An almost sure CLT for stretched polymers

Dmitry Ioffe (Technion Haifa)
Yvan Velenik (Université de Genève)


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 $d+1\geq 4$.

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Pages: 1-20

Publication Date: November 11, 2013

DOI: 10.1214/EJP.v18-2231


  • 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.

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