Communications in Mathematical Physics

, Volume 317, Issue 2, pp 347–362

# Continuum Statistics of the Airy2 Process

Article

## Abstract

We develop an exact determinantal formula for the probability that the Airy_2 process is bounded by a function g on a finite interval. As an application, we provide a direct proof that $${\sup(\mathcal{A}_{2}(x)-x^2)}$$ is distributed as a GOE random variable. Both the continuum formula and the GOE result have applications in the study of the end point of an unconstrained directed polymer in a disordered environment. We explain Johansson’s (Commun. Math. Phys. 242(1–2):277–329, 2003) observation that the GOE result follows from this polymer interpretation and exact results within that field. In a companion paper (Moreno Flores et al. in Commun. Math. Phys. 2012) these continuum statistics are used to compute the distribution of the endpoint of directed polymers.

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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)
2. AGZ10.
Anderson, G.W., Guionnet, A., Zeitouni, O.: An introduction to random matrices., Vol. 118. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 2010Google Scholar
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. BFP08.
Bornemann F ., Ferrari P., Prähofer M.: The Airy_1 Process is not the Limit of the Largest Eigenvalue in GOE Matrix Diffusion. J. Stat. Phys. 133, 405–415 (2008)
5. BFS08.
Borodin A., Ferrari P.L., Sasamoto T.: Large time asymptotics of growth models on space-like paths. II. PNG and parallel TASEP. Commun. Math. Phys. 283(2), 417–449 (2008)
6. BR01.
Baik, J., Rains, E.M.: Symmetrized random permutations. In: Random matrix models and their applications. Vol. 40. Math. Sci. Res. Inst. Publ., Cambridge: Cambridge Univ. Press, 2001, pp. 1–19Google Scholar
7. BS02.
Borodin, A.N., Salminen, P.: Handbook of Brownian motion—facts and formulae. Second Edition. Probability and its Applications. Basel: Birkhäuser Verlag, 2002Google Scholar
8. CH11.
Corwin, I., Hammond, A.: Brownian Gibbs property for Airy line ensembles. http://arxiv.org/abs/1108.2291v1 [math.PR], 2011
9. Dys62.
Dyson F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198 (1962)
10. Erd56.
Erdélyi, A.: Asymptotic expansions. New York: Dover Publications Inc., 1956Google Scholar
11. Fei09.
Feierl, T.: The Height and Range of Watermelons without Wall. In: Combinatorial Algorithms. Vol. 5874. Lecture Notes in Computer Science. Berlin-Heidelberg: Springer, 2009, pp. 242–253Google Scholar
12. Fis84.
Fisher M.E.: Walks, walls, wetting, and melting. J. Stat. Phys. 34, 667–729 (1984)
13. 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)
14. FNH99.
Forrester P.J., Nagao T., Honner G.: Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges. Nucl. Phys. B. 553(3), 601–643 (1999)
15. FS05.
Ferrari P.L., Spohn H.: A determinantal formula for the GOE Tracy-Widom distribution. J. Phys. A. 38, L557–L561 (2005)
16. Joh03.
Johansson K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242(1-2), 277–329 (2003)
17. Mac94.
Macêdo A.M.S.: Universal Parametric Correlations at the Soft Edge of the Spectrum of Random Matrix Ensembles. Europhys. Lett. 26(9), 641 (1994)
18. MQR11.
Moreno Flores, G., Quastel, J., Remenik, D.: Endpoint distribution of directed polymers in 1+1 dimensions. Commun. Math. Phys. (2012). doi:10.1007/s00220-012-1583-z
19. PS02.
Prähofer M., Spohn H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108(5-6), 1071–1106 (2002)
20. PS11.
Prolhac S., Spohn H.: The one-dimensional KPZ equation and the Airy process. J. Stat. Mech. Theor. Exp. 2011(03), P03020 (2011)
21. QR12.
Quastel, J., Remenik, D.: Local behavior and hitting probabilities of the Airy1 process. http://arxiv.org/abs/1201.4709v3 [math.PR], 2012
22. RS10.
Rambeau J., Schehr G.: Extremal statistics of curved growing interfaces in 1+1 dimensions. EPL (Europhysics Letters) 91(6), 60006 (2010)
23. RS11.
Rambeau J., Schehr G.: Distribution of the time at which N vicious walkers reach their maximal height. Phys. Rev. E 83, 061146 (2011)
24. Sas05.
Sasamoto T.: Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A: Math. and Gen. 38, L549 (2005)
25. Sim05.
Simon, B.: Trace ideals and their applications. Second ed., Vol. 120. Mathematical Surveys and Monographs. Providence, RI: Amer. Math. Soc., 2005Google Scholar
26. TW96.
Tracy C.A., Widom H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177(3), 727–754 (1996)
27. VS10.
Vallée, O., Soares, M.: Airy functions and applications to physics. Second ed., London: Imperial College Press, 2010Google Scholar