Communications in Mathematical Physics

, Volume 346, Issue 2, pp 741–779 | Cite as

Variational Formulas and Cocycle solutions for Directed Polymer and Percolation Models

  • Nicos Georgiou
  • Firas Rassoul-Agha
  • Timo Seppäläinen


We discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting free energy for directed polymers. The results are valid for models in arbitrary dimension, steps of the admissible paths can be general, the environment process is ergodic under spatial translations, and the potential accumulated along a path can depend on the environment and the next step of the path. The variational formulas come in two types: one minimizes over gradient-like cocycles, and another one maximizes over invariant measures on the space of environments and paths. Minimizing cocycles can be obtained from Busemann functions when these can be proved to exist. The results are illustrated through 1+1 dimensional exactly solvable examples, periodic examples, and polymers in weak disorder.


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  1. 1.
    Aldous D., Diaconis P.: Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Relat. Fields 103(2), 199–213 (1995)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Armstrong S.N., Souganidis P.E.: Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments. J. Math. Pures Appl. (9) 97(5), 460–504 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Auffinger A., Damron M.: Differentiability at the edge of the percolation cone and related results in first-passage percolation. Probab. Theory Relat. Fields 156(1-2), 193–227 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Louis Baccelli F., Cohen G., Olsder G.J., Quadrat J.-P.: Synchronization and Linearity. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Ltd., Chichester (1992)Google Scholar
  5. 5.
    Baik J., Deift P., Johansson K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12(4), 1119–1178 (1999)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bakhtin Y., Cator E., Khanin K.: Space-time stationary solutions for the Burgers equation. J. Am. Math. Soc. 27(1), 193–238 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Berman, A., Plemmons, R.J.: Nonnegative matrices in the mathematical sciences, vol. 9 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1994) (Revised reprint of the 1979 original)Google Scholar
  8. 8.
    Carmona P., Hu Y.: On the partition function of a directed polymer in a Gaussian random environment. Probab. Theory Relat. Fields 124(3), 431–457 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cator E., Pimentel L.P.R.: A shape theorem and semi-infinite geodesics for the Hammersley model with random weights. ALEA Lat. Am. J. Probab. Math. Stat. 8, 163–175 (2011)MathSciNetMATHGoogle Scholar
  10. 10.
    Cator E., Pimentel L.P.R.: Busemann functions and equilibrium measures in last passage percolation models. Probab. Theory Relat. Fields 154(1-2), 89–125 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cator E., Pimentel L.P.R.: Busemann functions and the speed of a second class particle in the rarefaction fan. Ann. Probab. 41(4), 2401–2425 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cohn H., Elkies N., Propp J.: Local statistics for random domino tilings of the Aztec diamond. Duke Math. J. 85(1), 117–166 (1996)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Comets F., Shiga T., Yoshida N.: Directed polymers in a random environment: path localization and strong disorder. Bernoulli 9(4), 705–723 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Comets, F., Shiga, T., Yoshida, N.: Probabilistic analysis of directed polymers in a random environment: a review. In: Stochastic analysis on large scale interacting systems, vol. 39 of Adv. Stud. Pure Math., pp. 115–142. Math. Soc. Japan, Tokyo (2004)Google Scholar
  15. 15.
    Comets F., Yoshida N.: Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34(5), 1746–1770 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl. 1(1), 1130001, 76 (2012)Google Scholar
  17. 17.
    Cox J.T., Durrett R.: Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9(4), 583–603 (1981)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Damron M., Hanson J.: Busemann functions and infinite geodesics in two-dimensional first-passage percolation. Comm. Math. Phys. 325(3), 917–963 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Dembo, A., Zeitouni, O.: Large deviations techniques and applications, vol. 38 of Applications of Mathematics 2nd (ed.). Springer, New York (1998)Google Scholar
  20. 20.
    den Hollander, F.: Random polymers, vol. 1974 of Lecture Notes in Mathematics. Springer, Berlin (2009) (Lectures from the 37th Probability Summer School held in Saint-Flour, (2007))Google Scholar
  21. 21.
