Source: Braz. J. Probab. Stat. Volume 24, Number 2
(2010), 279-299.
We describe some recent results concerning the statistical properties of a self-interacting polymer stretched by an external force. We concentrate mainly on the cases of purely attractive or purely repulsive self-interactions, but our results are stable under suitable small perturbations of these pure cases. We provide in particular a precise description of the stretched phase (local limit theorems for the endpoint and local observables, invariance principle, microscopic structure). Our results also characterize precisely the (nontrivial, direction-dependent) critical force needed to trigger the collapsed/stretched phase transition in the attractive case. We also describe some recent progress: first, the determination of the order of the phase transition in the attractive case; second, a proof that a semi-directed polymer in quenched random environment is diffusive in dimensions 4 and higher when the temperature is high enough. In addition, we correct an incomplete argument from Ioffe and Velenik [In Analysis and Stochastics of Growth Processes and Interface Models (2008) 55–79].
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References
[1] Bolthausen, E. (1994). Localization of a two-dimensional random walk with an attractive path interaction., The Annals of Probability 22 875–918.
[2] de Gennes, P.-G. (1979)., Scaling Concepts in Polymer Physics. Cornell Univ. Press, Ithaca, NY.
[3] Eisele, T. and Lang, R. (1987). Asymptotics for the Wiener sausage with drift., Probability Theory and Related Fields 74 125–140.
Mathematical Reviews (MathSciNet):
MR863722
[4] Flury, M. (2007a). Coincidence of lyapunov exponents for random walks in weak random potentials. Preprint. Available at,
arXiv:math/0608357.
[5] Flury, M. (2007b). Large deviations and phase transition for random walks in random nonnegative potentials., Stochastic Processes and Their Applications 117 596–612.
[6] Grassberger, P. and Procaccia, I. (1982). Diffusion and drift in a medium with randomly distributed traps., Physical Review A 26 3686–3688.
[7] Ioffe, D. and Velenik, Y. (2008). Ballistic phase of self-interacting random walks. In, Analysis and Stochastics of Growth Processes and Interface Models (P. Morters, R. Moser, M. Penrose, H. Schwetlick and J. Zimmer, eds.) 55–79. Oxford Univ. Press.
[8] Madras, N. and Slade, G. (1993)., The Self-Avoiding Walk. Probability and Its Applications. Birkhäuser Boston, Boston, MA.
[9] Maggs, A. C., Leibler, S., Fisher, M. E. and Camacho, C. J. (1990). Size of an inflated vesicle in two dimensions., Physical Review A 42 691–695.
[10] Mehra, V. and Grassberger, P. (2002). Transition to localization of biased walkers in a randomly absorbing environment., Physica D-Nonlinear Phenomena 168 244–257.
[11] Povel, T. (1995). On weak convergence of conditional survival measure of one-dimensional Brownian motion with a drift., The Annals of Applied Probability 5 222–238.
[12] Povel, T. (1997). Critical large deviations of one-dimensional annealed Brownian motion in a Poissonian potential., The Annals of Probability 25 1735–1773.
[13] Povel, T. (1998). The one-dimensional annealed, δ-Lyapunov exponent. Annales de l’Institute Henri Poincaré. Probabilités et Statistiques 34 61–72.
[14] Povel, T. (1999). Confinement of Brownian motion among Poissonian obstacles in, Rd, d≥3. Probability Theory and Related Fields 114 177–205.
[15] Rubinstein, M. and Colby, R. H. (2003)., Polymer Physics. Oxford Univ. Press, New York.
[16] Sznitman, A.-S. (1991). On the confinement property of two-dimensional Brownian motion among Poissonian obstacles., Communications on Pure and Applied Mathematics 44 1137–1170.
[17] Sznitman, A.-S. (1998)., Brownian Motion, Obstacles and Random Media. Springer-Verlag, Berlin.
[18] Zygouras, N. (2009). Lyapounov norms for random walks in low disorder and dimension greater than three., Probability Theory and Related Fields 143 615–642.
