Electronic Journal of Probability

The hard-edge tacnode process for Brownian motion

Patrik L. Ferrari and Bálint Vető

Full-text: Open access


We consider $N$ non-intersecting Brownian bridges conditioned to stay below a fixed threshold. We consider a scaling limit where the limit shape is tangential to the threshold. In the large $N$ limit, we determine the limiting distribution of the top Brownian bridge conditioned to stay below a function as well as the limiting correlation kernel of the system. It is a one-parameter family of processes which depends on the tuning of the threshold position on the natural fluctuation scale. We also discuss the relation to the six-vertex model and to the Aztec diamond on restricted domains.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 79, 32 pp.

Received: 25 January 2017
Accepted: 21 August 2017
First available in Project Euclid: 2 October 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60G55: Point processes
Secondary: 60J65: Brownian motion [See also 58J65] 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

non-colliding walks determinantal processes tacnode process edge scaling limit

Creative Commons Attribution 4.0 International License.


Ferrari, Patrik L.; Vető, Bálint. The hard-edge tacnode process for Brownian motion. Electron. J. Probab. 22 (2017), paper no. 79, 32 pp. doi:10.1214/17-EJP97. https://projecteuclid.org/euclid.ejp/1506931229

Export citation


  • [1] M. Adler, S. Chhita, K. Johansson, and P. van Moerbeke, Tacnode GUE-minor processes and double Aztec diamonds, Probab. Theory Related Fields 162 (2015), 275–325.
  • [2] M. Adler, J. Delépine, and P. van Moerbeke, Dyson’s nonintersecting Brownian motions with a few outliers, Comm. Pure Appl. Math. 62 (2010), 334–395.
  • [3] M. Adler, P.L. Ferrari, and P. van Moerbeke, Non-intersecting random walks in the neighborhood of a symmetric tacnode, Ann. Probab. 41 (2013), 2599–2647.
  • [4] M. Adler, K. Johansson, and P. van Moerbeke, Double Aztec diamonds and the tacnode process, Adv. Math. 252 (2014), 518–571.
  • [5] P. Bleher and A. Kuijlaars, Large $n$ limit of Gaussian random matrices with external source, Part III: Double scaling limit, Comm. Math. Phys. 270 (2007), 481–517.
  • [6] A. Borodin, Biorthogonal ensembles, Nucl. Phys. B 536 (1999), 704–732.
  • [7] A. Borodin, Schur dynamics of the Schur processes, Adv. Math. 228 (2011), 2268–2291.
  • [8] A. Borodin, I. Corwin, and D. Remenik, Multiplicative functionals on ensembles of non-intersecting paths, Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), 28–58.
  • [9] A. Borodin and P.L. Ferrari, Random tilings and Markov chains for interlacing particles, preprint: arXiv:1506.03910 (2015).
  • [10] A. Borodin, P.L. Ferrari, M. Prähofer, T. Sasamoto, and J. Warren, Maximum of Dyson Brownian motion and non-colliding systems with a boundary, Electron. Comm. Probab. 14 (2009), 486–494.
  • [11] A. Borodin, P.L. Ferrari, and T. Sasamoto, Transition between Airy$_1$ and Airy$_2$ processes and TASEP fluctuations, Comm. Pure Appl. Math. 61 (2008), 1603–1629.
  • [12] A. Borodin and G. Olshanski, Stochastic dynamics related to Plancherel measure, AMS Transl.: Representation Theory, Dynamical Systems, and Asymptotic Combinatorics (V. Kaimanovich and A. Lodkin, eds.), 2006, pp. 9–22.
  • [13] A. Borodin and E.M. Rains, Eynard-Mehta theorem, Schur process, and their Pfaffian analogs, J. Stat. Phys. 121 (2006), 291–317.
  • [14] E. Cator and L. Pimentel, On the local fluctuations of last-passage percolation models, Stoch. Proc. Appl. 125 (2015), 879–903.
  • [15] F. Colomo and A. Pronko, Third-order phase transition in random tilings, Phys. Rev. E 88 (2013), 042125.
  • [16] F. Colomo and A. Pronko, Thermodynamics of the six-vertex model in an $L$-shaped domain, Comm. Math. Phys. 339 (2015), 699–728.
  • [17] F. Colomo, A. Pronko, and A. Sportiello, Generalized emptiness formation probability in the six-vertex model, J. Phys. A: Math. Theor. 49 (2016), 415203.
  • [18] F. Colomo and A. Sportiello, Arctic curves of the six-vertex model on generic domains: the Tangent Method, J. Stat. Phys. 164 (2016), 1488.
  • [19] I. Corwin and A. Hammond, Brownian Gibbs property for Airy line ensembles, Invent. Math. 195 (2013), 441–508.
  • [20] I. Corwin, J. Quastel, and D. Remenik, Continuum statistics of the Airy$_2$ process, Comm. Math. Phys. 317 (2013), 347–362.
  • [21] S. Delvaux, The tacnode kernel: equality of Riemann-Hilbert and Airy resolvent formulas, preprint: arXiv:1211.4845 (2012).
  • [22] S. Delvaux, Non-Intersecting Squared Bessel Paths at a Hard-Edge Tacnode, Comm. Math. Phys. 324 (2013), 715–766.
  • [23] S. Delvaux, A. Kuijlaars, and L. Zhang, Critical behavior of non-intersecting Brownian motions at a tacnode, Comm. Pure Appl. Math. 64 (2011), 1305–1383.
