • Published: 11 January 2007

Phase coexistence of gradient Gibbs states

  • Marek Biskup1 &
  • Roman Kotecký2 

Probability Theory and Related Fields volume 139, pages 1–39 (2007)Cite this article

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Abstract

We consider the (scalar) gradient fields η a  = (η b )—with b denoting the nearest-neighbor edges in \(\mathbb{Z}^{2}\)—that are distributed according to the Gibbs measure proportional to \(\hbox{e}^{-\beta H(\eta)}\nu(d \eta)\). Here H = ∑ b V(η b ) is the Hamiltonian, V is a symmetric potential, β > 0 is the inverse temperature, and ν is the Lebesgue measure on the linear space defined by imposing the loop condition \(\eta_{b_1}+\eta_{b_2}=\eta_{b_3}+\eta_{b_4}\) for each plaquette (b 1,b 2,b 3,b 4) in \(\mathbb{Z}^{2}\). For convex V, Funaki and Spohn have shown that ergodic infinite-volume Gibbs measures are characterized by their tilt. We describe a mechanism by which the gradient Gibbs measures with non-convex V undergo a structural, order-disorder phase transition at some intermediate value of inverse temperature β. At the transition point, there are at least two distinct gradient measures with zero tilt, i.e., E η b  = 0.

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Authors and Affiliations

  1. Department of Mathematics, UCLA, Los Angeles, CA, 90095-1555, USA

    Marek Biskup

  2. Center for Theoretical Study, Charles University, 11000, Prague, Czech Republic

    Roman Kotecký

Authors
  1. Marek Biskup
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  2. Roman Kotecký
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Correspondence to Marek Biskup.

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Biskup, M., Kotecký, R. Phase coexistence of gradient Gibbs states. Probab. Theory Relat. Fields 139, 1–39 (2007). https://doi.org/10.1007/s00440-006-0013-6

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  • Received: 22 February 2005

  • Revised: 05 April 2006

  • Published: 11 January 2007

  • Issue Date: September 2007

  • DOI: https://doi.org/10.1007/s00440-006-0013-6

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Keywords

  • Partition Function
  • Thermodynamic Limit
  • Gibbs Measure
  • Phase Coexistence
  • Loop Condition
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