## The Annals of Probability

### Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase

#### Abstract

In this paper, we investigate a model for a $1+1$ dimensional self-interacting and partially directed self-avoiding walk, usually referred to by the acronym IPDSAW. The interaction intensity and the free energy of the system are denoted by $\beta$ and $f$, respectively. The IPDSAW is known to undergo a collapse transition at $\beta_{c}$. We provide the precise asymptotic of the free energy close to criticality, that is, we show that $f(\beta_{c}-\varepsilon)\sim\gamma\varepsilon^{3/2}$ where $\gamma$ is computed explicitly and interpreted in terms of an associated continuous model. We also establish some path properties of the random walk inside the collapsed phase $(\beta>\beta_{c})$. We prove that the geometric conformation adopted by the polymer is made of a succession of long vertical stretches that attract each other to form a unique macroscopic bead and we establish the convergence of the region occupied by the path properly rescaled toward a deterministic Wulff shape.

#### Article information

Source
Ann. Probab., Volume 44, Number 5 (2016), 3234-3290.

Dates
Received: February 2014
Revised: July 2015
First available in Project Euclid: 21 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1474462097

Digital Object Identifier
doi:10.1214/15-AOP1046

Mathematical Reviews number (MathSciNet)
MR3551196

Zentralblatt MATH identifier
1360.60173

#### Citation

Carmona, Philippe; Nguyen, Gia Bao; Pétrélis, Nicolas. Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase. Ann. Probab. 44 (2016), no. 5, 3234--3290. doi:10.1214/15-AOP1046. https://projecteuclid.org/euclid.aop/1474462097

#### References

• Alexander, K. S. and Sidoravicius, V. (2006). Pinning of polymers and interfaces by random potentials. Ann. Appl. Probab. 16 636–669.
• Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Applications of Mathematics (New York) 51. Springer, New York.
• Brak, R., Guttmann, A. J. and Whittington, S. G. (1992). A collapse transition in a directed walk model. J. Phys. A 25 2437–2446.
• Brak, R., Owczarek, A. L. and Prellberg, T. (1993). The tricritical behavior of self-interacting partially directed walks. J. Stat. Phys. 72 737–772.
• Brak, R., Owczarek, A. L., Prellberg, T. and Guttmann, A. J. (1993). Finite-length scaling of collapsing directed walks. Phys. Rev. E 48 2386–2396.
• Brak, R., Dyke, P., Lee, J., Owczarek, A. L., Prellberg, T., Rechnitzer, A. and Whittington, S. G. (2009). A self-interacting partially directed walk subject to a force. J. Phys. A 42 085001, 30.
• Caravenna, F., den Hollander, F. and Pétrélis, N. (2012). Lectures on random polymers. In Probability and Statistical Physics in Two and More Dimensions. Clay Math. Proc. 15 319–393. Amer. Math. Soc., Providence, RI.
• Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer, Berlin.
• Derrida, B., Hakim, V. and Vannimenus, J. (1992). Effect of disorder on two-dimensional wetting. J. Stat. Phys. 66 1189–1213.
• Derrida, B., Giacomin, G., Lacoin, H. and Toninelli, F. L. (2009). Fractional moment bounds and disorder relevance for pinning models. Comm. Math. Phys. 287 867–887.
• Dobrushin, R. and Hryniv, O. (1996). Fluctuations of shapes of large areas under paths of random walks. Probab. Theory Related Fields 105 423–458.
• Dobrushin, R., Kotecký, R. and Shlosman, S. (1992). Wulff Construction: A Global Shape from Local Interaction. Translations of Mathematical Monographs 104. Amer. Math. Soc., Providence, RI.
• Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Univ. Press, Cambridge.
• Giacomin, G. (2007). Random Polymer Models. Imperial College Press, London.
• Giacomin, G. (2011). Disorder and Critical Phenomena Through Basic Probability Models. Lecture Notes in Math. 2025. Springer, Heidelberg.
• Hryniv, O. and Velenik, Y. (2004). Universality of critical behaviour in a class of recurrent random walks. Probab. Theory Related Fields 130 222–258.
• Ioffe, D. (1994). Large deviations for the $2$D Ising model: A lower bound without cluster expansions. J. Stat. Phys. 74 411–432.
• Ioffe, D. (1995). Exact large deviation bounds up to $T_{c}$ for the Ising model in two dimensions. Probab. Theory Related Fields 102 313–330.
• Ioffe, D. and Schonmann, R. H. (1998). Dobrushin–Kotecký–Shlosman theorem up to the critical temperature. Comm. Math. Phys. 199 117–167.
• Janson, S. (2007). Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas. Probab. Surv. 4 80–145.
• Kac, M. (1946). On the average of a certain Wiener functional and a related limit theorem in calculus of probability. Trans. Amer. Math. Soc. 59 401–414.
• Nguyen, G. B. and Pétrélis, N. (2013). A variational formula for the free energy of the partially directed polymer collapse. J. Stat. Phys. 151 1099–1120.
• Owczarek, A. L. and Prellberg, T. (2007). Exact solution of semi-flexible and super-flexible interacting partially directed walks. J. Stat. Mech. Theory Exp. P11010 1–14.
• Sakhanenko, A. I. (1980). On unimprovable estimates of the rate of convergence in the invariance principle. In Colloq. Math. Soc. János Bolyai. Nonparametric Statistical Inference 32 779–783. North Holland Publishing Company, Amsterdam.
• Samanta, H. S. and Thirumalai, D. (2013). Exact solution of the Zwanzig–Lauritzen model of polymer crystallization under tension. J. Chem. Phys. 138 104901.
• Shao, Q. M. (1995). Strong approximation theorems for independent random variables and their applications. J. Multivariate Anal. 52 107–130.
• Takács, L. (1993). On the distribution of the integral of the absolute value of the Brownian motion. Ann. Appl. Probab. 3 186–197.
• Torri, N. (2015). Pinning model with heavy tailed disorder. Stochastic Process. Appl. Avalaible online 21 September 2015.
• van der Hofstad, R., den Hollander, F. and König, W. (2003). Weak interaction limits for one-dimensional random polymers. Probab. Theory Related Fields 125 483–521.
• Wulff, G. (1901). Zur frage der Geschwindigkeit des Wachstums und der auflosüng der krystallflagen. Z. Kryst. Mineral. 34 449.
• Zwanzig, R. and Lauritzen, J. I. (1968). Exact calculation of the partition function for a model of two dimensional polymer crystallization by chain folding. J. Chem. Phys. 48 3351.