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Variational Formulas and Cocycle solutions for Directed Polymer and Percolation Models

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Abstract

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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References

  1. Aldous D., Diaconis P.: Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Relat. Fields 103(2), 199–213 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Comets F., Shiga T., Yoshida N.: Directed polymers in a random environment: path localization and strong disorder. Bernoulli 9(4), 705–723 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  15. Comets F., Yoshida N.: Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34(5), 1746–1770 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl. 1(1), 1130001, 76 (2012)

  17. Cox J.T., Durrett R.: Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9(4), 583–603 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Damron M., Hanson J.: Busemann functions and infinite geodesics in two-dimensional first-passage percolation. Comm. Math. Phys. 325(3), 917–963 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Dembo, A., Zeitouni, O.: Large deviations techniques and applications, vol. 38 of Applications of Mathematics 2nd (ed.). Springer, New York (1998)

  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))

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Durrett R., Liggett T.M.: The shape of the limit set in Richardson’s growth model. Ann. Probab. 9(2), 186–193 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ferrari P.A., Pimentel L.P.R.: Competition interfaces and second class particles. Ann. Probab. 33(4), 1235–1254 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gandolfi A., Kesten H.: Greedy lattice animals. II. Linear growth. Ann. Appl. Probab. 4(1), 76–107 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Geodesics and the competition interface for the corner growth model. arXiv:1510.00860 (2015)

  27. Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Stationary cocycles and Busemann functions for the corner growth model. arXiv:1510.00859 (2015)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  32. Hoffman C.: Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab. 15(1B), 739–747 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hoffman C.: Geodesics in first passage percolation. Ann. Appl. Probab. 18(5), 1944–1969 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Howard C.D., Newman C.M.: Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Probab. 29(2), 577–623 (2001)

    MathSciNet  MATH  Google Scholar 

  35. Jockusch, W., Propp, J., Shor, P.: Random domino tilings and the arctic circle theorem. arXiv:math/9801068

  36. Johansson K.: Shape fluctuations and random matrices. Comm. Math. Phys. 209(2), 437–476 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  42. Krishnan, A.: Variational formula for the time-constant of first-passage percolation. Comm. Pure Appl. Math. arXiv:1311.0316 (2016) (To appear)

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Licea C., Newman C.M.: Geodesics in two-dimensional first-passage percolation. Ann. Probab. 24(1), 399–410 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  46. Marchand R.: Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab. 12(3), 1001–1038 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  47. Martin J.B.: Limiting shape for directed percolation models. Ann. Probab. 32(4), 2908–2937 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

  51. Pimentel L.P.R.: Multitype shape theorems for first passage percolation models. Adv. Appl. Probab. 39(1), 53–76 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Rockafellar, R.T.: Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J (1970)

  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)

  61. Rost H.: Nonequilibrium behaviour of a many particle process: density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58(1), 41–53 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. Seneta, E.: Nonnegative matrices and Markov chains. Springer Series in Statistics 2nd (ed.). Springer, New York (1981)

  63. Seppäläinen T.: Large deviations for lattice systems. I. Parametrized independent fields. Probab. Theory Relat. Fields 96(2), 241–260 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  64. Seppäläinen, T.: A microscopic model for the Burgers equation and longest increasing subsequences. Electron. J. Probab., 1(5), approx. pp. 51 (1996) (electronic)

  65. Seppäläinen T.: Hydrodynamic scaling, convex duality and asymptotic shapes of growth models. Markov Process. Relat. Fields 4(1), 1–26 (1998)

    MathSciNet  MATH  Google Scholar 

  66. Seppäläinen T.: Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40(1), 19–73 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  67. Spitzer, F.: Principles of random walks, 2nd (ed.). Graduate Texts in Mathematics, Vol. 34. Springer, New York (1976)

  68. Spohn, H.: Stochastic integrability and the KPZ equation. arXiv:1204.2657 (2012)

  69. Stroock, D.W.: An introduction to the theory of large deviations. Universitext. Springer, New York (1984)

  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)

  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)

  72. Vargas V.: Strong localization and macroscopic atoms for directed polymers. Probab. Theory Relat. Fields 138(3-4), 391–410 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematics, University of Sussex, Falmer Campus, Brighton, BN1 9QH, UK

    Nicos Georgiou

  2. Mathematics, University of Utah, 155S 1400E, Salt Lake City, UT, 84112, USA

    Firas Rassoul-Agha

  3. Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Dr., Madison, WI, 53706-1388, USA

    Timo Seppäläinen

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  1. Nicos Georgiou
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  2. Firas Rassoul-Agha
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  3. Timo Seppäläinen
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Correspondence to Timo Seppäläinen.

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Communicated by F. Toninelli

F. Rassoul-Agha and N. Georgiou were partially supported by National Science Foundation Grant DMS-0747758.

F. Rassoul-Agha was partially supported by National Science Foundation Grant DMS-1407574 and by Simons Foundation Grant 306576.

T. Seppäläinen was partially supported by National Science Foundation Grant DMS-1306777, by Simons Foundation Grant 338287, and by the Wisconsin Alumni Research Foundation.

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Georgiou, N., Rassoul-Agha, F. & Seppäläinen, T. Variational Formulas and Cocycle solutions for Directed Polymer and Percolation Models. Commun. Math. Phys. 346, 741–779 (2016). https://doi.org/10.1007/s00220-016-2613-z

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