### An almost sure CLT for stretched polymers

**Dmitry Ioffe**

*(Technion Haifa)*

**Yvan Velenik**

*(Université de Genève)*

#### Abstract

We prove an almost sure central limit theorem (CLT) for spatial extension of stretched (meaning subject to a non-zero pulling force) polymers at very weak disorder in all dimensions $d+1\geq 4$.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-20

Publication Date: November 11, 2013

DOI: 10.1214/EJP.v18-2231

#### References

- Bolthausen, Erwin. A note on the diffusion of directed polymers in a random
environment.
*Comm. Math. Phys.*123 (1989), no. 4, 529--534. MR1006293 - Comets, Francis; Yoshida, Nobuo. Directed polymers in random environment are diffusive at weak
disorder.
*Ann. Probab.*34 (2006), no. 5, 1746--1770. MR2271480 - Comets, Francis; Vargas, Vincent. Majorizing multiplicative cascades for directed polymers in random
media.
*ALEA Lat. Am. J. Probab. Math. Stat.*2 (2006), 267--277. MR2249671 - Comets, Francis; Cranston, Michael. Overlaps and pathwise localization in the Anderson polymer model.
*Stochastic Process. Appl.*123 (2013), no. 6, 2446--2471. MR3038513 - Flury, Markus. Large deviations and phase transition for random walks in random
nonnegative potentials.
*Stochastic Process. Appl.*117 (2007), no. 5, 596--612. MR2320951 - Flury, Markus. Coincidence of Lyapunov exponents for random walks in weak random
potentials.
*Ann. Probab.*36 (2008), no. 4, 1528--1583. MR2435858 - David A. Huse and Christopher L. Henley. Pinning and roughening of domain walls in Ising systems due to random impurities. Phys. Rev. Lett., 54, 2708--2711, 1985.
- Kosygina, Elena; Mountford, Thomas. Crossing velocities for an annealed random walk in a random
potential.
*Stochastic Process. Appl.*122 (2012), no. 1, 277--304. MR2860450 - Imbrie, J. Z.; Spencer, T. Diffusion of directed polymers in a random environment.
*J. Statist. Phys.*52 (1988), no. 3-4, 609--626. MR0968950 - Ioffe, Dmitry; Velenik, Yvan. Ballistic phase of self-interacting random walks.
*Analysis and stochastics of growth processes and interface models,*55--79,*Oxford Univ. Press, Oxford,*2008. MR2603219 - Ioffe, Dmitry; Velenik, Yvan. Crossing random walks and stretched polymers at weak disorder.
*Ann. Probab.*40 (2012), no. 2, 714--742. MR2952089 - Dmitry Ioffe and Yvan Velenik. Stretched polymers in random environment, in
*Probability in Complex Physical Systems*, Springer Proceedings in Mathematics, Vol. 11, 339--369, 2012. - Ioffe, Dmitry; Velenik, Yvan. Self-attractive random walks: the case of critical drifts.
*Comm. Math. Phys.*313 (2012), no. 1, 209--235. MR2928223 - Lacoin, Hubert. New bounds for the free energy of directed polymers in dimension $1+1$
and $1+2$.
*Comm. Math. Phys.*294 (2010), no. 2, 471--503. MR2579463 - McLeish, D. L. A maximal inequality and dependent strong laws.
*Ann. Probability*3 (1975), no. 5, 829--839. MR0400382 - Sinai, Yakov G. A remark concerning random walks with random potentials.
*Fund. Math.*147 (1995), no. 2, 173--180. MR1341729 - Michael Trachsler. Phase Transitions and Fluctuations for Random Walks with Drift in Random Potentials. PhD thesis, Universität Zürich, 1999.
- Vargas, Vincent. A local limit theorem for directed polymers in random media: the
continuous and the discrete case.
*Ann. Inst. H. Poincaré Probab. Statist.*42 (2006), no. 5, 521--534. MR2259972 - Zerner, Martin P. W. Directional decay of the Green's function for a random nonnegative
potential on ${\bf Z}^ d$.
*Ann. Appl. Probab.*8 (1998), no. 1, 246--280. MR1620370 - Zygouras, N. Lyapounov norms for random walks in low disorder and dimension greater
than three.
*Probab. Theory Related Fields*143 (2009), no. 3-4, 615--642. MR2475675 - Nikos Zygouras. Strong disorder in semidirected random polymers. Ann. Inst. Henri Poincaré (B) Prob. Stat., 49(3), 753--780, 2013.

This work is licensed under a Creative Commons Attribution 3.0 License.