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

Abstract

top
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.

How to cite

top

Birkner, 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. [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. [2] M. Birkner. A condition for weak disorder for directed polymers in random environment. Electron. Comm. Probab. 9 (2004) 22–25. Zbl1067.82030MR2041302
  3. [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. [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. [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. [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. [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. [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. [9] R. Durrett. Probability: Theory and Examples, 2nd edition. Duxbury Press, Belmont, CA, 1996. Zbl1202.60002MR1609153
  10. [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. [11] J. Gärtner and W. König. The parabolic Anderson model. In Interacting Stochastic Systems 153–179. Springer, Berlin, 2005. Zbl1111.82011
  12. [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. [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. [14] G. Giacomin. Random Polymer Models. Imperial College Press, World Scientific, London, 2007. Zbl1125.82001MR2380992
  15. [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. [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. [17] F. L. Toninelli. Coarse graining, fractional moments and the critical slope of random copolymers. Electron. J. Probab. 14 (2009) 531–547. Zbl1189.60186MR2480552

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.