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Stationary cocycles and Busemann functions for the corner growth model

Abstract

We study the directed last-passage percolation model on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside of the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles serve as boundary conditions for stationary last-passage percolation, solve variational formulas that characterize limit shapes, and yield existence of Busemann functions in directions where the shape has some regularity. In a sequel to this paper the cocycles are used to prove results about semi-infinite geodesics and the competition interface.

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References

  1. 1.

    Alm, S.E.: A note on a problem by Welsh in first-passage percolation. Comb. Probab. Comput. 7(1), 11–15 (1998)

  2. 2.

    Alm, S.E., Wierman, J.C.: Inequalities for means of restricted first-passage times in percolation theory. Comb. Probab. Comput. 8(4), 307–315 (1999). (Random graphs and combinatorial structures (Oberwolfach, 1997))

  3. 3.

    Armstrong, S.N., Souganidis, P.E.: Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments. J. Math. Pure Appl. (9) 97(5), 460–504 (2012)

  4. 4.

    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)

  5. 5.

    Bakhtin, Y.: Burgers equation with random boundary conditions. Proc. Am. Math. Soc. 135(7), 2257–2262 (2007). (electronic)

  6. 6.

    Bakhtin, Y.: The Burgers equation with Poisson random forcing. Ann. Probab. 41(4), 2961–2989 (2013)

  7. 7.

    Bakhtin, Y.: Inviscid Burgers equation with random kick forcing in noncompact setting (2014). Preprint arXiv:1406.5660

  8. 8.

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

  9. 9.

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

  10. 10.

    Balázs, M., Cator, E., Seppäläinen, T.: Cube root fluctuations for the corner growth model associated to the exclusion process. Electron. J. Probab. 11(42), 1094–1132 (2006). (electronic)

  11. 11.

    Burkholder, D.L.: Distribution function inequalities for martingales. Ann. Probab. 1, 19–42 (1973)

  12. 12.

    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)

  13. 13.

    Cator, E., Groeneboom, P.: Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab. 34(4), 1273–1295 (2006)

  14. 14.

    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)

  15. 15.

    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)

  16. 16.

    Cohn, H., Elkies, N., Propp, J.: Local statistics for random domino tilings of the Aztec diamond. Duke Math. J. 85(1), 117–166 (1996)

  17. 17.

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

  18. 18.

    Corwin, I.: The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1(1), 1130001, 76 (2012). doi:10.1142/S2010326311300014

  19. 19.

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

  20. 20.

    Durrett, R.: Oriented percolation in two dimensions. Ann. Probab. 12(4), 999–1040 (1984)

  21. 21.

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

  22. 22.

    E, W., Khanin, K., Mazel, A., Sinai, Y.: Invariant measures for Burgers equation with stochastic forcing. Ann. Math (2) 151(3), 877–960 (2000)

  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)

  24. 24.

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

  25. 25.

    Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Stationary cocycles for the corner growth model (2014). Preprint arXiv:1404.7786

  26. 26.

    Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Geodesics and the competition interface for the corner growth model. Probab. Theory Relat. Fields (2016). doi:10.1007/s00440-016-0734-0

  27. 27.

    Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Variational formulas and cocycle solutions for directed polymer and percolation models. Commun. Math. Phys. (2016). To appear (arXiv:1311.0316)

  28. 28.

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

  29. 29.

    Glynn, P.W., Whitt, W.: Departures from many queues in series. Ann. Appl. Probab. 1(4), 546–572 (1991)

  30. 30.

    Hoang, V.H., Khanin, K.: Random Burgers equation and Lagrangian systems in non-compact domains. Nonlinearity 16(3), 819–842 (2003)

  31. 31.

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

  32. 32.

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

  33. 33.

    den Hollander, F.: Random polymers. Lecture Notes in Mathematics, vol. 1974. Springer-Verlag, Berlin (2009)

  34. 34.

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

  35. 35.

    Iturriaga, R., Khanin, K.: Burgers turbulence and random Lagrangian systems. Commun. Math. Phys. 232(3), 377–428 (2003)

  36. 36.

    Jockusch, W., Propp, J., Shor, P.: Random domino tilings and the arctic circle theorem (1998). arXiv:math/9801068

  37. 37.

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

  38. 38.

    Johansson, K.: Random matrices and determinantal processes. In: Mathematical statistical physics, pp. 1–55. Elsevier B. V., Amsterdam (2006)

  39. 39.

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

  40. 40.

    Loynes, R.M.: The stability of a queue with non-independent interarrival and service times. Proc. Camb. Philos. Soc. 58, 497–520 (1962)

  41. 41.

    Mairesse, J., Prabhakar, B.: The existence of fixed points for the \(\cdot /GI/1\) queue. Ann. Probab. 31(4), 2216–2236 (2003)

  42. 42.

