Abstract
We study the long-range directed polymer model on \(\mathbbm {Z}\) in a random environment, where the underlying random walk lies in the domain of attraction of an \(\alpha \)-stable process for some \(\alpha \in (0,2]\). Similar to the more classic nearest-neighbor directed polymer model, as the inverse temperature \(\beta \) increases, the model undergoes a transition from a weak disorder regime to a strong disorder regime. We extend most of the important results known for the nearest-neighbor directed polymer model on \(\mathbbm {Z}^d\) to the long-range model on \(\mathbbm {Z}\). More precisely, we show that in the entire weak disorder regime, the polymer satisfies an analogue of invariance principle, while in the so-called very strong disorder regime, the polymer end point distribution contains macroscopic atoms and under some mild conditions, the polymer has a super-\(\alpha \)-stable motion. Furthermore, for \(\alpha \in (1,2]\), we show that the model is in the very strong disorder regime whenever \(\beta >0\), and we give explicit bounds on the free energy.
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Alexander, K.S., Yıldırım, G.: Directed polymers in a random environment with a defect line. Electron. J. Probab. 20(6), 1–20 (2015)
Atlagh, M., Weber, M.: Le théoreme central limite presque sûr. Expos. Math. 18(2), 097–126 (2000)
Berger, Q., Lacoin, H.: The high-temperature behavior for the directed polymer in dimension 1+2. Annales de l’Institut Henri Poincaré
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 (to appear)
Bezerra, S., Tindel, S., Viens, F., et al.: Superdiffusivity for a brownian polymer in a continuous gaussian environment. Ann. Probab. 36(5), 1642–1675 (2008)
Billingsley, P.: Convergence of Probability Measures. Wiley, New York (2013)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: 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)
F. Caravenna, R. Sun, and N. Zygouras. Polynomial chaos and scaling limits of disordered systems. J. Eur. Math. Soc. (2015, to appear)
Caravenna, F., Toninelli, F.L., Torri, N.: Universality for the pinning model in the weak coupling regime. arXiv:1505.04927 (2015)
Carmona, P., Hu, Y.: On the partition function of a directed polymer in a gaussian random environment. Probab. Theory Relat. Fields 124(3), 431–457 (2002)
Chung, K.L.: A Course in Probability Theory. Academic Press, New York (2001)
Comets, F.: Weak disorder for low dimensional polymers: the model of stable laws. Markov Process. Relat. Fields 13(4), 681–696 (2007)
Comets, F., Shiga, T., Yoshida, N.: Directed polymers in a random environment: path localization and strong disorder. Bernoulli 9(4), 705–723 (2003)
Comets, F., Shiga, T., Yoshida, N.: Probabilistic analysis of directed polymers in a random environment: a review. Adv. Stud. Pure Math. 39, 115–142 (2004)
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. 34, 1746–1770 (2006)
Hollander, F.D.: Random Polymers: École dÉté de Probabilités de Saint-Flour XXXVII–2007. Springer, Berlin (2009)
Dudley, R.M.: Real Analysis and Probability, vol. 74. Cambridge University Press, Cambridge (2002)
Giacomin, G., Toninelli, F., Lacoin, H.: Marginal relevance of disorder for pinning models. Commun. Pure Appl. Math. 63(2), 233–265 (2010)
Gut, A.: Probability: A Graduate Course. Springer Science & Business Media, New York (2012)
Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Application. Academic Press, New York (2014)
Huse, D.A., Henley, C.L.: Pinning and roughening of domain walls in ising systems due to random impurities. Phys. Rev. Lett. 54(25), 2708 (1985)
Imbrie, J.Z., Spencer, T.: Diffusion of directed polymers in a random environment. J. Stat. Phys. 52(3–4), 609–626 (1988)
Jonsson, F.: Almost sure central limit theory. Department of Mathematics, Project Report, Uppsala University (2007)
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)
Lacoin, H.: Influence of spatial correlation for directed polymers. Ann. Probab. 39(1), 139–175 (2011)
Miura, M., Tawara, Y., Tsuchida, K.: Strong and weak disorder for lévy directed polymers in random environment. Stoch. Anal. Appl. 26(5), 1000–1012 (2008)
Newell, G., Gnedenko, B.V., Kolmogorov, A.N., Chung, K.L.: Limit distributions for sums of independent random variables. JSTOR (1955)
Resnick, S.I.: Point processes, regular variation and weak convergence. Adv. Appl. Probab. 18, 66–138 (1986)
Toninelli, F.L.: Coarse graining, fractional moments and the critical slope of random copolymers. Electron. J. Probab 14(20), 531–547 (2009)
Vargas, V.: Strong localization and macroscopic atoms for directed polymers. Probab. Theory Relat. Fields 138(3–4), 391–410 (2007)
Acknowledgments
I would like to acknowledge support from AcRF Tier 1 Grant R-146-000-220-112. I am deeply indebted to my supervisor Prof. Rongfeng Sun, who introduced this topic to me, provided suggestions and discussed with me and I am grateful to my classmate Jinjiong Yu for helpful discussion. I also want to thank two referees, whose comments help me add the super-\(\alpha \)-stable result and correct some mistakes, which improve an early version of this paper.
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Wei, R. On the Long-Range Directed Polymer Model. J Stat Phys 165, 320–350 (2016). https://doi.org/10.1007/s10955-016-1612-y
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DOI: https://doi.org/10.1007/s10955-016-1612-y
Keywords
- Long-range directed polymer
- Free energy
- Strong disorder
- Weak disorder
- Invariance principle
- Coarse graining
- Localization
- Super-\(\alpha \)-stable motion