The Annals of Probability

Superdiffusivity for a Brownian polymer in a continuous Gaussian environment

Sérgio Bezerra, Samy Tindel, and Frederi Viens

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This paper provides information about the asymptotic behavior of a one-dimensional Brownian polymer in random medium represented by a Gaussian field W on ℝ+×ℝ which is white noise in time and function-valued in space. According to the behavior of the spatial covariance of W, we give a lower bound on the power growth (wandering exponent) of the polymer when the time parameter goes to infinity: the polymer is proved to be superdiffusive, with a wandering exponent exceeding any α<3/5.

Article information

Ann. Probab. Volume 36, Number 5 (2008), 1642-1675.

First available in Project Euclid: 11 September 2008

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Zentralblatt MATH identifier

Primary: 82D60: Polymers 60K37: Processes in random environments 60G15: Gaussian processes

Polymer model random medium Gaussian field free energy wandering exponent


Bezerra, Sérgio; Tindel, Samy; Viens, Frederi. Superdiffusivity for a Brownian polymer in a continuous Gaussian environment. Ann. Probab. 36 (2008), no. 5, 1642--1675. doi:10.1214/07-AOP363.

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  • [1] Albeverio, S. and Zhou, X. Y. (1996). A martingale approach to directed polymers in a random environment. J. Theoret. Probab. 9 171–189.
  • [2] Bolthausen, E. (1989). A note on the diffusion of directed polymers in a random environment. Comm. Math. Phys. 123 529–534.
  • [3] Carmona, P. and Hu, Y. (2002). On the partition function of a directed polymer in a Gaussian random environment. Probab. Theory Related Fields 124 431–457.
  • [4] Carmona, R., Koralov, L. and Molchanov, S. (2001). Asymptotics for the almost sure Lyapunov exponent for the solution of the parabolic Anderson problem. Random Oper. Stochastic Equations 9 77–86.
  • [5] Carmona, R. A. and Viens, F. G. (1998). Almost-sure exponential behavior of a stochastic Anderson model with continuous space parameter. Stochastics Stochastics Rep. 62 251–273.
  • [6] Comets, F. and Yoshida, N. (2005). Brownian directed polymers in random environment. Comm. Math. Phys. 254 257–287.
  • [7] Derrida, B. and Spohn, H. (1988). Polymers on disordered trees, spin glasses, and traveling waves. J. Statist. Phys. 51 817–840.
  • [8] Fisher, D. S. and Huse, D. A. (1991). Directed paths in random potential. Phys. Rev. B 43 10,728–10,742.
  • [9] Florescu, I. and Viens, F. (2006). Sharp estimation of the almost-sure Lyapunov exponent for the Anderson model in continuous space. Probab. Theory Related Fields 135 603–644.
  • [10] Huse, D. A. and Henley, C. L. (1985). Pinning and roughening of domain wall in Ising systems due to random impurities. Phys. Rev. Lett. 54 2708–2711.
  • [11] Krug, H. and Spohn, H. (1991). Kinetic roughening of growing surfaces. In Solids Far from Equilibrium (C. Godrèche, ed.). Cambridge Univ. Press.
  • [12] Imbrie, J. Z. and Spencer, T. (1988). Diffusion of directed polymers in a random environment. J. Statist. Phys. 52 609–626.
  • [13] Mejane, O. (2004). Upper bound of a volume exponent for directed polymers in a random environment. Ann. Inst. H. Poincaré Probab. Statist. 40 299–308.
  • [14] Petermann, M. (2000). Superdiffusivity of polymers in random environment. Ph.D. thesis, Univ. Zürich.
  • [15] Piza, M. S. T. (1997). Directed polymers in a random environment: Some results on fluctuations. J. Statist. Phys. 89 581–603.
  • [16] Rovira, C. and Tindel, S. (2005). On the Brownian-directed polymer in a Gaussian random environment. J. Funct. Anal. 222 178–201.
  • [17] Tindel, S. and Viens, F. (1999). On space-time regularity for the stochastic heat equation on Lie groups. J. Funct. Anal. 169 559–603.
  • [18] Wüthrich, M. V. (1998). Superdiffusive behavior of two-dimensional Brownian motion in a Poissonian potential. Ann. Probab. 26 1000–1015.