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

  43. 43.

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

  44. 44.

    Muth, E.J.: The reversibility property of production lines. Management Sci. 25(2), 152–158 (1979/80)

  45. 45.

    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. Birkhäuser, Basel (1995)

  46. 46.

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

  47. 47.

    Pimentel, L.P.R.: Duality between coalescence times and exit points in last-passage percolation models. Ann. Probab. (2015). To appear (arXiv:1307.7769)

  48. 48.

    Prabhakar, B.: The attractiveness of the fixed points of a \(\cdot /GI/1\) queue. Ann. Probab. 31(4), 2237–2269 (2003)

  49. 49.

    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)

  50. 50.

    Rassoul-Agha, F., Seppäläinen, T., Yılmaz, A.: Quenched free energy and large deviations for random walks in random potentials. Commun. Pure Appl. Math. 66(2), 202–244 (2013)

  51. 51.

    Rassoul-Agha, F., Seppäläinen, T., Yılmaz, A.: Variational formulas and disorder regimes of random walks in random potentials. Bernoulli (2016). To appear (arXiv:1410.4474)

  52. 52.

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

  53. 53.

    Seppäläinen, T.: Coupling the totally asymmetric simple exclusion process with a moving interface. Markov Process. Relat. Fields 4(4), 593–628 (1998). (I Brazilian School in Probability (Rio de Janeiro, 1997))

  54. 54.

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

  55. 55.

    Seppäläinen, T.: Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40(1), 19–73 (2012). Corrected version available at arXiv:0911.2446

  56. 56.

    Wüthrich, M.V.: Asymptotic behaviour of semi-infinite geodesics for maximal increasing subsequences in the plane. In: In and out of equilibrium (Mambucaba, 2000), Progr. Probab., vol. 51, pp. 205–226. Birkhäuser Boston, Boston (2002)

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Acknowledgments

The authors thank Yuri Bakhtin and Michael Damron for useful discussions and two anonymous referees for valuable comments. N. Georgiou was partially supported by a Wylie postdoctoral fellowship at the University of Utah and the Strategic Development Fund (SDF) at the University of Sussex. 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 grants DMS-1306777 and DMS-1602486, by Simons Foundation grant 338287, and by the Wisconsin Alumni Research Foundation.

Author information

Correspondence to Timo Seppäläinen.

Appendix: Ergodic theorem for cocycles

Appendix: Ergodic theorem for cocycles

Cocycles satisfy a uniform ergodic theorem. The following is a special case of Theorem 9.3 of [28]. Note that a one-sided bound suffices for a hypothesis. Recall Definition 2.1 for the space \(\mathscr {K}_0\) of centered cocycles.

Theorem 7.8

Assume \(\mathbb {P}\) is ergodic under the transformations \(\{T_{e_i}:i\in \{1,2\}\}\). Let \(F\in \mathscr {K}_0\). Assume there exists a function V such that for \(\mathbb {P}\)-a.e. \(\omega \)

$$\begin{aligned} \varlimsup _{\varepsilon \searrow 0}\;\varlimsup _{n\rightarrow \infty } \;\max _{x: \vert x\vert _1\le n}\;\frac{1}{n} \sum _{0\le k\le \varepsilon n} \vert V(T_{x+ke_i}\omega )\vert =0\qquad \text {for }i\in \{1,2\} \end{aligned}$$
(7.23)

and \(\max _{i\in \{1,2\}} F(\omega ,0,e_i)\le V(\omega )\). Then

$$\begin{aligned} \lim _{n\rightarrow \infty }\;\max _{\begin{array}{c} x=z_1+\cdots +z_n\\ z_{1,n}\in \{e_1, e_2\}^n \end{array}} \;\frac{\vert F(\omega ,0,x)\vert }{n}=0 \qquad \text {for } \mathbb {P}\text {-a.e. } \omega .\end{aligned}$$

If the process \(\{V(T_x\omega ):x\in \mathbb {Z}^2\}\) is i.i.d., then a sufficient condition for (7.23) is \(\mathbb {E}(\vert V\vert ^p)<\infty \) for some \(p>2\) [50, Lemma A.4].

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Georgiou, N., Rassoul-Agha, F. & Seppäläinen, T. Stationary cocycles and Busemann functions for the corner growth model. Probab. Theory Relat. Fields 169, 177–222 (2017). https://doi.org/10.1007/s00440-016-0729-x

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Keywords

  • Busemann function
  • Cocycle
  • Competition interface
  • Directed percolation
  • Geodesic
  • Last-passage percolation
  • Percolation cone
  • Queueing fixed point
  • Variational formula

Mathematics Subject Classification

  • 60K35
  • 65K37