  • [24] S. Delvaux and B. Vető, The hard edge tacnode process and the hard edge Pearcey process with non-intersecting squared Bessel paths, Random Matrices Theory Appl. 4 (2015), 1550008.
  • [25] P. Desrosiers and P.J. Forrester, A note on biorthogonal ensembles, J. Approx. Theory 152 (2008), 167–187.
  • [26] F.J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix, J. Math. Phys. 3 (1962), 1191–1198.
  • [27] B. Eynard and M.L. Mehta, Matrices coupled in a chain. I. Eigenvalue correlations, J. Phys. A 31 (1998), 4449–4456.
  • [28] P.L. Ferrari, Shape fluctuations of crystal facets and surface growth in one dimension, Ph.D. thesis, Technische Universität München, http://tumb1.ub.tum.de/publ/diss/ma/2004/ferrari.html, 2004.
  • [29] P.L. Ferrari and H. Spohn, Step fluctations for a faceted crystal, J. Stat. Phys. 113 (2003), 1–46.
  • [30] P.L. Ferrari and H. Spohn, Domino tilings and the six-vertex model at its free fermion point, J. Phys. A: Math. Gen. 39 (2006), 10297–10306.
  • [31] P.L. Ferrari, H. Spohn, and T. Weiss, Brownian motions with one-sided collisions: the stationary case, Electron. J. Probab. 20 (2015), 1–41.
  • [32] P.L. Ferrari and B. Vető, Non-colliding Brownian bridges and the asymmetric tacnode process, Electron. J. Probab. 44 (2012), 1–17.
  • [33] J. Hägg, Local Gaussian fluctuations in the Airy and discrete PNG processes, Ann. Probab. 36 (2008), 1059–1092.
  • [34] K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. 153 (2001), 259–296.
  • [35] K. Johansson, Non-intersecting paths, random tilings and random matrices, Probab. Theory Related Fields 123 (2002), 225–280.
  • [36] K. Johansson, The arctic circle boundary and the Airy process, Ann. Probab. 33 (2005), 1–30.
  • [37] K. Johansson, Random matrices and determinantal processes, Mathematical Statistical Physics, Session LXXXIII: Lecture Notes of the Les Houches Summer School 2005 (A. Bovier, F. Dunlop, A. van Enter, F. den Hollander, and J. Dalibard, eds.), Elsevier Science, 2006, pp. 1–56.
  • [38] K. Johansson, Non-colliding Brownian Motions and the extended tacnode process, Commun. Math. Phys. 319 (2013), 231–267.
  • [39] S. Karlin and L. McGregor, Coincidence probabilities, Pacific J. 9 (1959), 1141–1164.
  • [40] M. Katori and H. Tanemura, Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems, J. Math. Phys. 45 (2004), 3058–3085.
  • [41] M. Katori and H. Tanemura, Infinite systems of noncolliding generalized meanders and Riemann-Liouville differintegrals, Probab. Theory Relat. Fields 138 (2007), 113–156.
  • [42] M. Katori and H. Tanemura, Noncolliding Brownian Motion and Determinantal Processes, J. Stat. Phys. 129 (2007), 1233–1277.
  • [43] V. Korepin and P. Zinn-Justin, Thermodynamic limit of the six-vertex model with domain wall boundary conditions, J. Phys. A 33 (2000), 7053–7066.
  • [44] I. Krasikov, New bounds on the Hermite polynomials, arXiv:math.CA/0401310 (2004).
  • [45] A. B. J. Kuijlaars, A. Martínez-Finkelshtein, and F. Wielonsky, Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit, Comm. Math. Phys. 308 (2011), 227–279.
  • [46] K. Liechty and D. Wang, Nonintersecting Brownian bridges between reflecting or absorbing walls, Adv. Math. 309 (2017), 155–208.
  • [47] T. Nagao, Dynamical Correlations for Vicious Random Walk with a Wall, Nucl. Phys. B 658 (2003), 373–396.
  • [48] T. Nagao and P.J. Forrester, Multilevel dynamical correlation functions for Dyson’s Brownian motion model of random matrices, Phys. Lett. A 247 (1998), 42–46.
  • [49] T. Nagao, M. Katori, and H. Tanemura, Dynamical correlations among vicious random walkers, Phys. Lett. A 307 (2003), 29–33.
  • [50] G.B. Nguyen and D. Remenik, Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble, Ann. Inst. Henri Poincaré Probab. Stat. to appear (2015).
  • [51] M. Prähofer and H. Spohn, Scale invariance of the PNG droplet and the Airy process, J. Stat. Phys. 108 (2002), 1071–1106.
  • [52] M. Reed and B. Simon, Methods of modern mathematical physics IV: Analysis of operators, Academic Press, New York, 1978.
  • [53] G. Szegő, Orthogonal polynomials, 3th ed., American Mathematical Society Providence, Rhode Island, 1967.
  • [54] C.A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), 151–174.
  • [55] C.A. Tracy and H. Widom, Differential equations for Dyson processes, Comm. Math. Phys. 252 (2004), 7–41.
  • [56] C.A. Tracy and H. Widom, The Pearcey Process, Comm. Math. Phys. 263 (2006), 381–400.
  • [57] C.A. Tracy and H. Widom, Nonintersecting Brownian Excursions, Ann. Appl. Prob. 17 (2007), 953–979.
  • [58] P. Zinn-Justin, Six-vertex model with domain wall boundary conditions and one-matrix models, Phys. Rev. E 62 (2000), 3411–3418.