Interaction Versus Entropic Repulsion for Low Temperature Ising Polymers

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• 作者：Dmitry Ioffe (1)
  Senya Shlosman (2) (3)
  Fabio Lucio Toninelli (4)
  
  1. Faculty of IE&M ; Technion ; 32000 ; Haifa ; Israel
  2. Aix Marseille Universit茅 ; Universit茅 de Toulon ; CNRS ; CPT UMR 7332 ; 13288 ; Marseille ; France
  3. Institute of the Information Transmission Problems ; RAS ; Moscow ; Russia
  4. Universit茅 de Lyon ; CNRS and Institut Camille Jordan ; Universit茅 Lyon 1 ; 43 bd du 11 novembre 1918 ; 69622 ; Villeurbanne ; France
  
• 关键词：Entropic repulsion ; Ising model ; SOS model ; Cluster expansion
• 刊名：Journal of Statistical Physics
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：158
• 期：5
• 页码：1007-1050
• 全文大小：608 KB
• 参考文献：1. Alili, L., Doney, R.A.: Wiener-Hopf factorization revisited and some applications. Stoch. Rep. 66(1鈥?), 87鈥?02 (1999) CrossRef
  2. Campanino, M., Ioffe, D., Velenik, Y.: Ornstein鈥揨ernike theory for finite range Ising models above \(T_c\) . Probab. Theory Relat. Fields 125(3), 305鈥?49 (2003) CrossRef
  3. Campanino, M., Ioffe, D., Louidor, O.: Finite connections for supercritical Bernoulli bond percolation in 2D. Mark. Proc. Relat. Fields 16, 225鈥?66 (2010)
  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., Koteck媒, R., Shlosman, S.: Wulff Construction: A Global Shape from Local Interaction. American Mathematical Society, Providence (1992)
  8. Dobrushin, R., Shlosman, S.: 鈥淣on- Gibbsian鈥?states and their Gibbs description. Commun. Math. Phys. 200, 125鈥?79 (1999) CrossRef
  9. Giacomin, G.: Random Polymer Models. Imperial College Press, World Scientific, London (2007) CrossRef
  10. McCoy, B., Wu, T.T.: The Two-Dimensional Ising Model. Harvard University Press, Cambridge (1973) CrossRef
  11. Ferrari, P.L., Spohn, H.: Constrained Brownian motion: fluctuations away from circular and parabolic barriers. Ann. Probab. 33(4), 1302鈥?325 (2005) CrossRef
  12. Hryniv, O., Koteck媒, R.: Surface tension and the Ornstein鈥揨ernike behaviour for the 2D Blume鈥揅apel model. J. Stat. Phys. 106(3鈥?), 431鈥?76 (2003)
  13. Ioffe, D., Shlosman, S., Velenik, Y.: An invariance principle to Ferrari鈥揝pohn diffusions, to appear on Comm. Math. Phys., http://arxiv.org/pdf/1403.5073v1.pdf
  14. Ioffe, D., Shlosman, S.: In preparation
  15. Ioffe, D., Velenik, Y.: Ballistic phase of self-interacting random walks. In: Analysis and Stochastics of Growth Processes and Interface Models, pp. 55鈥?9. Oxford University Press, Oxford (2008)
  16. Malyshev, V.A., Minlos, R.A.: Gibbs Random Fields: Cluster Expansions. Mathematics and Its Applications (Soviet Series). Springer, Dordrecht (1991) CrossRef
  17. Pfister, C.-E., Velenik, Y.: Interface, surface tension and reentrant pinning transition in the 2D Ising model. Commun. Math. Phys. 204(2), 269鈥?12 (1999) CrossRef
  18. Schonmann, R.H., Shlosman, S.B.: Constrained variational problem with applications to the Ising model. J. Stat. Phys. 83(5鈥?), 867鈥?05 (1996) CrossRef
  19. Isozaki, Y., Yoshida, N.: Weakly pinned random walk on the wall: pathwise descriptions of the phase transition. Stoch. Process. Appl. 96, 261鈥?84 (2001) CrossRef
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Statistical Physics
  Mathematical and Computational Physics
  Physical Chemistry
  Quantum Physics
  
• 出版者：Springer Netherlands
• ISSN：1572-9613

文摘

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.