The Annals of Probability

Pfaffian Schur processes and last passage percolation in a half-quadrant

Jinho Baik, Guillaume Barraquand, Ivan Corwin, and Toufic Suidan

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We study last passage percolation in a half-quadrant, which we analyze within the framework of Pfaffian Schur processes. For the model with exponential weights, we prove that the fluctuations of the last passage time to a point on the diagonal are either GSE Tracy–Widom distributed, GOE Tracy–Widom distributed or Gaussian, depending on the size of weights along the diagonal. Away from the diagonal, the fluctuations of passage times follow the GUE Tracy–Widom distribution. We also obtain a two-dimensional crossover between the GUE, GOE and GSE distribution by studying the multipoint distribution of last passage times close to the diagonal when the size of the diagonal weights is simultaneously scaled close to the critical point. We expect that this crossover arises universally in KPZ growth models in half-space. Along the way, we introduce a method to deal with diverging correlation kernels of point processes where points collide in the scaling limit.

Article information

Ann. Probab., Volume 46, Number 6 (2018), 3015-3089.

Received: July 2016
Revised: August 2017
First available in Project Euclid: 25 September 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C23: Exactly solvable dynamic models [See also 37K60]
Secondary: 60G55: Point processes 05E05: Symmetric functions and generalizations 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Last passage percolation KPZ universality class Tracy–Widom distributions Schur process Fredholm Pfaffian phase transition


Baik, Jinho; Barraquand, Guillaume; Corwin, Ivan; Suidan, Toufic. Pfaffian Schur processes and last passage percolation in a half-quadrant. Ann. Probab. 46 (2018), no. 6, 3015--3089. doi:10.1214/17-AOP1226.

