Advertisement

Tracy-Widom Asymptotics for a River Delta Model

  • Guillaume BarraquandEmail author
  • Mark Rychnovsky
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 282)

Abstract

We study an oriented first passage percolation model for the evolution of a river delta. This model is exactly solvable and occurs as the low temperature limit of the beta random walk in random environment. We analyze the asymptotics of an exact formula from [13] to show that, at any fixed positive time, the width of a river delta of length L approaches a constant times \(L^{2/3}\) with Tracy-Widom GUE fluctuations of order \(L^{4/9}\). This result can be rephrased in terms of particle systems. We introduce an exactly solvable particle system on the integer half line and show that after running the system for only finite time the particle positions have Tracy-Widom fluctuations.

Keywords

KPZ universality First passage percolation Exclusion processes Tracy-Widom distribution Integrable probability 

Notes

Acknowledgements

The authors thank Ivan Corwin for many helpful discussions and for useful comments on an earlier draft of the paper. The authors thank an anonymous reviewer for detailed and helpful comments on the manuscript. G. B. was partially supported by the NSF grant DMS:1664650. M. R. was partially supported by the Fernholz Foundation’s “Summer Minerva Fellow” program, and also received summer support from Ivan Corwin’s NSF grant DMS:1811143.

References

  1. 1.
    Aggarwal, A.: Current fluctuations of the stationary ASEP and six-vertex model. Duke Math. J. 167(2), 269–384 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aggarwal, A.: Dynamical stochastic higher spin vertex models. Selecta Math. 24, 2659–2735 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Aggarwal, A., Borodin, A.: Phase transitions in the ASEP and stochastic six-vertex model. Ann. Probab. 47(2), 613–689 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Auffinger, A., Baik, J., Corwin, I.: Universality for directed polymers in thin rectangles. arXiv preprint arXiv:1204.4445 (2012)
  5. 5.
    Auffinger, A., Damron, M., Hanson, J.: 50 Years of First-Passage Percolation, University Lecture Series, vol. 68. American Mathematical Society, Providence (2017)Google Scholar
  6. 6.
    Baik, J., Barraquand, G., Corwin, I., Suidan, T.: Facilitated exclusion process. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H.Z. (eds.) Abel Symposium 2016: Computation and Combinatorics in Dynamics, Stochastics and Control, pp. 1–35, Springer, Cham (2018)Google Scholar
  7. 7.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Baik, J., Suidan, T.M.: A GUE central limit theorem and universality of directed first and last passage site percolation. Int. Math. Res. Not. 2005(6), 325–337 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Balázs, M., Rassoul-Agha, F., Seppäläinen, T.: Large deviations and wandering exponent for random walk in a dynamic beta environment. arXiv preprint arXiv:1801.08070 (2018)
  10. 10.
    Barraquand, G.: A phase transition for q-TASEP with a few slower particles. Stoch. Process. Appl. 125(7), 2674–2699 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Barraquand, G., Borodin, A., Corwin, I., Wheeler, M.: Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process. Duke Math. J. 167(13), 2457–2529 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Barraquand, G., Corwin, I.: The \(q\)-Hahn asymmetric exclusion process. Ann. Appl. Probab. 26(4), 2304–2356 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Barraquand, G., Corwin, I.: Random-walk in Beta-distributed random environment. Probab. Theor. Relat. Fields 167(3–4), 1057–1116 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bodineau, T., Martin, J.: A universality property for last-passage percolation paths close to the axis. Electron. Commun. Probab. 10, 105–112 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Borodin, A.: On a family of symmetric rational functions. Adv. Math. 306, 973–1018 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Borodin, A., Corwin, I.: Macdonald processes. Probab. Theor. Relat. Fields 158(1–2), 225–400 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Borodin, A., Corwin, I., Ferrari, P.: Free energy fluctuations for directed polymers in random media in 1+1 dimension. Commun. Pure Appl. Math. 67(7), 1129–1214 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Borodin, A., Corwin, I., Ferrari, P., Vető, B.: Height fluctuations for the stationary KPZ equation. Math. Phys. Anal. Geom. 18(1), Art. 20, 95 (2015)Google Scholar
  19. 19.
