Skip to main content

Dynamic polymers: invariant measures and ordering by noise

Abstract

We develop a dynamical approach to infinite volume directed polymer measures in random environments. We define polymer dynamics in \(1+1\) dimension as a stochastic gradient flow on polymers pinned at the origin, for energy involving quadratic nearest neighbor interaction and local interaction with random environment. We prove existence and uniqueness of the solution, continuity of the flow, the order-preserving property with respect to the coordinatewise partial order, and the invariance of the asymptotic slope. We establish ordering by noise which means that if two initial conditions have distinct slopes, then the associated solutions eventually get ordered coordinatewise. This, along with the shear-invariance property and existing results on static infinite volume polymer measures, allows to prove that for a fixed asymptotic slope and almost every realization of the environment, the polymer dynamics has a unique invariant distribution given by a unique infinite volume polymer measure, and, moreover, One Force—One Solution principle holds. We also prove that every polymer measure is concentrated on paths with well-defined asymptotic slopes and give an estimate on deviations from straight lines.

This is a preview of subscription content, access via your institution.

References

  1. Albeverio, S., Kondratiev, Y.G., Rƶckner, M., Tsikalenko, T.V.: Glauber dynamics for quantum lattice systems. Rev. Math. Phys. 13(1), 51–124 (2001)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  2. Alberts, T., Rassoul-Agha, F., Simper, M.: Busemann functions and semi-infinite O’Connell-Yor polymers. Bernoulli 26(3), 1927–1955 (2020)

  3. Arnold, L.: Random Dynamical Systems. Springer Monographs in Mathematics. Springer, Berlin (1998)

    BookĀ  Google ScholarĀ 

  4. Bakhtin, Y.: Inviscid Burgers equation with random kick forcing in noncompact setting. Electron. J. Probab. 21, 50 (2016)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  5. Bates, E., Chatterjee, S.: The endpoint distribution of directed polymers. Ann. Probab. 48(2), 817–871 (2020)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  6. Bakhtin, Y., Cator, E., Khanin, K.: Space-time stationary solutions for the Burgers equation. J. Am. Math. Soc. 27(1), 193–238 (2014)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  7. Bakhtin, Y., Khanin, K.: Localization and Perron–Frobenius theory for directed polymers. Mosc. Math. J. 10(4), 667–686, 838 (2010)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  8. Bakhtin, Y., Khanin, K.: On global solutions of the random Hamilton-Jacobi equations and the KPZ problem. Nonlinearity 31(4), R93–R121 (2018)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  9. Bakhtin, Y., Li, L.: Zero temperature limit for directed polymers and inviscid limit for stationary solutions of stochastic Burgers equation. J. Stat. Phys. 172(5), 1358–1397 (2018)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  10. Bakhtin, Y., Li, L.: Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation. Commun. Pure Appl. Math. 72(3), 536–619 (2019)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  11. Bogachev, V.I., Rƶckner, M., Wang, F.-Y.: Invariance implies Gibbsian: some new results. Commun. Math. Phys. 248(2), 335–355 (2004)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  12. Bakhtin, Y., Seo, D.: Localization of directed polymers in continuous space (2020). arXiv:1905.00930

  13. Comets, F.: Directed Polymers in Random Environments, Volume 2175 of Lecture Notes in Mathematics. Springer, Cham (2017). Lecture notes from the 46th Probability Summer School held in Saint-Flour (2016)

  14. Crauel, H.: Markov measures for random dynamical systems. Stoch. Stoch. Rep. 37(3), 153–173 (1991)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  15. Dunlap, A., Graham, C., Ryzhik, L.: Stationary solutions to the stochastic Burgers equation on the line (2019). arXiv:1910.07464

  16. den Hollander, F.: Random Polymers, Volume 1974 of Lecture Notes in Mathematics. Springer, Berlin (2009). Lectures from the 37th Probability Summer School held in Saint-Flour (2007)

  17. Dunlap, A.: Existence of stationary stochastic Burgers evolutions on \( {R}^2\) and \( {R}^3\). Nonlinearity 33(12), 6480–6501 (2020)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  18. Flandoli, F., Gess, B., Scheutzow, M.: Synchronization by noise for order-preserving random dynamical systems. Ann. Probab. 45(2), 1325–1350 (2017)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  19. Fritz, J.: Stationary measures of stochastic gradient systems, infinite lattice models. Z. Wahrsch. Verw. Gebiete 59(4), 479–490 (1982)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  20. Funaki, T., Spohn, H.: Motion by mean curvature from the Ginzburg–Landau \(\nabla \phi \) interface model. Commun. Math. Phys. 185(1), 1–36 (1997)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  21. Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. Adaptive Computation and Machine Learning. MIT Press, Cambridge (2016)

