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Stationary cocycles and Busemann functions for the corner growth model

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

    Article  MathSciNet  MATH  Google Scholar 

  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))

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

  8. 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 

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

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. 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 

  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)

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

  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. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

  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. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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))

    MathSciNet  MATH  Google Scholar 

  54. 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 

  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. 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.

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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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