The Annals of Probability

Crossing random walks and stretched polymers at weak disorder

Dmitry Ioffe and Yvan Velenik

Full-text: Open access

Abstract

We consider a model of a polymer in ℤd+1, constrained to join 0 and a hyperplane at distance N. The polymer is subject to a quenched nonnegative random environment. Alternatively, the model describes crossing random walks in a random potential (see Zerner [Ann Appl. Probab. 8 (1998) 246–280] or Chapter 5 of Sznitman [Brownian Motion, Obstacles and Random Media (1998) Springer] for the original Brownian motion formulation). It was recently shown [Ann. Probab. 36 (2008) 1528–1583; Probab. Theory Related Fields 143 (2009) 615–642] that, in such a setting, the quenched and annealed free energies coincide in the limit N → ∞, when d ≥ 3 and the temperature is sufficiently high. We first strengthen this result by proving that, under somewhat weaker assumptions on the distribution of disorder which, in particular, enable a small probability of traps, the ratio of quenched and annealed partition functions actually converges. We then conclude that, in this case, the polymer obeys a diffusive scaling, with the same diffusivity constant as the annealed model.

Article information

Source
Ann. Probab., Volume 40, Number 2 (2012), 714-742.

Dates
First available in Project Euclid: 26 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1332772718

Digital Object Identifier
doi:10.1214/10-AOP625

Mathematical Reviews number (MathSciNet)
MR2952089

Zentralblatt MATH identifier
1251.60074

Keywords
Polymer central limit theorem diffusivity Ornstein–Zernike theory quenched random environment

Citation

Ioffe, Dmitry; Velenik, Yvan. Crossing random walks and stretched polymers at weak disorder. Ann. Probab. 40 (2012), no. 2, 714--742. doi:10.1214/10-AOP625. https://projecteuclid.org/euclid.aop/1332772718


Export citation

References

  • [1] Bolthausen, E. (1989). A note on the diffusion of directed polymers in a random environment. Comm. Math. Phys. 123 529–534.
  • [2] Campanino, M., Ioffe, D. and Louidor, O. (2010). Finite connections for supercritical Bernoulli bond percolation in 2D. Markov Process. Related Fields 16 225–266.
  • [3] Carmona, P. and Hu, Y. (2006). Strong disorder implies strong localization for directed polymers in a random environment. ALEA Lat. Am. J. Probab. Math. Stat. 2 217–229.
  • [4] Comets, F. and Yoshida, N. (2006). Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 1746–1770.
  • [5] Flury, M. (2008). Coincidence of Lyapunov exponents for random walks in weak random potentials. Ann. Probab. 36 1528–1583.
  • [6] Ioffe, D. and Velenik, Y. (2008). Ballistic phase of self-interacting random walks. In Analysis and Stochastics of Growth Processes and Interface Models 55–79. Oxford Univ. Press, Oxford.
  • [7] Kaup, L. and Kaup, B. (1983). Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory. de Gruyter Studies in Mathematics 3. de Gruyter, Berlin.
  • [8] Sinai, Y. G. (1995). A remark concerning random walks with random potentials. Fund. Math. 147 173–180.
  • [9] Sznitman, A.-S. (1998). Brownian Motion, Obstacles and Random Media. Springer, Berlin.
  • [10] Zerner, M. P. W. (1998). Directional decay of the Green’s function for a random nonnegative potential on Zd. Ann. Appl. Probab. 8 246–280.
  • [11] Zygouras, N. (2009). Lyapounov norms for random walks in low disorder and dimension greater than three. Probab. Theory Related Fields 143 615–642.