Communications in Mathematical Physics

, Volume 317, Issue 2, pp 363–380

Endpoint Distribution of Directed Polymers in 1 + 1 Dimensions

  • Gregorio Moreno Flores
  • Jeremy Quastel
  • Daniel Remenik
Article

Abstract

We give an explicit formula for the joint density of the max and argmax of the Airy2 process minus a parabola. The argmax has a universal distribution which governs the rescaled endpoint for large time or temperature of directed polymers in 1+1 dimensions.

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References

  1. ACQ11.
    Amir G., Corwin I., Quastel J.: Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions. Comm. Pure Appl. Math. 64(4), 466–537 (2011)MathSciNetMATHCrossRefGoogle Scholar
  2. AKQ12.
    Alberts, T., Khanin, K., Quastel, J.: The continuum directed random polymer. http://arxiv.org/abs/1202.4403v1 [math.PR], 2012
  3. AS64.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Vol. 55. Washington, DC: National Bureau of Standards Applied Mathematics Series, 1964Google Scholar
  4. Bor10a.
    Bornemann F.: On the numerical evaluation of distributions in random matrix theory: A review. Markov Process. Related Fields 16(4), 803–866 (2010)MathSciNetMATHGoogle Scholar
  5. Bor10b.
    Bornemann F.: On the numerical evaluation of Fredholm determinants. Math. Comp. 79(270), 871–915 (2010)MathSciNetADSMATHCrossRefGoogle Scholar
  6. BR01.
    Baik, J., Rains, E.M.: Symmetrized random permutations. In: Random matrix models and their applications. Vol. 40 of Math. Sci. Res. Inst. Publ. Cambridge: Cambridge Univ. Press, 2001, pp. 1–19Google Scholar
  7. CH11.
    Corwin, I., Hammond, A.: Brownian Gibbs property for Airy line ensembles. http://arxiv.org/abs/1108.2291v1 [math.PR], 2011
  8. COSZ11.
    Corwin, I., O’Connell, N., Seppäläinen, T., Zygouras, N.: Tropical combinatorics and Whittaker functions. http://arxiv.org/abs/1110.3489v3 [math.PR], 2011
  9. CQR12.
    Corwin, I., Quastel, J., Remenik, D.: Continuum statistics of the Airy 2 process. Commun. Math. Phys. doi:10.1007/s00220-012-1582-0. http://arxiv.org/abs/1106.2717v4 [math.PR], 2012
  10. Fei09.
    Feierl, T.: The height and range of watermelons without wall. In: Combinatorial Algorithms. Vol. 5874 of Lecture Notes in Computer Science. Berlin-Heidelberg: Springer/Heidelberg, 2009, pp. 242–253Google Scholar
  11. Fis84.
    Fisher M.E.: Walks, walls, wetting, and melting. J. Stat. Phys. 34, 667–729 (1984)ADSMATHCrossRefGoogle Scholar
  12. FMS11.
    Forrester P.J., Majumdar S.N., Schehr G.: Non-intersecting brownian walkers and Yang-Mills theory on the sphere. Nucl. Phys. B 844(3), 500–526 (2011)MathSciNetADSMATHCrossRefGoogle Scholar
  13. FS05.
    Ferrari P.L., Spohn H.: A determinantal formula for the GOE Tracy-Widom distribution. J. Phys. A 38(33), L557–L561 (2005)MathSciNetADSCrossRefGoogle Scholar
  14. HHZ95.
    Halpin-Healy T., Zhang Y.-C.: Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Phys. Rep. 254(4-6), 215–414 (1995)ADSCrossRefGoogle Scholar
  15. Joh03.
    Johansson K.: Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242(1-2), 277–329 (2003)MathSciNetADSMATHGoogle Scholar
  16. MP92.
    Mézard M., Parisi G.: A variational approach to directed polymers. J. Phys. A 25(17), 4521–4534 (1992)MathSciNetADSMATHCrossRefGoogle Scholar
  17. OY01.
    O’Connell N., Yor M.: Brownian analogues of Burke’s theorem. Stochastic Process. Appl. 96(2), 285–304 (2001)MathSciNetMATHCrossRefGoogle Scholar
  18. PS11.
    Prolhac S., Spohn H.: The one-dimensional KPZ equation and the Airy process. J. Stat. Mech. Theor. Exp. 2011(03), P03020 (2011)MathSciNetCrossRefGoogle Scholar
  19. QR12.
    Quastel, J., Remenik, D.: Tails of the endpoint distribution of directed polymers. Ann IHP Prob. Stat. http://arxiv.org/abs/1203.2907v2 [math.PR] (2012, to appear)
  20. RS10.
    Rambeau J., Schehr G.: Extremal statistics of curved growing interfaces in 1+1 dimensions. EPL (Europhysics Letters) 91(6), 60006 (2010)ADSCrossRefGoogle Scholar
  21. RS11.
    Rambeau J., Schehr G.: Distribution of the time at which N vicious walkers reach their maximal height. Phys. Rev. E83, 061146 (2011)ADSGoogle Scholar
  22. Sas05.
    Sasamoto T.: Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A:Math. Gen. 38(33), L549 (2005)MathSciNetADSCrossRefGoogle Scholar
  23. Sep12.
    Seppäläinen T.: Scaling for a one-dimensional directed polymer with boundary. Ann. Probab. 40(1), 19–73 (2012)MathSciNetMATHCrossRefGoogle Scholar
  24. Sim05.
    Simon, B.: Trace ideals and their applications. Second ed., Vol. 120. Mathematical Surveys and Monographs. Amer. Math. Soc., 2005Google Scholar
  25. SMCRF08.
    Schehr G., Majumdar S.N., Comtet A., Randon-Furling J.: Exact distribution of the maximal height of p vicious walkers. Phys. Rev. Lett. 101(15), 150601 (2008)MathSciNetADSCrossRefGoogle Scholar
  26. SS10.
    Sasamoto T., Spohn H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nucl. Phys. B 834(3), 523–542 (2010)MathSciNetADSMATHCrossRefGoogle Scholar
  27. TW96.
    Tracy C.A., Widom H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177(3), 727–754 (1996)MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Gregorio Moreno Flores
    • 1
  • Jeremy Quastel
    • 2
  • Daniel Remenik
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada
  3. 3.Departamento de Ingeniería MatemáticaUniversidad de ChileSantiagoChile

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