# Interaction Versus Entropic Repulsion for Low Temperature Ising Polymers

## 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.

## Keywords

Entropic repulsion Ising model SOS model Cluster expansion## Notes

### 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.

## References

- 1.Alili, L., Doney, R.A.: Wiener-Hopf factorization revisited and some applications. Stoch. Rep.
**66**(1–2), 87–102 (1999)CrossRefMATHMathSciNetGoogle Scholar - 2.Campanino, M., Ioffe, D., Velenik, Y.: Ornstein–Zernike theory for finite range Ising models above \(T_c\). Probab. Theory Relat. Fields
**125**(3), 305–349 (2003)CrossRefMATHMathSciNetGoogle Scholar - 3.Campanino, M., Ioffe, D., Louidor, O.: Finite connections for supercritical Bernoulli bond percolation in 2D. Mark. Proc. Relat. Fields
**16**, 225–266 (2010)MATHMathSciNetGoogle Scholar - 4.Caputo, P., Lubetzky, E., Martinelli, F., Sly, A., Toninelli, F. L.: Dynamics of 2+1 dimensional SOS surfaces above a wall: slow mixing induced by entropic repulsion, to appear on Ann. Probab., arXiv:1205.6884
- 5.Caputo, P., Lubetzky, E., Martinelli, F., Sly, A., Toninelli, F. L.: Scaling limit and cube-root fluctuations in SOS surfaces above a wall, to appear on J. Eur. Math. Soc., arXiv:1302.6941
- 6.Caputo, P., Martinelli, F., Toninelli, F. L.: On the probability of staying above a wall for the (2+1)-dimensional SOS model at low temperature, arXiv:1406.1206
- 7.Dobrushin, R., Kotecký, R., Shlosman, S.: Wulff Construction: A Global Shape from Local Interaction. American Mathematical Society, Providence (1992)MATHGoogle Scholar
- 8.Dobrushin, R., Shlosman, S.: “Non- Gibbsian” states and their Gibbs description. Commun. Math. Phys.
**200**, 125–179 (1999)CrossRefADSMATHMathSciNetGoogle Scholar - 9.Giacomin, G.: Random Polymer Models. Imperial College Press, World Scientific, London (2007)CrossRefMATHGoogle Scholar
- 10.McCoy, B., Wu, T.T.: The Two-Dimensional Ising Model. Harvard University Press, Cambridge (1973)CrossRefMATHGoogle Scholar
- 11.Ferrari, P.L., Spohn, H.: Constrained Brownian motion: fluctuations away from circular and parabolic barriers. Ann. Probab.
**33**(4), 1302–1325 (2005)CrossRefMATHMathSciNetGoogle Scholar - 12.Hryniv, O., Kotecký, R.: Surface tension and the Ornstein–Zernike behaviour for the 2D Blume–Capel model. J. Stat. Phys.
**106**(3–4), 431–476 (2003)Google Scholar - 13.Ioffe, D., Shlosman, S., Velenik, Y.: An invariance principle to Ferrari–Spohn diffusions, to appear on Comm. Math. Phys., http://arxiv.org/pdf/1403.5073v1.pdf
- 14.Ioffe, D., Shlosman, S.: In preparationGoogle Scholar
- 15.Ioffe, D., Velenik, Y.: Ballistic phase of self-interacting random walks. In: Analysis and Stochastics of Growth Processes and Interface Models, pp. 55–79. Oxford University Press, Oxford (2008)Google Scholar
- 16.Malyshev, V.A., Minlos, R.A.: Gibbs Random Fields: Cluster Expansions. Mathematics and Its Applications (Soviet Series). Springer, Dordrecht (1991)CrossRefMATHGoogle Scholar
- 17.Pfister, C.-E., Velenik, Y.: Interface, surface tension and reentrant pinning transition in the 2D Ising model. Commun. Math. Phys.
**204**(2), 269–312 (1999)CrossRefADSMATHMathSciNetGoogle Scholar - 18.Schonmann, R.H., Shlosman, S.B.: Constrained variational problem with applications to the Ising model. J. Stat. Phys.
**83**(5–6), 867–905 (1996)CrossRefADSMATHMathSciNetGoogle Scholar - 19.Isozaki, Y., Yoshida, N.: Weakly pinned random walk on the wall: pathwise descriptions of the phase transition. Stoch. Process. Appl.
**96**, 261–284 (2001)CrossRefMATHMathSciNetGoogle Scholar