Journal of Statistical Physics

, Volume 158, Issue 5, pp 1007–1050 | Cite as

Interaction Versus Entropic Repulsion for Low Temperature Ising Polymers

  • Dmitry Ioffe
  • Senya Shlosman
  • Fabio Lucio Toninelli


Contours associated to many interesting low-temperature statistical mechanics models (2D Ising model, (2+1)D SOS interface model, etc) can be described as self-interacting and self-avoiding walks on \(\mathbb Z^2\). When the model is defined in a finite box, the presence of the boundary induces an interaction, that can turn out to be attractive, between the contour and the boundary of the box. On the other hand, the contour cannot cross the boundary, so it feels entropic repulsion from it. In various situations of interest (in Caputo et al. Ann. Probab., arXiv:1205.6884, J. Eur. Math. Soc., arXiv:1302.6941, arXiv:1406.1206, Ioffe and Shlosman, in preparation), a crucial technical problem is to prove that entropic repulsion prevails over the pinning interaction: in particular, the contour-boundary interaction should not modify significantly the contour partition function and the related surface tension should be unchanged. Here we prove that this is indeed the case, at least at sufficiently low temperature, in a quite general framework that applies in particular to the models of interest mentioned above.


Entropic repulsion Ising model SOS model Cluster expansion 



F. L. T. is very grateful to P. Caputo and to F. Martinelli for countless discussions on these issues. The research of D.I. was supported by Israeli Science Foundation Grants 817/09, 1723/14 and by the Meitner Humboldt Award. The hospitality of Bonn University during the academic year 2012–2013 is gratefully acknowledged.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Dmitry Ioffe
    • 1
  • Senya Shlosman
    • 2
    • 3
  • Fabio Lucio Toninelli
    • 4
  1. 1.Faculty of IE&MTechnionHaifaIsrael
  2. 2.Aix Marseille Université, Université de Toulon, CNRS, CPT UMR 7332MarseilleFrance
  3. 3.Institute of the Information Transmission Problems, RASMoscowRussia
  4. 4.Université de Lyon, CNRS and Institut Camille JordanUniversité Lyon 1VilleurbanneFrance

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