The Annals of Probability

Airy processes with wanderers and new universality classes

Mark Adler, Patrik L. Ferrari, and Pierre van Moerbeke

Full-text: Open access

Abstract

Consider n+m nonintersecting Brownian bridges, with n of them leaving from 0 at time t=−1 and returning to 0 at time t=1, while the m remaining ones (wanderers) go from m points ai to m points bi. First, we keep m fixed and we scale ai, bi appropriately with n. In the large-n limit, we obtain a new Airy process with wanderers, in the neighborhood of $\sqrt{2n}$, the approximate location of the rightmost particle in the absence of wanderers. This new process is governed by an Airy-type kernel, with a rational perturbation.

Letting the number m of wanderers tend to infinity as well, leads to two Pearcey processes about two cusps, a closing and an opening cusp, the location of the tips being related by an elliptic curve. Upon tuning the starting and target points, one can let the two tips of the cusps grow very close; this leads to a new process, which might be governed by a kernel, represented as a double integral involving the exponential of a quintic polynomial in the integration variables.

Article information

Source
Ann. Probab., Volume 38, Number 2 (2010), 714-769.

Dates
First available in Project Euclid: 9 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1268143531

Digital Object Identifier
doi:10.1214/09-AOP493

Mathematical Reviews number (MathSciNet)
MR2642890

Zentralblatt MATH identifier
1200.60069

Subjects
Primary: 60G60: Random fields 60G65 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 60G10: Stationary processes 35Q58

Keywords
Dyson’s Brownian motion Airy process Pearcey process extended kernels random Hermitian ensembles quintic kernel coupled random matrices

Citation

Adler, Mark; Ferrari, Patrik L.; van Moerbeke, Pierre. Airy processes with wanderers and new universality classes. Ann. Probab. 38 (2010), no. 2, 714--769. doi:10.1214/09-AOP493. https://projecteuclid.org/euclid.aop/1268143531