    Donsker M.D., Varadhan S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time. III. Comm. Pure Appl. Math. 29(4), 389–461 (1976)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Durrett R., Liggett T.M.: The shape of the limit set in Richardson’s growth model. Ann. Probab. 9(2), 186–193 (1981)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Ferrari P.A., Martin J.B., Pimentel L.P.R.: A phase transition for competition interfaces. Ann. Appl. Probab. 19(1), 281–317 (2009)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Ferrari P.A., Pimentel L.P.R.: Competition interfaces and second class particles. Ann. Probab. 33(4), 1235–1254 (2005)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Gandolfi A., Kesten H.: Greedy lattice animals. II. Linear growth. Ann. Appl. Probab. 4(1), 76–107 (1994)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Geodesics and the competition interface for the corner growth model. arXiv:1510.00860 (2015)
  27. 27.
    Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Stationary cocycles and Busemann functions for the corner growth model. arXiv:1510.00859 (2015)
  28. 28.
    Georgiou N., Rassoul-Agha F., Seppäläinen T., Yilmaz A.: Ratios of partition functions for the log-gamma polymer. Ann. Probab. 43(5), 2282–2331 (2015)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Georgiou N., Seppäläinen T.: Large deviation rate functions for the partition function in a log-gamma distributed random potential. Ann. Probab. 41(6), 4248–4286 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Hammersley, J.M.: A few seedlings of research. In: Proceedings of the sixth Berkeley symposium on mathematical statistics and probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. I: Theory of statistics, pp. 345–394, University of California Press, Berkeley, California (1972)Google Scholar
  31. 31.
    Heidergott, B., Oldser, G.J., van der Woude, J.: Max Plus at Work. In: Princeton Series in Applied Mathematics. Modeling and analysis of synchronized systems: a course on max-plus algebra and its applications. Princeton University Press, Princeton, NJ (2006)Google Scholar
  32. 32.
    Hoffman C.: Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab. 15(1B), 739–747 (2005)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Hoffman C.: Geodesics in first passage percolation. Ann. Appl. Probab. 18(5), 1944–1969 (2008)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Howard C.D., Newman C.M.: Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Probab. 29(2), 577–623 (2001)MathSciNetMATHGoogle Scholar
  35. 35.
    Jockusch, W., Propp, J., Shor, P.: Random domino tilings and the arctic circle theorem. arXiv:math/9801068
  36. 36.
    Johansson K.: Shape fluctuations and random matrices. Comm. Math. Phys. 209(2), 437–476 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Kenyon, R.: Lectures on dimers. In: Statistical mechanics, vol. 16 of IAS/Park City Math. Ser., pp. 191–230. Am. Math. Soc., Providence, RI (2009)Google Scholar
  38. 38.
    Kosygina, E.: Homogenization of stochastic Hamilton-Jacobi equations: brief review of methods and applications. In: Stochastic analysis and partial differential equations, volume 429 of Contemp. Math., pp. 189–204. Amer. Math. Soc., Providence, RI (2007)Google Scholar
  39. 39.
    Kosygina E., Rezakhanlou F., Varadhan S.R.S.: Stochastic homogenization of Hamilton-Jacobi-Bellman equations. Comm. Pure Appl. Math. 59(10), 1489–1521 (2006)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Kosygina E., Varadhan S.R.S.: Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium. Comm. Pure Appl. Math. 61(6), 816–847 (2008)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Krishnan, A.: Variational formula for the time-constant of first-passage percolation. ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)–New York University (2014)Google Scholar
  42. 42.
    Krishnan, A.: Variational formula for the time-constant of first-passage percolation. Comm. Pure Appl. Math. arXiv:1311.0316 (2016) (To appear)
  43. 43.
    Lacoin H.: New bounds for the free energy of directed polymers in dimension 1+1 and 1+2. Comm. Math. Phys. 294(2), 471–503 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Licea C., Newman C.M.: Geodesics in two-dimensional first-passage percolation. Ann. Probab. 24(1), 399–410 (1996)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Lions P.-L., Souganidis P.E.: Homogenization of “viscous” Hamilton-Jacobi equations in stationary ergodic media. Comm. Partial Differ. Equ. 30(1-3), 335–375 (2005)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Marchand R.: Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab. 12(3), 1001–1038 (2002)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Martin J.B.: Limiting shape for directed percolation models. Ann. Probab. 32(4), 2908–2937 (2004)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Moreno G.: Convergence of the law of the environment seen by the particle for directed polymers in random media in the L 2 region. J. Theoret. Probab. 23(2), 466–477 (2010)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Moriarty J., O’Connell N.: On the free energy of a directed polymer in a Brownian environment. Markov Process. Relat. Fields 13(2), 251–266 (2007)MathSciNetMATHGoogle Scholar
