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Interaction Versus Entropic Repulsion for Low Temperature Ising Polymers

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Abstract

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.

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Notes

  1. Indeed, the polymer-wall contact fraction in the thermodynamic limit can be obtained in the following way. One has to add to \(\tilde{\Phi }\) a term equal to the number of polymer-wall contacts times the parameter \(\alpha >0\), and then compute the derivative w.r.t. \(\alpha \), at the point \(\alpha =0\), of the infinite-volume surface tension. It easily follows from (2.4) that such derivative is zero.

  2. Note that since steps of effective random walk may have a negative components, the events \({\mathfrak {A}_{m}} (0 ,\mathsf {v})\) and \({\mathfrak {A}_{m}} ( \mathsf {v}, 0 )\) may be non-empty even if \(m > \left| \mathsf {v}\right| _1\).

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Acknowledgments

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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Correspondence to Fabio Lucio Toninelli.

Appendix: A Correction to [7]

Appendix: A Correction to [7]

The first motivation for one of the authors of the present paper (S.S.) was to correct the mistake in the Wulff construction book [7]. Namely, one statement in that book—the Theorem 4.16, dealing with spatial sensitivity of the surface tension—is not correct; more precisely, the upper bound statement 4.19 is erroneous. This mistake was uncovered by the authors of the paper [4]. But the reader of the present paper should not think that some forty pages have to be added to [7] in order to correct it, because a weaker version of the Theorem 4.16 is quite sufficient to get all other results of [7]. We will give here the formulation of this weaker statement, in the notations of the book [7]:

Theorem 29

Theorem 4.16 of [7] holds for \(\bar{V}_{N}=U_{N,d,\varkappa },\) with \(d<\bar{d}/2,\) i.e. when the change from the interaction \(\Phi \) to \(\tilde{\Phi }\) happens far away from the range \(\bar{V}\) of the random contour.

In terms of the present paper, the meaning of the above statement is that the surface tension does not change if the interaction is perturbed far from the range of the contour. For example, if we compute the surface tension over the polymers \(\Gamma _{N}\) fitting a strip \(\left\{ x,y:\left| y-\varkappa x\right| <\frac{1}{2}N^{\alpha }\right\} ,\) but perturb the interactions \(\Phi _{\beta }\left( \mathcal {C}\right) \) only if \(\mathcal {C}\) does not fit the wider strip \(\left\{ x,y:\left| y-\varkappa x\right| <N^{\alpha }\right\} ,\) then the claim that the surface tension is unaffected by the perturbation holds true, and is easy to prove. For a motivated reader of [7], who reached Theorem 4.16 of it, the proof of the above statement and the check that it is sufficient for all the needs of the book, will be an easy exercise.

But the problem of spatial sensitivity of the surface tension in its stronger form of Theorem 1 is important in various applications and is of independent interest.

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Ioffe, D., Shlosman, S. & Toninelli, F.L. Interaction Versus Entropic Repulsion for Low Temperature Ising Polymers. J Stat Phys 158, 1007–1050 (2015). https://doi.org/10.1007/s10955-014-1153-1

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