Journal of Statistical Physics

, Volume 162, Issue 6, pp 1587–1607 | Cite as

A Generalised Airy Distribution Function for the Accumulated Area Swept by N Vicious Brownian Paths



In this work exact expressions for the distribution function of the accumulated area swept by excursions and meanders of N vicious Brownian particles up to time T are derived. The results are expressed in terms of a generalised Airy distribution function, containing the Vandermonde determinant of the Airy roots. By mapping the problem to an Random Matrix Theory ensemble we are able to perform Monte Carlo simulations finding perfect agreement with the theoretical results.


Random walkers Vicious walkers Random matrices  Airy distribution function 



The authors warmly thank N. Kobayashi and M. Katori for email correspondence regarding the simulations. We also thank E. Barkai for pointing out some references.


  1. 1.
    Adler, M., van Moerbeke, P., Vanderstichelen, D.: Non-intersecting brownian motions leaving from and going to several points. Physica D 241(5), 443–460 (2012)CrossRefADSMathSciNetMATHGoogle Scholar
  2. 2.
    Baik, J.: Random vicious walks and random matrices. Commun. Pure Appl. Math. 53(11), 1385–1410 (2000)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Barkai, E., Aghion, E., Kessler, D.: From the area under the bessel excursion to anomalous diffusion of cold atoms. Phys. Rev. X 4, 021036 (2014)Google Scholar
  4. 4.
    Bleher, P., Delvaux, S., Kuijlaars, A.B.J.: Random matrix model with external source and a constrained vector equilibrium problem. Commun. Pure Appl. Math. 64(1), 116–160 (2011)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Borodin, A., Kuan, J.: Random surface growth with a wall and plancherel measures for \(o(\infty )\). Commun. Pure Appl. Math. 63(7), 831–894 (2010)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Castillo, I.P., Dupic, T.: Reunion probabilities of \( n \) one-dimensional random walkers with mixed boundary conditions. arXiv preprint arXiv:1311.0654 (2013)
  7. 7.
    Daems, E., Kuijlaars, A.: Multiple orthogonal polynomials of mixed type and non-intersecting brownian motions. J. Approx. Theory 146(1), 91–114 (2007)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Darling, D.A.: On the supremum of a certain gaussian process. Ann. Probab. 11, 803 (1983)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Ferrari, P.L., Praehofer, M.: One-dimensional stochastic growth and Gaussian ensembles of random matrices. Markov Processes Relat. Fields 12, 203–234 (2006)Google Scholar
  10. 10.
    Ferrari, P.L., Prähofer, M., Spohn, H.: Fluctuations of an atomic ledge bordering a crystalline facet. Phys. Rev. E 69, 035102 (2004)CrossRefADSGoogle Scholar
  11. 11.
    Flajolet, P., Louchard, G.: Analytic variations on the airy distribution. Algorithmica 31(3), 361–377 (2001)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Forrester, P.J.: Random walks and random permutations. J. Phys. A 34(31), L417 (2001)CrossRefADSMathSciNetMATHGoogle Scholar
  13. 13.
    Forrester, P.J.: Log-gases and random matrices (LMS-34). Princeton University Press, Princeton (2010)CrossRefMATHGoogle Scholar
  14. 14.
    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)CrossRefADSMathSciNetMATHGoogle Scholar
  15. 15.
    Janson, S.: Brownian excursion area, Wright’s constants in graph enumeration, and other brownian areas. Probab. Surv. 4, 80–145 (2007)MathSciNetMATHGoogle Scholar
  16. 16.
    Johansson, K.: Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242(1–2), 277–329 (2003)CrossRefADSMathSciNetMATHGoogle Scholar
  17. 17.
