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Reunion Probability of N Vicious Walkers: Typical and Large Fluctuations for Large N

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Abstract

We consider three different models of N non-intersecting Brownian motions on a line segment [0,L] with absorbing (model A), periodic (model B) and reflecting (model C) boundary conditions. In these three cases we study a properly normalized reunion probability, which, in model A, can also be interpreted as the maximal height of N non-intersecting Brownian excursions (called “watermelons” with a wall) on the unit time interval. We provide a detailed derivation of the exact formula for these reunion probabilities for finite N using a Fermionic path integral technique. We then analyze the asymptotic behavior of this reunion probability for large N using two complementary techniques: (i) a saddle point analysis of the underlying Coulomb gas and (ii) orthogonal polynomial method. These two methods are complementary in the sense that they work in two different regimes, respectively for \(L\ll O(\sqrt{N})\) and \(L\geq O(\sqrt{N})\). A striking feature of the large N limit of the reunion probability in the three models is that it exhibits a third-order phase transition when the system size L crosses a critical value \(L=L_{c}(N)\sim\sqrt{N}\). This transition is akin to the Douglas-Kazakov transition in two-dimensional continuum Yang-Mills theory. While the central part of the reunion probability, for LL c (N), is described in terms of the Tracy-Widom distributions (associated to GOE and GUE depending on the model), the emphasis of the present study is on the large deviations of these reunion probabilities, both in the right [LL c (N)] and the left [LL c (N)] tails. In particular, for model B, we find that the matching between the different regimes corresponding to typical LL c (N) and atypical fluctuations in the right tail LL c (N) is rather unconventional, compared to the usual behavior found for the distribution of the largest eigenvalue of GUE random matrices. This paper is an extended version of (Schehr et al. in Phys. Rev. Lett. 101:150601, 2008) and (Forrester et al. in Nucl. Phys. B 844:500–526, 2011).

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Notes

  1. Note that we have corrected a sign error from Ref. [29] and reproduced also in Ref. [19].

References

  1. de Gennes, P.G.: Soluble model for fibrous structures with steric constraints. J. Chem. Phys. 48, 2257–2259 (1968)

    Article  ADS  Google Scholar 

  2. Fisher, M.E.: Walks, walls, wetting, and melting. J. Stat. Phys. 34, 667–728 (1984)

    Article  ADS  MATH  Google Scholar 

  3. Krattenthaler, C., Guttmann, A.J., Viennot, X.G.: Vicious walkers, friendly walkers and Young tableaux: II. With a wall. J. Phys. A, Math. Gen. 33, 8835–8866 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Baik, J.: Random vicious walks and random matrices. Commun. Pure Appl. Math. 53, 1385–1410 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Forrester, P.J.: Random walks and random permutations. J. Phys. A 34, L417–L423 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329 (2003)

    MathSciNet  ADS  MATH  Google Scholar 

  7. Nagao, T.: Dynamical correlations for vicious random walk with a wall. Nucl. Phys. B 658, 373–396 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Katori, M., Tanemura, H.: Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. J. Math. Phys. 45, 3058–3086 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Ferrari, P., Praehofer, M.: One-dimensional stochastic growth and Gaussian ensembles of random matrices. Markov Process. Relat. Fields 12, 203–234 (2006). Proc. Inhomogeneous Random Systems 2005

    MATH  Google Scholar 

  10. Tracy, C.A., Widom, H.: Non-intersecting Brownian excursions. Ann. Appl. Probab. 17, 953–979 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Daems, E., Kuijlaars, A.B.J.: Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions. J. Approx. Theory 146, 91–114 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  ADS  Google Scholar 

  13. Nadal, C., Majumdar, S.N.: Non-intersecting Brownian interfaces and Wishart random matrices. Phys. Rev. E 79, 061117 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  14. Novak, J.: Vicious walkers and random contraction matrices. Int. Math. Res. Not. 2009, 3310–3327 (2009)

    MATH  Google Scholar 

  15. Rambeau, J., Schehr, G.: Extremal statistics of curved growing interfaces in 1+1 dimensions. Europhys. Lett. 91, 60006 (2010)