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  • [1] Baik, J. (2002). Painlevé expressions for LOE, LSE, and interpolating ensembles. Int. Math. Res. Not. IMRN 33 1739–1789.
  • [2] Baik, J., Barraquand, G., Corwin, I. and Suidan, T. (2017). Facilitated exclusion process. Preprint. Available at arXiv:1707.01923.
  • [3] 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.
  • [4] Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119–1178.
  • [5] Baik, J. and Rains, E. M. (2001). Algebraic aspects of increasing subsequences. Duke Math. J. 109 1–65.
  • [6] Baik, J. and Rains, E. M. (2001). The asymptotics of monotone subsequences of involutions. Duke Math. J. 109 205–281.
  • [7] Baik, J. and Rains, E. M. (2001). Symmetrized random permutations. In Random Matrix Models and Their Applications. Math. Sci. Res. Inst. Publ. 40 1–19. Cambridge Univ. Press, Cambridge.
  • [8] Barraquand, G., Borodin, A. and Corwin, I. Half-space Macdonald processes. Preprint.
  • [9] Barraquand, G., Borodin, A., Corwin, I. and Wheeler, M. (2017). Stochastic six-vertex model in a half-quadrant and half-line open ASEP. Preprint. Available at arXiv:1704.04309.
  • [10] Ben Arous, G. and Corwin, I. (2011). Current fluctuations for TASEP: A proof of the Prähofer–Spohn conjecture. Ann. Probab. 39 104–138.
  • [11] Bleher, P. M. and Kuijlaars, A. B. J. (2005). Integral representations for multiple Hermite and multiple Laguerre polynomials. Ann. Inst. Fourier (Grenoble) 55 2001–2014.
  • [12] Bloemendal, A. and Virág, B. (2013). Limits of spiked random matrices I. Probab. Theory Related Fields 156 795–825.
  • [13] Borodin, A. (2011). Schur dynamics of the Schur processes. Adv. Math. 4 2268–2291.
  • [14] Borodin, A., Bufetov, A. and Corwin, I. (2016). Directed random polymers via nested contour integrals. Ann. Physics 368 191–247.
  • [15] Borodin, A. and Corwin, I. (2014). Macdonald processes. Probab. Theory Related Fields 158 225–400.
  • [16] 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.
  • [17] Borodin, A. and Ferrari, P. L. (2014). Anisotropic growth of random surfaces in $2+1$ dimensions. Comm. Math. Phys. 325 603–684.
  • [18] Borodin, A., Ferrari, P. L. and Sasamoto, T. (2008). Large time asymptotics of growth models on space-like paths II: PNG and parallel TASEP. Comm. Math. Phys. 283 417–449.
  • [19] Borodin, A. and Gorin, V. (2012). Lectures on integrable probability, Lectures notes for a summer school in Saint Petersburg. Available at arXiv:1212.3351.
  • [20] Borodin, A. and Petrov, L. (2016). Nearest neighbor Markov dynamics on Macdonald processes. Adv. Math. 300 71–155.
  • [21] Borodin, A. and Rains, E. M. (2005). Eynard–Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121 291–317.
  • [22] Borodin, A. and Strahov, E. (2006). Averages of characteristic polynomials in random matrix theory. Comm. Pure Appl. Math. 59 161–253.
  • [23] Ferrari, P. L. (2004). Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues. Comm. Math. Phys. 252 77–109.
  • [24] Forrester, P. J. (2010). Log-Gases and Random Matrices. London Mathematical Society Monographs Series 34. Princeton Univ. Press, Princeton, NJ.
  • [25] Forrester, P. J., Nagao, T. and Honner, G. (1999). Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges. Nuclear Phys. B 553 601–643.
  • [26] Forrester, P. J., Nagao, T. and Rains, E. M. (2006). Correlation functions for random involutions. Int. Math. Res. Not. IMRN 2006 89796.
  • [27] Ghosal, P. (2017). Correlation functions of the Pfaffian Schur process using Macdonald difference operators. Preprint. Available at arXiv:1705.05859.
  • [28] Gueudré, T. and Le Doussal, P. (2012). Directed polymer near a hard wall and KPZ equation in the half-space. Euro. Phys. Lett. 100 26006.
  • [29] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476.
  • [30] Kerov, S. V. (2003). Asymptotic Representation Theory of the Symmetric Group and Its Applications in Analysis. Translations of Mathematical Monographs 219. Amer. Math. Soc., Providence, RI.
  • [31] Macdonald, I. G. (1995). Symmetric Functions and Hall Polynomials, 2nd ed. Clarendon Press, New York.
  • [32] Matveev, K. and Petrov, L. (2017). $q$-randomized Robinson–Schensted–Knuth correspondences and random polymers. Ann. Inst. Henri Poincaré D 4 1–123.
  • [33] O’Connell, N., Seppäläinen, T. and Zygouras, N. (2014). Geometric RSK correspondence, Whittaker functions and symmetrized random polymers. Invent. Math. 197 361–416.
  • [34] Okounkov, A. (2001). Infinite wedge and random partitions. Selecta Math. (N.S.) 7 57–81.
  • [35] Okounkov, A. and Reshetikhin, N. (2003). Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Amer. Math. Soc. 16 581–603.
  • [36] Ortmann, J., Quastel, J. and Remenik, D. (2017). A Pfaffian representation for flat ASEP. Comm. Pure Appl. Math. 70 3–89.
  • [37] Rains, E. M. (2000). Correlation functions for symmetrized increasing subsequences. Preprint. Available at arXiv:math/0006097 [math.CO].
  • [38] Sasamoto, T. and Imamura, T. (2004). Fluctuations of the one-dimensional polynuclear growth model in half-space. J. Stat. Phys. 115 749–803.
  • [39] Stembridge, J. R. (1990). Nonintersecting paths, Pfaffians, and plane partitions. Adv. Math. 83 96–131.
  • [40] Tracy, C. A. and Widom, H. (2005). Matrix kernels for the Gaussian orthogonal and symplectic ensembles. Ann. Inst. Fourier (Grenoble) 55 2197–2207.
  • [41] Tracy, C. A. and Widom, H. (2013). The asymmetric simple exclusion process with an open boundary. J. Math. Phys. 54 103301, 16.
  • [42] Wang, D. (2009). The largest sample eigenvalue distribution in the rank 1 quaternionic spiked model of Wishart ensemble. Ann. Probab. 37 1273–1328.