# Annealed vs quenched critical points for a random walk pinning model

Matthias Birkner; Rongfeng Sun

Annales de l'I.H.P. Probabilités et statistiques (2010)

- Volume: 46, Issue: 2, page 414-441
- ISSN: 0246-0203

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topBirkner, Matthias, and Sun, Rongfeng. "Annealed vs quenched critical points for a random walk pinning model." Annales de l'I.H.P. Probabilités et statistiques 46.2 (2010): 414-441. <http://eudml.org/doc/244025>.

@article{Birkner2010,

abstract = {We study a random walk pinning model, where conditioned on a simple random walk Y on ℤd acting as a random medium, the path measure of a second independent simple random walk X up to time t is Gibbs transformed with hamiltonian −Lt(X, Y), where Lt(X, Y) is the collision local time between X and Y up to time t. This model arises naturally in various contexts, including the study of the parabolic Anderson model with moving catalysts, the parabolic Anderson model with brownian noise, and the directed polymer model. It falls in the same framework as the pinning and copolymer models, and exhibits a localization-delocalization transition as the inverse temperature β varies. We show that in dimensions d=1, 2, the annealed and quenched critical values of β are both 0, while in dimensions d≥4, the quenched critical value of β is strictly larger than the annealed critical value (which is positive). This implies the existence of certain intermediate regimes for the parabolic Anderson model with brownian noise and the directed polymer model. For d≥5, the same result has recently been established by Birkner, Greven and den Hollander [Quenched LDP for words in a letter sequence (2008)] via a quenched large deviation principle. Our proof is based on a fractional moment method used recently by Derrida et al. [Comm. Math. Phys.287 (2009) 867–887] to establish the non-coincidence of annealed and quenched critical points for the pinning model in the disorder-relevant regime. The critical case d=3 remains open.},

author = {Birkner, Matthias, Sun, Rongfeng},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {random walks; pinning models; annealed and quenched critical points; collision local time; disordered system},

language = {eng},

number = {2},

pages = {414-441},

publisher = {Gauthier-Villars},

title = {Annealed vs quenched critical points for a random walk pinning model},

url = {http://eudml.org/doc/244025},

volume = {46},

year = {2010},

}

TY - JOUR

AU - Birkner, Matthias

AU - Sun, Rongfeng

TI - Annealed vs quenched critical points for a random walk pinning model

JO - Annales de l'I.H.P. Probabilités et statistiques

PY - 2010

PB - Gauthier-Villars

VL - 46

IS - 2

SP - 414

EP - 441

AB - We study a random walk pinning model, where conditioned on a simple random walk Y on ℤd acting as a random medium, the path measure of a second independent simple random walk X up to time t is Gibbs transformed with hamiltonian −Lt(X, Y), where Lt(X, Y) is the collision local time between X and Y up to time t. This model arises naturally in various contexts, including the study of the parabolic Anderson model with moving catalysts, the parabolic Anderson model with brownian noise, and the directed polymer model. It falls in the same framework as the pinning and copolymer models, and exhibits a localization-delocalization transition as the inverse temperature β varies. We show that in dimensions d=1, 2, the annealed and quenched critical values of β are both 0, while in dimensions d≥4, the quenched critical value of β is strictly larger than the annealed critical value (which is positive). This implies the existence of certain intermediate regimes for the parabolic Anderson model with brownian noise and the directed polymer model. For d≥5, the same result has recently been established by Birkner, Greven and den Hollander [Quenched LDP for words in a letter sequence (2008)] via a quenched large deviation principle. Our proof is based on a fractional moment method used recently by Derrida et al. [Comm. Math. Phys.287 (2009) 867–887] to establish the non-coincidence of annealed and quenched critical points for the pinning model in the disorder-relevant regime. The critical case d=3 remains open.

LA - eng

KW - random walks; pinning models; annealed and quenched critical points; collision local time; disordered system

UR - http://eudml.org/doc/244025

ER -

## References

top- [1] K. S. Alexander and N. Zygouras. Quenched and annealed critical points in polymer pinning models, 2008. Available at arXiv:0805.1708v1. Zbl1188.82154MR2534789
- [2] M. Birkner. A condition for weak disorder for directed polymers in random environment. Electron. Comm. Probab. 9 (2004) 22–25. Zbl1067.82030MR2041302
- [3] M. Birkner, A. Greven and F. den Hollander. Quenched LDP for words in a letter sequence. Preprint, 2008. Available at arXiv:0807.2611v1. Zbl1243.60027
- [4] T. Bodineau, G. Giacomin, H. Lacoin and F. L. Toninelli. Copolymers at selective interfaces: New bounds on the phase diagram. J. Statist. Phys. 132 (2008) 603–626. Zbl1157.82025MR2429695
- [5] A. Camanes and P. Carmona. The critical temperature of a directed polymer in a random environment. Markov Process. Related Fields 15 (2009) 105–116. Zbl1203.60148MR2509426
- [6] F. Comets and N. Yosida. Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 (2006) 1746–1770. Zbl1104.60061MR2271480
- [7] 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. Zbl1114.82017MR2073332
- [8] 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. Zbl1226.82028MR2486665
- [9] R. Durrett. Probability: Theory and Examples, 2nd edition. Duxbury Press, Belmont, CA, 1996. Zbl1202.60002MR1609153
- [10] T. Garel and C. Monthus. Freezing transitions of the directed polymer in a 1+d random medium: Location of the critical temperature and unusual critical properties. Phys. Rev. E 74 (2006) 011101.
- [11] J. Gärtner and W. König. The parabolic Anderson model. In Interacting Stochastic Systems 153–179. Springer, Berlin, 2005. Zbl1111.82011
- [12] J. Gärtner and M. Heydenreich. Annealed asymptotics for the parabolic Anderson model with a moving catalyst. Stochastic Process. Appl. 116 (2006) 1511–1529. Zbl1102.60058MR2269214
- [13] J. Gärtner and R. Sun. A quenched limit theorem for the local time of random walks on ℤ2. Stochastic Process. Appl. 119 (2009) 1198–1215. Zbl1163.60018MR2508570
- [14] G. Giacomin. Random Polymer Models. Imperial College Press, World Scientific, London, 2007. Zbl1125.82001MR2380992
- [15] G. Giacomin, H. Lacoin and F. L. Toninelli. Marginal relevance of disorder for pinning models, 2008. Available at arXiv:0811.0723v1. Zbl1189.60173
- [16] A. Greven and F. den Hollander. Phase transitions for the long-time behaviour of interacting diffusions. Ann. Probab. 35 (2007) 1250–1306. Zbl1126.60085MR2330971
- [17] F. L. Toninelli. Coarse graining, fractional moments and the critical slope of random copolymers. Electron. J. Probab. 14 (2009) 531–547. Zbl1189.60186MR2480552

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