    Article  ADS  Google Scholar 

  16. Borodin, A., Kuan, J.: Random surface growth with a wall and Plancherel measures for O(∞). Commun. Pure Appl. Math. 63, 831–894 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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, 116–160 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Adler, M., van Moerbeke, P., Vanderstichelen, D.: Non-intersecting Brownian motions leaving and going to several points. Physica D 241, 443–460 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Forrester, P.J., Majumdar, S.N., Schehr, G.: Non-intersecting Brownian walkers and Yang-Mills theory on the sphere. Nucl. Phys. B 844, 500–526 (2011). Erratum: Nucl. Phys. B 857, 424–427 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. de Haro, S., Tierz, M.: Brownian motion, Chern-Simons theory, and 2d Yang-Mills. Phys. Lett. B 201, 201–208 (2004)

    Google Scholar 

  21. Nadal, C., Majumdar, S.N.: A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix. J. Stat. Mech. P04001 (2011)

  22. Vivo, P., Majumdar, S.N., Bohigas, O.: Distributions of conductance and shot noise and associated phase transitions. Phys. Rev. Lett. 101, 216809 (2008)

    Article  ADS  Google Scholar 

  23. Vivo, P., Majumdar, S.N., Bohigas, O.: Probability distributions of linear statistics in chaotic cavities and associated phase transitions. Phys. Rev. B 81, 104202 (2010)

    Article  ADS  Google Scholar 

  24. Damle, K., Majumdar, S.N., Tripathi, V., Vivo, P.: Phase transitions in the distribution of the Andreev conductance of superconductor-metal junctions with many transverse modes. Phys. Rev. Lett. 107, 177206 (2011)

    Article  ADS  Google Scholar 

  25. Nadal, C., Majumdar, S.N., Vergassola, M.: Phase transitions in the distribution of bipartite entanglement of a random pure state. Phys. Rev. Lett. 104, 110501 (2010)

    Article  ADS  Google Scholar 

  26. Nadal, C., Majumdar, S.N., Vergassola, M.: Statistical distribution of quantum entanglement for a random bipartite state. J. Stat. Phys. 142, 403–438 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. Douglas, M.R., Kazakov, V.A.: Large N phase transition in continuum QCD2. Phys. Lett. B 319, 219–230 (1993)

    Article  ADS  Google Scholar 

  30. Gross, D.J., Matytsin, A.: Instanton induced large N phase transitions in two and four dimensional QCD. Nucl. Phys. B 429, 50–74 (1994)

    Article  ADS  Google Scholar 

  31. Crescimanno, M., Naculich, S.G., Schnitzer, H.J.: Evaluation of the free energy of two-dimensional Yang-Mills theory. Phys. Rev. D 54, 1809–1813 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  32. Dean, D.S., Majumdar, S.N.: Large deviations of extreme eigenvalues of random matrices. Phys. Rev. Lett. 97, 160201 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  33. Vivo, P., Majumdar, S.N., Bohigas, O.: Large deviations of the maximum eigenvalue in Wishart random matrices. J. Phys. A, Math. Theor. 40, 4317–4337 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Dean, D.S., Majumdar, S.N.: Extreme value statistics of eigenvalues of Gaussian random matrices. Phys. Rev. E 77, 041108 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  35. Majumdar, S.N., Vergassola, M.: Large deviations of the maximum eigenvalue for Wishart and Gaussian random matrices. Phys. Rev. Lett. 102, 060601 (2009)

    Article  ADS  Google Scholar 

  36. Borot, G., Eynard, B., Majumdar, S.N., Nadal, C.: Large deviations of the maximal eigenvalue of random matrices. J. Stat. Mech. P11024 (2011)

  37. Forrester, P.J.: Spectral density asymptotics for Gaussian and Laguerre β-ensembles in the exponentially small region. J. Phys. A 45, 075206 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  38. Bonichon, N., Mosbah, M.: Watermelon uniform random generation with applications. Theor. Comput. Sci. 307, 241–256 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. Grabiner, D.J.: Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. Henri Poincaré B, Probab. Stat. 35, 177–204 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. Grabiner, D.J.: Random walk in an alcove of an affine Weyl group, and non-colliding random walks on an interval. J. Comb. Theory, Ser. A 97, 285–306 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  41. Fulmek, M.: Asymptotics of the average height of 2 watermelons with a wall. Electron. J. Comb. 14(1), R64/1–20 (2007)