    MATHĀ  Google ScholarĀ 

  22. Georgii, H.-O.: Gibbs Measures and Phase Transitions. De Gruyter Studies in Mathematics, vol. 9. Walter de Gruyter & Co., Berlin (1988)

    BookĀ  Google ScholarĀ 

  23. Giacomin, G.: Random Polymer Models. Imperial College Press, London (2007)

    BookĀ  Google ScholarĀ 

  24. Georgiou, N., Rassoul-Agha, F., SeppƤlƤinen, T.: Variational formulas and cocycle solutions for directed polymer and percolation models. Commun. Math. Phys. 346(2), 741–779 (2016)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  25. Georgiou, N., Rassoul-Agha, F., SeppƤlƤinen, T., Yilmaz, A.: Ratios of partition functions for the log-gamma polymer. Ann. Probab. 43(5), 2282–2331 (2015)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  26. Howard, C.D., Newman, C.M.: Euclidean models of first-passage percolation. Probab. Theory Relat. Fields 108, 153–170 (1997). https://doi.org/10.1007/s004400050105

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  27. Jahnel, B., Külske, C.: Attractor properties for irreversible and reversible interacting particle systems. Commun. Math. Phys. 366(1), 139–172 (2019)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  28. Janjigian, C., Rassoul-Agha, F.: Busemann functions and Gibbs measures in directed polymer models on \({\mathbb{Z}}^2\). Ann. Probab. 48(2), 778–816 (2020)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  29. Janjigian, C., Rassoul-Agha, F.: Uniqueness and ergodicity of stationary directed polymers on \(\mathbb{Z}^2\). J. Stat. Phys. 179(3), 672–689 (2020)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  30. Janjigian, C., Rassoul-Agha, F., SeppƤlƤinen, T.: Geometry of geodesics through Busemann measures in directed last-passage percolation (2020). arXiv:1908.09040

  31. Kolmogoroff, A.: Zur Umkehrbarkeit der statistischen Naturgesetze. Math. Ann. 113(1), 766–772 (1937)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  32. Koonin, E.V.: The Logic of Chance: The Nature and Origin of Biological Evolution. FT Press, Upper Saddle River (2011)

    Google ScholarĀ 

  33. Liggett, T.M.: Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 276. Springer, New York (1985)

    Google ScholarĀ 

  34. Ledrappier, F., Young, L.-S.: Entropy formula for random transformations. Probab. Theory Relat. Fields 80(2), 217–240 (1988)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  35. Mukherjee, C., Varadhan, S.R.S.: Brownian occupation measures, compactness and large deviations. Ann. Probab. 44(6), 3934–3964 (2016)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  36. Roberts, G.O., Tweedie, R.L., et al.: Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli 2(4), 341–363 (1996)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  37. Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales, Volume 1: Foundations. Wiley, Chichester (1994)

    MATHĀ  Google ScholarĀ 

  38. Sinaĭ, Ya.G.: Theory of Phase Transitions: Rigorous Results, Volume 108 of International Series in Natural Philosophy. Pergamon Press, Oxford (1982). Translated from the Russian by J. Fritz, A. KrƔmli, P. Major and D. SzƔsz

  39. Teschl, G.: Ordinary Differential Equations and Dynamical Systems, vol. 140. American Mathematical Society, Providence (2012)

    BookĀ  Google ScholarĀ 

  40. von Foerster, H.: On Self-Organizing Systems and Their Environments, pp. 1–19. Springer New York, New York (2003). Originally published in: Yovits, M.C., Cameron, S., (eds.) Self-Organizing Systems. Pergamon Press, London, pp. 31–50 (1960)

  41. Winkler, G.: Image Analysis, Random Fields and Dynamic Monte Carlo Methods, Volume 27 of Applications of Mathematics (New York). Springer, Berlin (1995). A Mathematical Introduction

Download references

Acknowledgements

Yuri Bakhtin is grateful to the National Science Foundation for partial support via grant DMS-1811444. Both authors thank the anonymous referee for the comments that helped to improve the paper significantly.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hong-Bin Chen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bakhtin, Y., Chen, HB. Dynamic polymers: invariant measures and ordering by noise. Probab. Theory Relat. Fields 183, 167–227 (2022). https://doi.org/10.1007/s00440-021-01099-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-021-01099-5

Mathematics Subject Classification

  • 37L40
  • 37L55
  • 35R60
  • 37H99
  • 60K35
  • 60K37
  • 82D60