Abstract
We study the Cauchy directed polymer model on \(\mathbb {Z}^{1+1}\), where the underlying random walk is in the domain of attraction to the 1-stable law. We show that, if the random walk satisfies certain regularity assumptions and its symmetrized version is recurrent, then the free energy is strictly negative at any inverse temperature \(\beta >0\). Moreover, under additional regularity assumptions on the random walk, we can identify the sharp asymptotics of the free energy in the high temperature limit, namely,
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berger, Q.: Notes on random walks in the Cauchy domain of attraction (2017). arXiv preprint arXiv:1706.07924
Berger, Q., Lacoin, H.: Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift. J. Inst. Math. Jussieu, 1–42 (2016)
Berger, Q., Lacoin, H.: The high-temperature behavior for the directed polymer in dimension 1+ 2. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 53(1), 430–450 (2017)
Bingham, N., Goldie, C., Teugels, J.: Regular Variation, vol. 27. Cambridge University Press, Cambridge (1989)
Bolthausen, E.: A note on the diffusion of directed polymers in a random environment. Commun. Math. Phys. 123(4), 529–534 (1989)
Caravenna, F., Toninelli, F.L., Torri, N.: Universality for the pinning model in the weak coupling regime (2015). arXiv:1505.04927
Comets, F.: Weak disorder for low dimensional polymers: the model of stable laws. Markov Process. Relat. Fields 13(4), 681–696 (2007)
Comets, F.: Directed Polymers in Random Environments: École d’Été de Probabilités de Saint-Flour XLVI–2016, vol. 2175. Springer (2017)
Comets, F., Vargas, V.: Majorizing multiplicative cascades for directed polymers in random media. Alea 2, 267–277 (2006)
Comets, F., Yoshida, N.: Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 1746–1770 (2006)
Comets, F., Shiga, T., Yoshida, N., et al.: Directed polymers in a random environment: path localization and strong disorder. Bernoulli 9(4), 705–723 (2003)
Comets, F., Shiga, T., Yoshida, N., et al.: Probabilistic analysis of directed polymers in a random environment: a review. Stoch. Anal. Large Scale Interact. Syst. 39, 115–142 (2004)
Den Hollander, F.: Random Polymers: École dÉté de Probabilités de Saint-Flour XXXVII-2007. Springer, New Year (2009)
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2. Wiley, New York (1966)
Lacoin, H.: New bounds for the free energy of directed polymers in dimension 1+ 1 and 1+ 2. Commun. Math. Phys. 294(2), 471–503 (2010)
Wei, R.: On the long-range directed polymer model. J. Stat. Phys. 165(2), 320–350 (2016)
Acknowledgements
The author would like to acknowledge support from AcRF Tier 1 grant R-146-000-220-112. The author also wants to thank Professor Rongfeng Sun for introducing and discussing this topic, and helping revise this paper. The author is especially grateful to Professor Quentin Berger for sharing his manuscript [1], in particular, the proof of Theorem 1.2 before publication. The author also thanks Professor Francesco Caravenna, who helped the author obtain more insight into this problem when he visited Singapore. Finally, the author would like to thank the unknown referees, who helped the author remove some unnatural assumptions on the underlying random walk S and improve the quality of this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wei, R. Free Energy of the Cauchy Directed Polymer Model at High Temperature. J Stat Phys 172, 1057–1085 (2018). https://doi.org/10.1007/s10955-018-2086-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-018-2086-x