Interaction Versus Entropic Repulsion for Low Temperature Ising Polymers
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.
KeywordsEntropic 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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