Export citation

References

  • [1] Adler, M., Delépine, J. and van Moerbeke, P. (2009). Dyson’s nonintersecting Brownian motions with a few outliers. Comm. Pure Appl. Math. 62 334–395.
  • [2] Adler, M. and van Moerbeke, P. (2005). PDEs for the joint distributions of the Dyson, Airy and sine processes. Ann. Probab. 33 1326–1361.
  • [3] Adler, M. and van Moerbeke, P. (2007). PDEs for the Gaussian ensemble with external source and the Pearcey distribution. Comm. Pure Appl. Math. 60 1261–1292.
  • [4] Aptekarev, A. I., Bleher, P. M. and Kuijlaars, A. B. J. (2005). Large n limit of Gaussian random matrices with external source. II. Comm. Math. Phys. 259 367–389.
  • [5] Baik, J., Ben Arous, G. and Péché, S. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 1643–1697.
  • [6] Baik, J. (2006). Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices. Duke Math. J. 133 205–235.
  • [7] Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B. (2006). Determinantal processes and independence. Probab. Surv. 3 206–229.
  • [8] Bleher, P. M. and Kuijlaars, A. B. J. (2004). Random matrices with external source and multiple orthogonal polynomials. Int. Math. Res. Not. 3 109–129.
  • [9] Bornemann, F. (2009). On the numerical evaluation of distributions in random matrix theory: A review with an invitation to experimental mathematics. Available at arXiv:0904.1581v4.
  • [10] Brézin, E. and Hikami, S. (1998). Level spacing of random matrices in an external source. Phys. Rev. E (3) 58 7176–7185.
  • [11] Borodin, A. (1999). Biorthogonal ensembles. Nuclear Phys. B 536 704–732.
  • [12] Borodin, A. and Ferrari, P. L. (2008). Large time asymptotics of growth models on space-like paths. I. PushASEP. Electron. J. Probab. 13 1380–1418.
  • [13] Borodin, A., Ferrari, P. L., Prähofer, M. and Sasamoto, T. (2007). Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 1055–1080.
  • [14] Borodin, A., Ferrari, P. L. and Sasamoto, T. (2009). Two speed TASEP. Available at arXiv:0904.4655.
  • [15] Borodin, A. and Péché, S. (2008). Airy kernel with two sets of parameters in directed percolation and random matrix theory. J. Stat. Phys. 132 275–290.
  • [16] Borodin, A. and Rains, E. M. (2005). Eynard–Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121 291–317.
  • [17] Daems, E. and Kuijlaars, A. B. J. (2007). Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions. J. Approx. Theory 146 91–114.
  • [18] Delvaux, S. and Kuijlaars, A. (2009). A phase transition for non-intersecting Brownian motions, and the Painlevé II equation. Available at arXiv:0809.1000v1.
  • [19] Dyson, F. J. (1962). A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 1191–1198.
  • [20] Eynard, B. and Mehta, M. L. (1998). Matrices coupled in a chain. I. Eigenvalue correlations. J. Phys. A 31 4449–4456.
  • [21] Ferrari, P. L. (2004). Shape fluctuation of crystal facets and surface growth in one dimension. Ph.D. thesis, Technical Univ. Munich. Available at http://tumb1.biblio.tu-muenchen.de/publ/diss/ma/2004/ferrari.html.
  • [22] Ferrari, P. L. and Spohn, H. (2003). Step fluctuations for a faceted crystal. J. Stat. Phys. 113 1–46.
  • [23] Imamura, T. and Sasamoto, T. (2007). Dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition. J. Stat. Phys. 128 799–846.
  • [24] Johansson, K. (2003). Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 277–329.
  • [25] Johansson, K. (2006). Random matrices and determinantal processes. In Mathematical Statistical Physics, Session LXXXIII (A. Bovier et al., eds.) 1–56. Elsevier, Amsterdam.
  • [26] Johansson, K. (2001). Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys. 215 683–705.
  • [27] Johansson, K. (2003). Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 277–329.
  • [28] Johansson, K. (2005). The arctic circle boundary and the Airy process. Ann. Probab. 33 1–30.
  • [29] Johansson, K. (2005). Non-intersecting, simple, symmetric random walks and the extended Hahn kernel. Ann. Inst. Fourier (Grenoble) 55 2129–2145.
  • [30] Karlin, S. and McGregor, J. (1959). Coincidence probabilities. Pacific J. Math. 9 1141–1164.
  • [31] Lyons, R. (2003). Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. 98 167–212.
  • [32] Nagao, T. and Forrester, P. J. (1998). Multilevel dynamical correlation functions for Dyson’s Brownian motion model of random matrices. Phys. Lett. A 247 42–46.
  • [33] Okounkov, A. and Reshetikhin, N. (2007). Random skew plane partitions and the Pearcey process. Comm. Math. Phys. 269 571–609.
  • [34] Pastur, L. A. (1972). The spectrum of random matrices. Teoret. Mat. Fiz. 10 102–112.
  • [35] Péché, S. (2006). The largest eigenvalue of small rank perturbations of Hermitian random matrices. Probab. Theory Related Fields 134 127–173.
  • [36] Prähofer, M. and Spohn, H. (2002). Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108 1071–1106.
  • [37] Tracy, C. A. and Widom, H. (1998). Correlation functions, cluster functions, and spacing distributions for random matrices. J. Stat. Phys. 92 809–835.
  • [38] Tracy, C. A. and Widom, H. (2004). Differential equations for Dyson processes. Comm. Math. Phys. 252 7–41.
  • [39] Tracy, C. A. and Widom, H. (2006). The Pearcey process. Comm. Math. Phys. 263 381–400.
  • [40] Soshnikov, A. (2000). Determinantal random point fields. Uspekhi Mat. Nauk 55 107–160.
  • [41] Spohn, H. (2006). Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals. Phys. A 369 71–99.
  • [42] Zinn-Justin, P. (1997). Random Hermitian matrices in an external field. Nuclear Phys. B 497 725–732.
  • [43] Zinn-Justin, P. (1998). Universality of correlation functions of Hermitian random matrices in an external field. Comm. Math. Phys. 194 631–650.