    Katori, M., Tanemura, H.: Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. J. Math. Phys. 45, 3058 (2004)CrossRefADSMathSciNetMATHGoogle Scholar
  18. 18.
    Kearney, M.J., Majumdar, S.N., Martin, R.J.: The first-passage area for drifted brownian motion and the moments of the airy distribution. J. Phys. A 40, F863 (2007)CrossRefADSMathSciNetMATHGoogle Scholar
  19. 19.
    Kessler, D., Medallion, S., Barkai, E.: The distribution of the area under a bessel excursion and its moments. J. Stat. Phys. 156, 686–706 (2014)CrossRefADSMathSciNetMATHGoogle Scholar
  20. 20.
    Kobayashi, N., Izumi, M., Katori, M.: Maximum distributions of bridges of noncolliding brownian paths. Phys. Rev. E 78, 051102 (2008)CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Laurenzi: Polynomials associated with the higher derivatives of the airy functions \(ai(z)\) and \(ai^{\prime }(z)\). arXiv:1110.2025 (2011)
  22. 22.
    Louchard, G.: Kac’s formula, levy’s local time and brownian excursion. J. Appl. Probab. 21, 479 (1984)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Majumdar, S., Comtet, A.: Exact maximal height distribution of fluctuating interfaces. Phys. Rev. Lett. 92, 225501 (2004)CrossRefADSGoogle Scholar
  24. 24.
    Majumdar, S.N.: Brownian functionals in physics and computer science. Curr. Sci. 89(12), 2076 (2005)MathSciNetGoogle Scholar
  25. 25.
    Majumdar, S.N., Comtet, A.: Airy distribution function: from the area under a brownian excursion to the maximal height of fluctuating interfaces. J. Stat. Phys. 119, 777–826 (2005)CrossRefADSMathSciNetMATHGoogle Scholar
  26. 26.
    Medalion, S., Aghion, E., Meirovitch, H., Barkai, E., Kessler, D.A.: Fluctuations of ring polymers. arXiv:1501.06143 (2015)
  27. 27.
    Nadal, C., Majumdar, S.N.: Nonintersecting brownian interfaces and wishart random matrices. Phys. Rev. E 79, 061117 (2009)CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Nagao, T.: Dynamical correlations for vicious random walk with a wall. Nucl. Phys. B 658(3), 373–396 (2003)CrossRefADSMathSciNetMATHGoogle Scholar
  29. 29.
    Novak, J.: Vicious walkers and random contraction matrices. Int. Math. Res. Not. 17, 3310–3327 (2009)MathSciNetMATHGoogle Scholar
  30. 30.
    Saito, N., Yukito, I., Hukushima, K.: Multicanonical sampling of rare events in random matrices. Phys. Rev. E 82, 031142 (2010)CrossRefADSGoogle Scholar
  31. 31.
    Rambeau, J., Schehr, G.: Extremal statistics of curved growing interfaces in 1+1 dimensions. Eur. Lett. 91(6), 60006 (2010)CrossRefADSGoogle Scholar
  32. 32.
    Schehr, G., Majumdar, S.N., Comtet, A., Forrester, P.J.: Reunion probability of n vicious walkers: typical and large fluctuations for large N. J. Stat. Phys. 2013, 1–40 (2013)MathSciNetMATHGoogle Scholar
  33. 33.
    Schehr, G., Majumdar, S.N., Comtet, A., Randon-Furling, J.: Exact distribution of the maximal height of \(p\) vicious walkers. Phys. Rev. Lett. 101, 150601 (2008)CrossRefADSMathSciNetMATHGoogle Scholar
  34. 34.
    Takács, L.: A bernoulli excursion and its various applications. Adv. Appl. Probab. 23, 557 (1991)CrossRefMathSciNetMATHGoogle Scholar
  35. 35.
    Takács, L.: On the distribution of the integral of the absolute value of the brownian motion. Ann. Appl. Probab. 3, 186 (1993)CrossRefMathSciNetMATHGoogle Scholar
  36. 36.
    Takacs, L.: Limit distributions for the bernoulli meander. J. Appl. Probab. 32, 375 (1995)CrossRefMathSciNetMATHGoogle Scholar
  37. 37.
    Tracy, C.A., Widom, H.: Nonintersecting brownian excursions. Ann. Appl. Probab. 17, 953–979 (2007)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Departamento de Sistemas ComplejosInstituto de Física, UNAMMexicoMexico

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