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**Strong disorder in semidirected random polymers.**
*(English.
French summary)*
Zbl 1290.82013

The present paper is devoted to a random walk in a random potential, which models a situation of a random polymer. The annealed and quenched costs to perform long crossings from a point to a hyperplane are studied. Note that the distribution of the random polymer is given by the Gibbs measure related to the walk. These costs are measured by the so-called Lyapunov norms. One of the main results of the paper is the identification of situations where the point-to-hyperplane annealed and quenched Lyapunov norms are different. Moreover, it is proved that in these cases the polymer path exhibits localization.

Reviewer: Farruh Mukhamedov (Kuantan)

### MSC:

82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |

82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |

60G50 | Sums of independent random variables; random walks |

82D60 | Statistical mechanics of polymers |

### Keywords:

random walks; random potential; Lyapunov norms; strong disorder; localization; fractional moments
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\textit{N. Zygouras}, Ann. Inst. Henri Poincaré, Probab. Stat. 49, No. 3, 753--780 (2013; Zbl 1290.82013)

### References:

[1] | E. Bolthausen. A note on the diffusion of directed polymers in a random environment. Comm. Math. Phys. 123 (1989) 529-534. · Zbl 0684.60013 |

[2] | E. Bolthausen and A. S. Sznitman. On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal. 9 (2002) 345-375. Special issue dedicated to Daniel W. Stroock and Srinivasa S. R. Varadhan on the occasion of their 60th birthday. · Zbl 1079.60079 |

[3] | M. Campanino, D. Ioffe and Y. Velenik. Ornstein-Zernike theory for finite range Ising models above \(T_{c}\). Probab. Theory Related Fields 125 (2003) 305-349. · Zbl 1032.60093 |

[4] | J. T. Chayes and L. Chayes. Ornstein-Zernike behavior for self-avoiding walks at all noncritical temperatures. Comm. Math. Phys. 105 (1986) 221-238. |

[5] | F. Comets and N. Yoshida. Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 (2006) 1746-1770. · Zbl 1104.60061 |

[6] | F. Comets, T. Shiga and N. Yoshida. Probabilistic analysis of directed polymers in a random environment: A review. In Stochastic Analysis on Large Scale Interacting Systems 115-142. Adv. Stud. Pure Math. 39 . Math. Soc. Japan, Tokyo, 2004. · Zbl 1114.82017 |

[7] | M. Flury. Coincidence of Lyapunov exponents for random walks in weak random potentials. Ann. Probab. 36 (2008) 1528-1583. · Zbl 1156.60076 |

[8] | G. Giacomin, H. Lacoin and F. L. Toninelli. Marginal relevance of disorder for pinning models. Comm. Pure Appl. Math. 63 (2010) 233-265. · Zbl 1189.60173 |

[9] | D. Ioffe and Y. Velenik. Crossing random walks and stretched polymers at weak disorder. Available at · Zbl 1251.60074 |

[10] | D. Ioffe and Y. Velenik. Ballistic phase of self-interacting random walks. In Analysis and Stochastics of Growth Processes and Interface Models 55-79. Oxford Univ. Press, Oxford, 2008. · Zbl 1255.60168 |

[11] | D. Ioffe and Y. Velenik. Stretched polymers in random environment. Available at · Zbl 1251.82070 |

[12] | H. Kesten. First passage percolation. In From Classical to Modern Probability 93-143. Progr. Probab. 54 . Birkhäuser, Basel, 2003. |

[13] | E. Kosygina, T. Mountford and M. P. W. Zerner. Lyapunov exponents of Green’s functions for random potentials tending to zero. Probab. Theory Related Fields 150 (2011) 43-59. · Zbl 1235.60147 |

[14] | H. Lacoin. New bounds for the free energy of directed polymer in dimension 1\(+\)1 and 1\(+\)2. Comm. Math. Phys. 294 (2010) 471-503. · Zbl 1227.82098 |

[15] | A. S. Sznitman. Annealed Lyapounov exponents and large deviations in a Poissonian potential. I. Ann. Sci. Éc. Norm. Supér. 28 (1995) 345-370. · Zbl 0826.60018 |

[16] | A. S. Sznitman. Brownian motion with a drift in a Poissonian potential. Comm. Pure Appl. Math. 47 (1994) 1283-1318. · Zbl 0814.60021 |

[17] | A. S. Sznitman. Brownian Motion, Obstacles and Random Media . Springer Monographs in Mathematics . Springer, Berlin, 1998. · Zbl 0973.60003 |

[18] | M. Trachsler. Phase transitions and fluctuations for random walks with drift in random potentials. Ph.D. thesis, Univ. Zurich. |

[19] | V. Vargas. Strong localization and macroscopic atoms for directed polymers. Probab. Theory Related Fields 138 (2007) 391-410. · Zbl 1113.60097 |

[20] | M. P. W. Zerner. Directional decay of the Green’s function for a random nonnegative potential on \(Z^{d}\). Ann. Appl. Probab. 8 (1998) 246-280. · Zbl 0938.60098 |

[21] | N. Zygouras. Lyapounov norms for random walks in low disorder and dimension greater than three. Probab. Theory Related Fields 143 (2009) 615-642. · Zbl 1163.60050 |

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