Tails of the endpoint distribution of directed polymers

Quastel, Jeremy; Remenik, Daniel

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 1-17.

Nous prouvons qu’une variable aléatoire $𝒯={arg\; max}_{t\in ℝ}\{{𝒜}_2\left(t\right)-t^2\}$, où ${𝒜}_2$ est un processus ${\mathrm{Airy}}_2$ a une queue qui décroît comme ${\mathrm{e}}^{-c{t}^3}$. La distribution de $𝒯$ est une distribution universelle qui gouverne la position du point final d’un polymère dirigé en dimension $1+1$ à temps grand ou à grande température.

We prove that the random variable $𝒯={arg\; max}_{t\in ℝ}\{{𝒜}_2\left(t\right)-t^2\}$, where ${𝒜}_2$ is the ${\mathrm{Airy}}_2$ process, has tails which decay like ${\mathrm{e}}^{-c{t}^3}$. The distribution of $𝒯$ is a universal distribution which governs the rescaled endpoint of directed polymers in $1+1$ dimensions for large time or temperature.

[1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. U.S. Government Printing Office, Washington, DC, 1964. | MR | Zbl

[2] J. Baik, R. Buckingham and J. Difranco. Asymptotics of Tracy–Widom distributions and the total integral of a Painlevé II function. Comm. Math. Phys. 280 (2) (2008) 463–497. | MR | Zbl

[3] J. Baik, K. Liechty and G. Schehr. On the joint distribution of the maximum and its position of the ${\mathrm{Airy}}_2$ process minus a parabola. J. Math. Phys. 53 (8) (2012) 1–13. | MR | Zbl

[4] J. Baik and E. M. Rains. Symmetrized random permutations. In Random Matrix Models and Their Applications. Math. Sci. Res. Inst. Publ. 40 1–19. Cambridge Univ. Press, Cambridge, 2001. | MR | Zbl

[5] A. Borodin, P. L. Ferrari and T. Sasamoto. Transition between ${\mathrm{Airy}}_1$ and ${\mathrm{Airy}}_2$ processes and TASEP fluctuations. Comm. Pure Appl. Math. 61 (11) (2008) 1603–1629. | MR | Zbl

[6] I. Corwin, J. Quastel and D. Remenik. Continuum statistics of the ${\mathrm{Airy}}_2$ process. Comm. Math. Phys. 317 (2) (2013) 347–362. | MR | Zbl

[7] I. Corwin and A. Hammond. Brownian Gibbs property for Airy line ensembles. Invent. Math. 195 (2) (2014) 441–508. | MR | Zbl

[8] P. L. Ferrari and H. Spohn. A determinantal formula for the GOE Tracy–Widom distribution. J. Phys. A 38 (33) (2005) L557–L561. | MR

[9] P. J. Forrester, T. Nagao and G. Honner. Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges. Nuclear Phys. B 553 (3) (1999) 601–643. | MR | Zbl

[10] P. Groeneboom. Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 (1989) 79–109. | DOI | MR

[11] T. Halpin-Healy and Y.-C. Zhang. Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Phys. Rep. 254 (4–6) (1995) 215–414.

[12] K. Johansson. Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 (1–2) (2003) 277–329. | MR | Zbl

[13] A. M. S. Macêdo. Universal parametric correlations at the soft edge of the spectrum of random matrix ensembles. Europhys. Lett. 26 (9) (1994) 641.

[14] M. Mézard and G. Parisi. A variational approach to directed polymers. J. Phys. A 25 (17) (1992) 4521–4534. | MR | Zbl

[15] G. Moreno Flores, J. Quastel and D. Remenik. Endpoint distribution of directed polymers in $1+1$ dimensions. Comm. Math. Phys. 317 (2) (2013) 363–380. | MR | Zbl

[16] M. Prähofer and H. Spohn. Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108 (5–6) (2002) 1071–1106. | MR | Zbl

[17] S. Prolhac and H. Spohn. The one-dimensional KPZ equation and the Airy process. J. Stat. Mech. Theory Exp. 2011 (03) (2011) 1–15. | MR | Zbl

[18] J. Quastel and D. Remenik. Local behavior and hitting probabilities of the ${\mathrm{Airy}}_1$ process. Probab. Theory Related Fields 157 (3–4) (2013) 605–634. | MR | Zbl

[19] J. Quastel and D. Remenik. Supremum of the ${\mathrm{Airy}}_2$ process minus a parabola on a half line. J. Stat. Phys. 150 (3) (2013) 442–456. | MR | Zbl

[20] T. Sasamoto. Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A Math. Gen. 38 (33) (2005) L549–L556. | DOI | MR

[21] G. Schehr. Extremes of $N$ vicious walkers for large $N$: Application to the directed polymer and KPZ interfaces. J. Stat. Phys. 149 (3) (2012) 385–410. | MR | Zbl

[22] B. Simon. Trace Ideals and Their Applications, 2nd edition. Mathematical Surveys and Monographs 120. Amer. Math. Soc., Providence, RI, 2005. | MR | Zbl

[23] C. A. Tracy and H. Widom. Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1) (1994) 151–174. | MR | Zbl

[24] C. A. Tracy and H. Widom. On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 (3) (1996) 727–754. | MR | Zbl

[25] H. Widom. On asymptotics for the Airy process. J. Stat. Phys. 115 (3–4) (2004) 1129–1134. | MR | Zbl