Strong disorder in semidirected random polymers
Zygouras, N.
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013), p. 753-780 / Harvested from Numdam
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Nous considérons une marche aléatoire dans un potentiel aléatoire qui modèle la situation d'un polymère aléatoire et nous étudions les coûts “annealed” et “quenched” pour réaliser de longues traversées d'un point à un hyperplan. Ces coûts sont mesurés en terme de normes de Lyapounov. Nous identifions des situations où les normes de Lyapounov d'un point à un hyperplan “annealed” et “quenched” sont différentes. Nous démontrons également que dans ces cas le chemin du polymère présente une localisation.

We consider a random walk in a random potential, which models a situation of a random polymer and we study the annealed and quenched costs to perform long crossings from a point to a hyperplane. These costs are measured by the so called Lyapounov norms. We identify situations where the point-to-hyperplane annealed and quenched Lyapounov norms are different. We also prove that in these cases the polymer path exhibits localization.

Publié le : 2013-01-01
DOI : https://doi.org/10.1214/12-AIHP483
Classification:  60xx 
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     author = {Zygouras, N.},
     title = {Strong disorder in semidirected random polymers},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {49},
     year = {2013},
     pages = {753-780},
     doi = {10.1214/12-AIHP483},
     mrnumber = {3112433},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2013__49_3_753_0}
}

Zygouras, N. Strong disorder in semidirected random polymers. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) pp. 753-780. doi : 10.1214/12-AIHP483. http://gdmltest.u-ga.fr/item/AIHPB_2013__49_3_753_0/

[1] E. Bolthausen. A note on the diffusion of directed polymers in a random environment. Comm. Math. Phys. 123 (1989) 529-534. | MR 1006293 | 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. | MR 2023130 | 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. | MR 1964456 | 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. | MR 849206

[5] F. Comets and N. Yoshida. Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 (2006) 1746-1770. | MR 2271480 | 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. | MR 2073332 | Zbl 1114.82017

[7] M. Flury. Coincidence of Lyapunov exponents for random walks in weak random potentials. Ann. Probab. 36 (2008) 1528-1583. | MR 2435858 | 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. | MR 2588461 | Zbl 1189.60173

[9] D. Ioffe and Y. Velenik. Crossing random walks and stretched polymers at weak disorder. Available at arXiv:1002.4289. | MR 2952089 | 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. | MR 2603219 | Zbl 1255.60168

[11] D. Ioffe and Y. Velenik. Stretched polymers in random environment. Available at arXiv:1011.0266. | Zbl 1251.82070

[12] H. Kesten. First passage percolation. In From Classical to Modern Probability 93-143. Progr. Probab. 54. Birkhäuser, Basel, 2003. | MR 2045986 | Zbl 1041.60077

[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. | MR 2800903 | 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. | MR 2579463 | 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. | Numdam | MR 1326672 | Zbl 0826.60018

[16] A. S. Sznitman. Brownian motion with a drift in a Poissonian potential. Comm. Pure Appl. Math. 47 (1994) 1283-1318. | MR 1295931 | Zbl 0814.60021

[17] A. S. Sznitman. Brownian Motion, Obstacles and Random Media. Springer Monographs in Mathematics. Springer, Berlin, 1998. | MR 1717054 | Zbl 0815.60077

[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. | MR 2299713 | 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. | MR 1620370 | 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. | MR 2475675 | Zbl 1163.60050