Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The high-temperature behavior for the directed polymer in dimension $1+2$

Abstract

We investigate the high-temperature behavior of the directed polymer model in dimension $1+2$. More precisely we study the difference $\Delta\mathtt{F}(\beta)$ between the quenched and annealed free energies for small values of the inverse temperature $\beta$. This quantity is associated to localization properties of the polymer trajectories, and is related to the overlap fraction of two replicas. Adapting recent techniques developed by the authors in the context of the disordered pinning model (Berger and Lacoin, 2015), we identify the sharp asymptotic high temperature behavior

$\lim_{\beta\to0}\beta^{2}\log\Delta \mathtt{F}(\beta)=-\pi.$

Résumé

Nous analysons le comportement du modèle de polymère dirigé en dimension $1+2$, dans la limite de haute température. Plus précisément, nous étudions la différence $\Delta\mathtt{F}(\beta)$ entre les énergies libres gelées et recuites, pour les petites valeurs de la température inverse $\beta$. Cette quantité est associée à des propriétés de localisation des trajectoires du polymère, et est reliée à la fraction de superposition de deux répliques. En adaptant des techniques récemment développées par les auteurs dans le contexte du modèle d’accrochage désordonné (Berger et Lacoin, 2015), nous identifions le comportement asymptotique précis dans la limite de haute température

$\lim_{\beta\to0}\beta^{2}\log\Delta\mathtt{F}(\beta)=-\pi.$

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 1 (2017), 430-450.

Dates
Revised: 23 September 2015
Accepted: 5 October 2015
First available in Project Euclid: 8 February 2017

https://projecteuclid.org/euclid.aihp/1486544897

Digital Object Identifier
doi:10.1214/15-AIHP721

Mathematical Reviews number (MathSciNet)
MR3606747

Zentralblatt MATH identifier
1362.82055

Citation

Berger, Quentin; Lacoin, Hubert. The high-temperature behavior for the directed polymer in dimension $1+2$. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 1, 430--450. doi:10.1214/15-AIHP721. https://projecteuclid.org/euclid.aihp/1486544897

References

• [1] T. Alberts, K. Khanin and J. Quastel. The intermediate disorder regime for directed polymers in dimension $1+1$. Ann. Probab. 42 (2014) 1212–1256.
• [2] K. S. Alexander and G. Yildirim. Directed polymers in a random environment with a defect line. Electron. J. Probab. 20 (2015) Article ID 6.
• [3] K. S. Alexander and N. Zygouras. Subgaussian concentration and rates of convergence in directed polymers. Electron. J. Probab. 18 (2013) Article ID 5.
• [4] Q. Berger and H. Lacoin Pinning on a defect line: Characterization of marginal disorder relevance and sharp asymptotics for the critical point shift. Preprint, 2015. Available at arXiv:1503.07315 [math-ph].
• [5] E. Bolthausen. A note on diffusion of directed polymers in a random environment. Comm. Math. Phys. 123 (1989) 529–534.
• [6] F. Caravenna, R. Sun and N. Zygouras. Polynomial chaos and scaling limits of disordered systems. J. Eur. Math. Soc. (JEMS). To appear.
• [7] F. Caravenna, R. Sun and N. Zygouras. Universality in marginally relevant disordered systems. Preprint, 2015. Available at arxiv:1510.06287 [math.PR].
• [8] F. Caravenna, F. Toninelli and N. Torri Universality for the pinning model in the weak coupling regime. Preprint, 2015. Available at arXiv:1505.04927v1 [math.PR].
• [9] P. Carmona and Y. Hu. On the partition function of a directed polymer in a random Gaussian environment. Probab. Theory Related Fields 124 (3) (2002) 431–457.
• [10] P. Carmona and Y. Hu. Strong disorder implies strong localization for directed polymers in a random environment. ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006) 217–229.
• [11] F. Comets. Weak disorder for low dimensional polymers: The model of stable laws. Markov Process. Related Fields 13 (4) (2007) 681–696.
• [12] F. Comets and V. Vargas. Majorizing multiplicative cascades for directed polymers in random media. ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006) 267–277.
• [13] F. Comets, T. Shiga and N. Yoshida. Directed polymers in a random environment: Strong disorder and path localization. Bernoulli 9 (4) (2003) 705–723.
• [14] F. Comets, T. Shiga and N. Yoshida. Probabilistic analysis of directed polymers in a random environment: A review. Adv. Stud. Pure Math. 39 (2004) 115–142.
• [15] F. Comets and N. Yoshida. Directed polymers in a random environment are diffusive at weak disorder. Ann. Probab. 34 (5) (2006) 1746–1770.
• [16] B. Derrida, G. Giacomin, H. Lacoin and F. L. Toninelli. Fractional moment bounds and disorder relevance for pinning models. Comm. Math. Phys. 287 (2009) 867–887.
• [17] G. Giacomin, H. Lacoin and F. L. Toninelli. Disorder relevance at marginality and critical point shift. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 148–175.
• [18] D. A. Huse and C. L. Henley. Pinning and roughening of domain wall in Ising systems due to random impurities. Phys. Rev. Lett. 54 (1985) 2708–2711.
• [19] H. Lacoin. New bounds for the free energy of directed polymers in dimension $1+1$ and $1+2$. Comm. Math. Phys. 294 (2010) 471–503.
• [20] M. Ledoux. The Concentration of Measure Phenomenon. American Mathematical Society, Providence, RI, 2005.
• [21] M. Nakashima. A remark on the bound for the free energy of directed polymers in random environment in $1+2$ dimension. J. Math. Phys. 55 (2014) 093304.
• [22] M. Miura, Y. Tawara and K. Tsuchida. Strong and weak disorder for Lévy directed polymers in random environment. Stoch. Anal. Appl. 26 (5) (2008) 1000–1012.
• [23] T. Sasamoto and H. Spohn. Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Phys. B 834 (2010) 523–542.
• [24] T. Sasamoto and H. Spohn. The one-dimensional KPZ equation: An exact solution and its universality. Phys. Rev. Lett. 104 (2010) 230602.
• [25] F. Watbled. Sharp asymptotics for the free energy of $1+1$ dimensional directed polymers in an infinitely divisible environment. Electron. Commun. Probab. 17 (2012) 53.