    Borodin, A., Corwin, I., Gorin, V.: Stochastic six-vertex model. Duke Math. J. 165(3), 563–624 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Borodin, A., Corwin, I., Petrov, L., Sasamoto, T.: Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz. Commun. Math. Phys. 339(3), 1167–1245 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Borodin, A., Corwin, I., Remenik, D.: Log-gamma polymer free energy fluctuations via a Fredholm determinant identity. Commun. Math. Phys. 324(1), 215–232 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Borodin, A., Corwin, I., Sasamoto, T.: From duality to determinants for q-TASEP and ASEP. Ann. Probab. 42(6), 2314–2382 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Borodin, A., Ferrari, P.: Large time asymptotics of growth models on space-like paths I: PushASEP. Electron. J. Probab. 13, 1380–1418 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Borodin, A., Olshanski, G.: The ASEP and determinantal point processes. Commun. Math. Phys. 353(2), 853–903 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Borodin, A., Petrov, L.: Higher spin six vertex model and symmetric rational functions. Selecta Math. 24(2), 751–874 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Chaumont, H., Noack, C.: Fluctuation exponents for stationary exactly solvable lattice polymer models via a Mellin transform framework. ALEA Lat. Am. J. Probab. Math. Stat. 15(1), 509–547 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl. 1(01), 1130001 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Corwin, I.: The q-Hahn Boson process and q-Hahn TASEP. Int. Math. Res. 2015(14), 5577–5603 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Corwin, I.: Kardar-Parisi-Zhang universality. Not. AMS 63, 230–239 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Corwin, I., Gu, Y.: Kardar-Parisi-Zhang equation and large deviations for random walks in weak random environments. J. Stat. Phys. 166(1), 150–168 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Corwin, I., Petrov, L.: Stochastic higher spin vertex models on the line. Commun. Math. Phys. 343(2), 651–700 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Corwin, I., Seppäläinen, T., Shen, H.: The strict-weak lattice polymer. J. Stat. Phys. 160, 1027–1053 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Ferrari, P., Vető, B.: Tracy-Widom asymptotics for q-TASEP, 51(4), 1465–1485 (2015)Google Scholar
  34. 34.
    Fontes, L., Isopi, M., Newman, C., Ravishankar, K.: The Brownian web. Proc. Nat. Acad. Sci. 99(25), 15888–15893 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Fontes, L., Isopi, M., Newman, C., Ravishankar, K.: The Brownian web: characterization and convergence. Ann. Probab. 32(4), 2857–2883 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Ghosal, P.: Hall-Littlewood-pushTASEP and its KPZ limit. arXiv preprint arXiv:1701.07308 (2017)
  37. 37.
    Hammersley, J.M., Welsh, D.J.A.: First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In: Proceedings of an International Research Seminar, Statistical Laboratory, University of California, Berkeley, California, pp. 61–110. Springer (1965)Google Scholar
  38. 38.
    Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209(2), 437–476 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)zbMATHCrossRefGoogle Scholar
  40. 40.
    Krishnan, A., Quastel, J.: Tracy-Widom fluctuations for perturbations of the log-gamma polymer in intermediate disorder. Ann. Appl. Probab. 28(6), 3736–3764 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Newman, C., Ravishankar, K., Schertzer, E.: Marking (1, 2) points of the Brownian web and applications, 46(2), 537–574 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    O’Connell, N., Ortmann, J.: Tracy-Widom asymptotics for a random polymer model with gamma-distributed weights. Electron. J. Probab. 20(25), 1–18 (2015)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Orr, D., Petrov, L.: Stochastic higher spin six vertex model and q-TASEPs. Adv. Math. 317, 473–525 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Povolotsky, A.: On the integrability of zero-range chipping models with factorized steady states. J. Phys. A 46, 465205 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Prähofer, M., Spohn, H.: Universal distributions for growth processes in 1+1 dimensions and random matrices. Phys. Rev. Lett. 84(21), 4882 (2000)CrossRefGoogle Scholar
  46. 46.
    Schertzer, E., Sun, R., Swart, J.: The Brownian web, the Brownian net, and their universality. In: Advances in Disordered Systems, Random Processes and Some Applications, pp. 270–368 (2015)Google Scholar
  47. 47.
    Sun, R., Swart, J.M.: The Brownian net. Ann. Probab. 36(3), 1153–1208 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Thiery, T.: Stationary measures for two dual families of finite and zero temperature models of directed polymers on the square lattice. J. Stat. Phys. 165(1), 44–85 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Thiery, T., Le Doussal, P.: On integrable directed polymer models on the square lattice. J. Phys. A 48(46), 465001 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Thiery, T., Le Doussal, P.: Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer. J. Phys. A 50(4), 045001 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Tracy, C.A., Widom, H.: A Fredholm determinant representation in ASEP. J. Stat. Phys. 132(2), 291–300 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Tracy, C.A., Widom, H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279(3), 815–844 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Tracy, C.A., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290(1), 129–154 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Vető, B.: Tracy-Widom limit of \(q\)-Hahn TASEP. Electron. J. Probab. 20, 1–22 (2015)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA

Personalised recommendations