    MathSciNet  Google Scholar 

  42. Katori, M., Izumi, M., Kobayashi, N.: Two Bessel bridges conditioned never to collide, double Dirichlet series, and Jacobi theta function. J. Stat. Phys. 131, 1067–1083 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. Kobayashi, N., Izumi, M., Katori, M.: Maximum distributions of bridges of noncolliding Brownian paths. Phys. Rev. E 78, 051102 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  44. Feierl, T.: The height of watermelons with wall. J. Phys. A, Math. Theor. 45, 095003 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  45. Rambeau, J., Schehr, G.: Distribution of the time at which N vicious walkers reach their maximal height. Phys. Rev. E 83, 061146 (2011)

    Article  ADS  Google Scholar 

  46. Liechty, K.: Non-intersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials. J. Stat. Phys. 147, 582–622 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. Borodin, A., Ferrari, P.L., Praehofer, M., Sasamoto, T., Warren, J.: Maximum of Dyson Brownian motion and non-colliding systems with a boundary. Electron. Commun. Probab. 14, 486–494 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. Prähofer, M., Spohn, H.: Universal distributions for growth processes in 1+1 dimensions and random matrices. Phys. Rev. Lett. 84, 4882–4885 (1999)

    Article  ADS  Google Scholar 

  49. Prähofer, M., Spohn, H.: Exact scaling functions for one-dimensional stationary KPZ growth. J. Stat. Phys. 115(1–2), 255–279 (2004)

    Article  ADS  MATH  Google Scholar 

  50. Corwin, I., Hammond, A.: Brownian Gibbs property for airy line ensembles. Preprint. arXiv:1108.2291

  51. Moreno Flores, G.R., Quastel, J., Remenik, D.: Endpoint distribution of directed polymers in 1+1 dimensions. Commun. Math. Phys. (to appear). Preprint arXiv:1106.2716

  52. Schehr, G.: Extremes of N vicious walkers for large N: application to the directed polymer and KPZ interfaces. arXiv:1203.1658

  53. Quastel, J., Remenik, D.: Tails of the endpoint distribution of directed polymers. arXiv:1203.2907

  54. Baik, J., Liechty, K., Schehr, G.: On the joint distribution of the maximum and its position of the Airy2 process minus a parabola. arXiv:1205.3665

  55. Takeuchi, K., Sano, M.: Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence. J. Stat. Phys. 147, 853–890 (2012)

    Article  ADS  MATH  Google Scholar 

  56. Karlin, S., McGregor, J.: Coincidence probabilities. Pac. J. Math. 9, 1141 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  57. Lindström, B.: On the vector representations of induced matroids. Bull. Lond. Math. Soc. 5, 85 (1973)

    Article  MATH  Google Scholar 

  58. Gessel, I., Viennot, G.: Determinants, paths, and plane partitions. Adv. Math. 58, 300 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  59. Gross, D.J., Witten, E.: Possible third-order phase transition in the large-n lattice gauge limit. Phys. Rev. D 21, 446–453 (1980)

    Article  ADS  Google Scholar 

  60. Wadia, S.R.: N=∞ phase transition in a class of exactly soluble model lattice gauge theories. Phys. Lett. B 93, 403–410 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  61. Periwal, V., Shevitz, D.: Unitary-matrix models as exactly solvable string theories. Phys. Rev. Lett. 64, 1326–1329 (1990)

    Article  ADS  Google Scholar 

  62. Dyson, F.J.: Statistical theory of the energy levels of complex systems. I. J. Math. Phys. 3, 140–156 (1962)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  63. Dyson, F.J.: Statistical theory of the energy levels of complex systems. II. J. Math. Phys. 3, 157–165 (1962)

    Article  MathSciNet  ADS  Google Scholar 

  64. Dyson, F.J.: Statistical theory of the energy levels of complex systems. III. J. Math. Phys. 3, 166–174 (1962)

    Article  MathSciNet  ADS  Google Scholar 

  65. Forrester, P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)

    MATH  Google Scholar 

  66. Gross, D.J., Matytsin, A.: Some properties of large N two dimensional Yang-Mills theory. Nucl. Phys. B 437, 541 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  67. Erdélyi, A., et al.: Higher Transcendental Functions. McGraw-Hill, New York (1953)

    Google Scholar 

  68. Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence (1975)

    MATH  Google Scholar 

  69. Baik, J., Jenkins, R.: Limiting distribution of maximal crossing and nesting of Poissonized random matchings. Ann. Probab. (to appear). Preprint arXiv:1111.0269

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Acknowledgements

This research was partially supported by ANR grant 2011-BS04-013-01 WALKMAT and in part by the Indo-French Centre for the Promotion of Advanced Research under Project 4604-3.

