The Annals of Applied Probability

A shape theorem for the scaling limit of the IPDSAW at criticality

Philippe Carmona and Nicolas Pétrélis

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Abstract

In this paper we give a complete characterization of the scaling limit of the critical Interacting Partially Directed Self-Avoiding Walk (IPDSAW) introduced in Zwanzig and Lauritzen [J. Chem. Phys. 48 (1968) 3351]. As the system size $L\in\mathbb{N}$ diverges, we prove that the set of occupied sites, rescaled horizontally by $L^{2/3}$ and vertically by $L^{1/3}$ converges in law for the Hausdorff distance toward a nontrivial random set. This limiting set is built with a Brownian motion $B$ conditioned to come back at the origin at $a_{1}$ the time at which its geometric area reaches $1$. The modulus of $B$ up to $a_{1}$ gives the height of the limiting set, while its center of mass process is an independent Brownian motion.

Obtaining the shape theorem requires to derive a functional central limit theorem for the excursion of a random walk with Laplace symmetric increments conditioned on sweeping a prescribed geometric area. This result is proven in a companion paper Carmona and Pétrélis (2017).

Article information

Source
Ann. Appl. Probab., Volume 29, Number 2 (2019), 875-930.

Dates
Received: September 2017
Revised: March 2018
First available in Project Euclid: 24 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1548298933

Digital Object Identifier
doi:10.1214/18-AAP1396

Mathematical Reviews number (MathSciNet)
MR3910020

Zentralblatt MATH identifier
07047441

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B26: Phase transitions (general) 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Keywords
Polymer collapse phase transition shape theorem Brownian motion Local limit theorem

Citation

Carmona, Philippe; Pétrélis, Nicolas. A shape theorem for the scaling limit of the IPDSAW at criticality. Ann. Appl. Probab. 29 (2019), no. 2, 875--930. doi:10.1214/18-AAP1396. https://projecteuclid.org/euclid.aoap/1548298933


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References

  • Billingsley, P. (2008). Convergence of Probability Measures. Wiley, 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., 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. and Deuschel, J.-D. (2009). Scaling limits of $(1+1)$-dimensional pinning models with Laplacian interaction. Ann. Probab. 37 903–945.
  • Caravenna, F., Sun, R. and Zygouras, N. (2016). The continuum disordered pinning model. Probab. Theory Related Fields 164 17–59.
  • Carmona, P., Nguyen, G. B. and Pétrélis, N. (2016). Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase. Ann. Probab. 44 3234–3290.
  • Carmona, P. and Pétrélis, N. (2016). Interacting partially directed self avoiding walk: Scaling limits. Electron. J. Probab. 21 Paper No. 49, 52.
  • Carmona, P. and Pétrélis, N. (2017). Limit theorems for random walk excursion conditioned to have a typical area. Available at https://arxiv.org/abs/1709.06448.
  • Denisov, D., Kolb, M. and Wachtel, V. (2015). Local asymptotics for the area of random walk excursions. J. Lond. Math. Soc. (2) 91 495–513.
  • Deuschel, J.-D., Giacomin, G. and Zambotti, L. (2005). Scaling limits of equilibrium wetting models in $(1+1)$-dimension. Probab. Theory Related Fields 132 471–500.
  • Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics 31. Cambridge Univ. Press, Cambridge.
  • Foster, D. P. (1990). Exact evaluation of the collapse phase boundary for two-dimensional directed polymers. J. Phys. A: Math. Gen. 23 L1135.
  • Foster, D. P. and Yeomans, J. (1991). Competition between self-attraction and adsorption in directed self-avoiding polymers. Phys. A 177 443–452.
  • Nguyen, G. B. (2013). Marche aléatoire auto-évitante en auto-interaction. Ph.D. thesis. Thèse de doctorat dirigée par Carmona, Philippe et Pétrélis, Nicolas Mathématiques et applications Nantes.
  • 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.
  • Sohier, J. (2013). The scaling limits of a heavy tailed Markov renewal process. Ann. Inst. Henri Poincaré Probab. Stat. 49 483–505.
  • 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.