  50. 50.
    Newman, C.M.: A surface view of first-passage percolation. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pp. 1017–1023, Basel, Birkhäuser (1995)Google Scholar
  51. 51.
    Pimentel L.P.R.: Multitype shape theorems for first passage percolation models. Adv. Appl. Probab. 39(1), 53–76 (2007)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Quastel, J.: Weakly asymmetric exclusion and KPZ. In: Proceedings of the International Congress of Mathematicians. Vol. IV, pp. 2310–2324. Hindustan Book Agency, New Delhi (2010)Google Scholar
  53. 53.
    Rassoul-Agha F., Seppäläinen T.: Process-level quenched large deviations for random walk in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 47(1), 214–242 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Rassoul-Agha, F., Seppäläinen, T.: Quenched point-to-point free energy for random walks in random potentials. arXiv:1202.2584, Version 1 (2012)
  55. 55.
    Rassoul-Agha F., Seppäläinen T.: Quenched point-to-point free energy for random walks in random potentials. Probab. Theory Relat. Fields 158(3-4), 711–750 (2014)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Rassoul-Agha, F., Seppäläinen, T.: A course on large deviations with an introduction to Gibbs measures, vol. 162 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2015)Google Scholar
  57. 57.
    Rassoul-Agha F., Seppäläinen T., Yılmaz A.: Quenched free energy and large deviations for random walks in random potentials. Comm. Pure Appl. Math. 66(2), 202–244 (2013)MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Rassoul-Agha, F., Seppäläinen, T., Yılmaz, A.: Variational formulas and disorder regimes of random walks in random potentials. Bernoulli. arXiv:1410.4474 (2016) (To appear)
  59. 59.
    Rockafellar, R.T.: Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J (1970)Google Scholar
  60. 60.
    Rosenbluth, J.M.: Quenched large deviation for multidimensional random walk in random environment: a variational formula. ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)–New York University (2006)Google Scholar
  61. 61.
    Rost H.: Nonequilibrium behaviour of a many particle process: density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58(1), 41–53 (1981)ADSMathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Seneta, E.: Nonnegative matrices and Markov chains. Springer Series in Statistics 2nd (ed.). Springer, New York (1981)Google Scholar
  63. 63.
    Seppäläinen T.: Large deviations for lattice systems. I. Parametrized independent fields. Probab. Theory Relat. Fields 96(2), 241–260 (1993)MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Seppäläinen, T.: A microscopic model for the Burgers equation and longest increasing subsequences. Electron. J. Probab., 1(5), approx. pp. 51 (1996) (electronic)Google Scholar
  65. 65.
    Seppäläinen T.: Hydrodynamic scaling, convex duality and asymptotic shapes of growth models. Markov Process. Relat. Fields 4(1), 1–26 (1998)MathSciNetMATHGoogle Scholar
  66. 66.
    Seppäläinen T.: Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40(1), 19–73 (2012)MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    Spitzer, F.: Principles of random walks, 2nd (ed.). Graduate Texts in Mathematics, Vol. 34. Springer, New York (1976)Google Scholar
  68. 68.
    Spohn, H.: Stochastic integrability and the KPZ equation. arXiv:1204.2657 (2012)
  69. 69.
    Stroock, D.W.: An introduction to the theory of large deviations. Universitext. Springer, New York (1984)Google Scholar
  70. 70.
    Tracy, C.A., Widom, H.: Distribution functions for largest eigenvalues and their applications. In: Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pp. 587–596. Higher Ed. Press, Beijing (2002)Google Scholar
  71. 71.
    Varadhan, S.R.S.: Large deviations for random walks in a random environment. Comm. Pure Appl. Math. 56(8), 1222–1245 (Dedicated to the memory of Jürgen K. Moser) (2003)Google Scholar
  72. 72.
    Vargas V.: Strong localization and macroscopic atoms for directed polymers. Probab. Theory Relat. Fields 138(3-4), 391–410 (2007)MathSciNetCrossRefMATHGoogle Scholar
  73. 73.
    Zerner M.P.W.: Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment. Ann. Probab. 26(4), 1446–1476 (1998)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Nicos Georgiou
    • 1
  • Firas Rassoul-Agha
    • 2
  • Timo Seppäläinen
    • 3
  1. 1.MathematicsUniversity of SussexBrightonUK
  2. 2.MathematicsUniversity of UtahSalt Lake CityUSA
  3. 3.MathematicsUniversity of Wisconsin-MadisonMadisonUSA

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