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Correspondence to Grégory Schehr.

Appendix: Details About the Large Deviation Regime

Appendix: Details About the Large Deviation Regime

In this appendix, we give some details concerning the calculation of \(\tilde{F}_{N}(L)\) in the large deviation regime.

1.1 A.1 Derivation of the Formula Given in Eq. (141)

We first provide a derivation of the formula given in Eq. (141) starting from (61) and (97). Indeed, from the definition of R k (α) one has

(221)

and more generally

(222)

Therefore, Ω N (α) in Eq. (96) can be rewritten as

(223)

which is the formula given in Eq. (141).

1.2 A.2 Derivation of the Formula Given in Eq. (142)

In this appendix we give a detailed derivation, starting from the exact expression of \(\tilde{F}_{N}(L)\) in Eqs. (61), (62), of the asymptotic estimate for \(\ln\tilde{F}_{N}(L)\), valid in the limit \(L \gg\sqrt{2N} \gg1\):

(224)

in terms of the variable ξ=π 2/α=2L 2 and the functions G 2N (ξ) given in Eq. (127) and I 2N (ξ) given by

$$ I_{2N}(\xi) = - 2 \xi \sum_{k=0}^{N-1} \frac{G_{2k+1}(\xi)}{2k+1}. $$
(225)

This formula (224) was given in Ref. [31] [see their Eq. (28)] without any detail: that is the purpose of this appendix to fill this gap by providing a detailed derivation of it.

To study the formula for \(\tilde{F}_{N}(L)\) starting from Eqs. (61), (62), we first write the product of the gamma functions in the denominator as

(226)

Note that the product of factorials can be written as

(227)

Similarly, the product of Γ(k+1/2) in Eq. (226) can be written as

(228)

where we have used

(229)

Finally, using Eqs. (227) and (228) we write Eq. (226) as

(230)

where we have used \(\varGamma(1/2) = \sqrt{\pi}\).

Using the ansatz for R k (α) given in Eq. (116) where we consider \(c_{k}(\alpha) e^{-\pi^{2}/\alpha} \ll1\), with α=π 2/2L 2, we write,

(231)

and for k≥1

(232)

so that, using the formula in Eq. (230) one obtains that \(\tilde{F}_{N}(L)\) in Eqs. (61), (62) can be written as

(233)

We now perform the expansion of this expression (233), considering \(c_{k}(\alpha) e^{-\pi^{2}/\alpha} \ll1\). It is more convenient to expand its logarithm \(\ln\tilde{F}_{N}(L)\) which using the explicit expression for c 1(α) in Eq. (117) together with the expression of c k (α) in terms of G k (ξ) in Eq. (120), can be written as a function of ξ=π 2/α as

(234)

where we have used G 1(ξ)=1. It is then possible to write these two sums in Eq. (234) as

(235)

Finally, using the following identity satisfied by the polynomials G k (ξ) [30]

(236)

which can be shown, for instance, by using their explicit expression of G k (ξ) (127), one obtains finally

(237)

as given in the text in Eq. (142).

1.3 A.3 An Integral Representation for I 2N (ξ) in Eq. (143)

We start with the integral representation for G k (ξ) in Eq. (126) to express I 2N (ξ) in Eq. (143) as

(238)

Performing an integration by part one obtains

(239)

By performing the sum over k in (239) and dropping an N-independent constant term, one finally arrives at the expression given in the text in Eq. (144).

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Schehr, G., Majumdar, S.N., Comtet, A. et al. Reunion Probability of N Vicious Walkers: Typical and Large Fluctuations for Large N . J Stat Phys 150, 491–530 (2013). https://doi.org/10.1007/s10955-012-0614-7

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