Disorder relevance for the random walk pinning model in dimension 3
Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 1, p. 259-293

We study the continuous time version of the random walk pinning model, where conditioned on a continuous time random walk (Ys)s≥0 on ℤd with jump rate ρ > 0, which plays the role of disorder, the law up to time t of a second independent random walk (Xs)0≤st with jump rate 1 is Gibbs transformed with weight eβLt(X,Y), where Lt(X, Y) is the collision local time between X and Y up to time t. As the inverse temperature β varies, the model undergoes a localization-delocalization transition at some critical βc ≥ 0. A natural question is whether or not there is disorder relevance, namely whether or not βc differs from the critical point βcann for the annealed model. In [3], it was shown that there is disorder irrelevance in dimensions d = 1 and 2, and disorder relevance in d ≥ 4. For d ≥ 5, disorder relevance was first proved in [2]. In this paper, we prove that if X and Y have the same jump probability kernel, which is irreducible and symmetric with finite second moments, then there is also disorder relevance in the critical dimension d = 3, and βc - βcann is at least of the order e-C(ζ)/ρζ, C(ζ) > 0, for any ζ > 2. Our proof employs coarse graining and fractional moment techniques, which have recently been applied by Lacoin [13] to the directed polymer model in random environment, and by Giacomin, Lacoin and Toninelli [10] to establish disorder relevance for the random pinning model in the critical dimension. Along the way, we also prove a continuous time version of Doney's local limit theorem [5] for renewal processes with infinite mean.

Nous étudions la version à temps continu du modèle de marche aléatoire avec accrochage, où conditionné sur une marche aléatoire à temps continu (Ys)s≥0 sur ℤd avec taux de saut ρ > 0, qui joue le rôle de désordre, la loi jusqu'au temps t d'une seconde marche aléatoire indépendante (Xs)0≤st avec taux de saut 1 est la transformée de Gibbs avec poids eβLt(X,Y), où Lt(X, Y) est le temps local de collision entre X et Y jusqu'au temps t. Lorsque la température inverse β varie, le modèle subit une transition de localisation-délocalisation à un βc ≥ 0 critique. Une question naturelle est de savoir s'il y a pertinence du désordre ou pas, i.e., si βc diffère ou pas du point critique βcann pour le modèle moyenné. Dans [3], il a été montré qu'il y avait non pertinence du désordre en dimensions d = 1 et 2, et pertinence du désordre lorsque d ≥ 4. Pour d ≥ 5, la pertinence du désordre fût d'abord prouvée dans [2]. Dans ce papier, nous prouvons que si X et Y ont le même noyau de probabilité de saut, qui est irréductible et symétrique avec des moments du second ordre finis, alors il y a également pertinence du désordre en dimension critique d = 3, et βc - βcann est au moins de l'ordre e-C(ζ)/ρζ, C(ζ) > 0, pour tout ζ > 2. Notre preuve utilise des techniques de coarse graining et de moment fractionnaire, qui ont été récemment appliquées par Lacoin [13] au modèle de polymère dirigé en milieu aléatoire, et par Giacomin, Lacoin et Toninelli [10] pour établir la pertinence du désordre pour le modèle d'accrochages aléatoires en dimension critique. En chemin, nous prouvons également une version en temps continu du théorème limite local de Doney [5] pour des processus de renouvellement avec moyenne infinie.

DOI : https://doi.org/10.1214/10-AIHP374
Classification:  60K35,  82B44
Keywords: collision local time, disordered pinning models, fractional moment method, local limit theorem, marginal disorder, random walks, renewal processes with infinite mean
@article{AIHPB_2011__47_1_259_0,
     author = {Birkner, Matthias and Sun, Rongfeng},
     title = {Disorder relevance for the random walk pinning model in dimension 3},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {1},
     year = {2011},
     pages = {259-293},
     doi = {10.1214/10-AIHP374},
     zbl = {1217.60085},
     mrnumber = {2779405},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2011__47_1_259_0}
}
Birkner, Matthias; Sun, Rongfeng. Disorder relevance for the random walk pinning model in dimension 3. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 1, pp. 259-293. doi : 10.1214/10-AIHP374. http://www.numdam.org/item/AIHPB_2011__47_1_259_0/

[1] Q. Berger and F. L. Toninelli. On the critical point of the Random Walk Pinning Model in dimension d=3. Electron. J. Probab. 15 (2010) 654-683. | MR 2650777 | Zbl 1226.82027 | Zbl pre05946914

[2] M. Birkner, A. Greven and F. Den Hollander. Collision local time of transient random walks and intermediate phases in interacting stochastic systems, EURANDOM Report 2008-49, 2008. Available at http://www.eurandom.nl/reports/2008/049-report.pdf. | Zbl pre05946986

[3] M. Birkner and R. Sun. Annealed vs quenched critical points for a random walk pinning model. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010) 414-441. | Numdam | MR 2667704 | Zbl 1206.60087

[4] B. Derrida, G. Giacomin, H. Lacoin and F. L. Toninelli. Fractional moment bounds and disorder relevance for pinning models. Comm. Math. Phys. 287 (2009) 867-887. | Zbl 1226.82028 | Zbl pre05789918

[5] R. A. Doney. One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Related Fields 107 (1997) 451-465. | MR 1440141 | Zbl 0883.60022

[6] R. Durrett. Probability: Theory and Examples, 2nd edition. Duxbury Press, Belmont, CA, 1996. | MR 1609153 | Zbl 1202.60002

[7] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York, 1966. | MR 210154 | Zbl 0219.60003

[8] G. Giacomin. Random Polymer Models. Imperial College Press, London, 2007. | Zbl 1125.82001

[9] 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

[10] G. Giacomin, H. Lacoin and F. L. Toninelli. Disorder relevance at marginality and critical point shift. Ann. Inst. H. Poincaré Probab. Statist. (2010). To appear. Available at arXiv:0906.1942v1. | Numdam | MR 2779401 | Zbl 1210.82036

[11] A. Greven and F. Den Hollander. Phase transitions for the long-time behaviour of interacting diffusions. Ann. Probab. 35 (2007) 1250-1306. | MR 2330971 | Zbl 1126.60085

[12] I. A. Ibragimov and Y. V. Linnik. Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen, 1971. | MR 322926 | Zbl 0219.60027

[13] H. Lacoin. New bounds for the free energy of directed polymers in dimension 1 + 1 and 1 + 2. Comm. Math. Phys. 294 (2010) 471-503. | Zbl 1227.82098 | Zbl pre05852664

[14] F. Spitzer. Principles of Random Walks, 2nd edition. Springer, New York, 1976. | MR 388547 | Zbl 0979.60002

[15] S. G. Tkačuk. Local limit theorems, allowing for large deviations, in the case of stable limit laws. Izv. Akad. Nauk USSR Ser. Fiz.-Mat. Nauk 17 (1973) 30-33, 70.

[16] A. Yilmaz and O. Zeitouni. Differing averaged and quenched large deviations for random walks in random environments in dimensions two and three. Comm. Math. Phys. (2010). To appear. Available at arXiv:0910.1169v1. | MR 2725188 | Zbl 1202.60163

[17] A. Zaigraev. Multivariate large deviations with stable limit laws. Probab. Math. Statist. 19 (1999) 323-335. | MR 1750906 | Zbl 0985.60027