The Annals of Applied Probability

Fluctuations of TASEP and LPP with general initial data

Ivan Corwin, Zhipeng Liu, and Dong Wang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove Airy process variational formulas for the one-point probability distribution of (discrete time parallel update) TASEP with general initial data, as well as last passage percolation from a general down-right lattice path to a point. We also consider variants of last passage percolation with inhomogeneous parameter geometric weights and provide variational formulas of a similar nature. This proves one aspect of the conjectural description of the renormalization fixed point of the Kardar–Parisi–Zhang universality class.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 4 (2016), 2030-2082.

Dates
Received: December 2014
Revised: June 2015
First available in Project Euclid: 1 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1472745451

Digital Object Identifier
doi:10.1214/15-AAP1139

Mathematical Reviews number (MathSciNet)
MR3543889

Zentralblatt MATH identifier
1356.82013

Subjects
Primary: C22 82B23: Exactly solvable models; Bethe ansatz 60H15: Stochastic partial differential equations [See also 35R60]

Keywords
TASEP last passage percolation Kardar–Parisi–Zhang

Citation

Corwin, Ivan; Liu, Zhipeng; Wang, Dong. Fluctuations of TASEP and LPP with general initial data. Ann. Appl. Probab. 26 (2016), no. 4, 2030--2082. doi:10.1214/15-AAP1139. https://projecteuclid.org/euclid.aoap/1472745451


Export citation

References

  • [1] Baik, J. (2006). Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices. Duke Math. J. 133 205–235.
    Mathematical Reviews (MathSciNet): MR2225691
    Digital Object Identifier: doi:10.1215/S0012-7094-06-13321-5
    Project Euclid: euclid.dmj/1148224038
  • [2] 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.
    Mathematical Reviews (MathSciNet): MR2165575
    Digital Object Identifier: doi:10.1214/009117905000000233
    Project Euclid: euclid.aop/1127395869
  • [3] Baik, J., Deift, P., McLaughlin, K. T.-R., Miller, P. and Zhou, X. (2001). Optimal tail estimates for directed last passage site percolation with geometric random variables. Adv. Theor. Math. Phys. 5 1207–1250.
    Mathematical Reviews (MathSciNet): MR1926668
    Digital Object Identifier: doi:10.4310/ATMP.2001.v5.n6.a7
  • [4] Baik, J., Ferrari, P. L. and Péché, S. (2010). Limit process of stationary TASEP near the characteristic line. Comm. Pure Appl. Math. 63 1017–1070.
    Mathematical Reviews (MathSciNet): MR2642384
  • [5] Baik, J. and Rains, E. M. (2000). Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100 523–541.
    Mathematical Reviews (MathSciNet): MR1788477
    Digital Object Identifier: doi:10.1023/A:1018615306992
  • [6] Baik, J. and Rains, E. M. (2001). Algebraic aspects of increasing subsequences. Duke Math. J. 109 1–65.
    Mathematical Reviews (MathSciNet): MR1844203
    Digital Object Identifier: doi:10.1215/S0012-7094-01-10911-3
    Project Euclid: euclid.dmj/1091737220
  • [7] Ben Arous, G. and Corwin, I. (2011). Current fluctuations for TASEP: A proof of the Prähofer–Spohn conjecture. Ann. Probab. 39 104–138.
    Mathematical Reviews (MathSciNet): MR2778798
    Digital Object Identifier: doi:10.1214/10-AOP550
    Project Euclid: euclid.aop/1291388298
  • [8] 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.
    Mathematical Reviews (MathSciNet): MR2363389
    Digital Object Identifier: doi:10.1007/s10955-007-9383-0
  • [9] 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.
    Mathematical Reviews (MathSciNet): MR2430639
    Digital Object Identifier: doi:10.1007/s00220-008-0515-4
  • [10] Borodin, A., Ferrari, P. L. and Sasamoto, T. (2008). Transition between $\mathrm{Airy}_{1}$ and $\mathrm{Airy}_{2}$ processes and TASEP fluctuations. Comm. Pure Appl. Math. 61 1603–1629.
    Mathematical Reviews (MathSciNet): MR2444377
    Digital Object Identifier: doi:10.1002/cpa.20234
  • [11] Borodin, A., Ferrari, P. L. and Sasamoto, T. (2009). Two speed TASEP. J. Stat. Phys. 137 936–977.
    Mathematical Reviews (MathSciNet): MR2570757
    Digital Object Identifier: doi:10.1007/s10955-009-9837-7
  • [12] Borodin, A. and Okounkov, A. (2000). A Fredholm determinant formula for Toeplitz determinants. Integral Equations Operator Theory 37 386–396.
    Mathematical Reviews (MathSciNet): MR1780118
    Digital Object Identifier: doi:10.1007/BF01192827
  • [13] Corwin, I. (2012). The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1 1130001, 76.
    Mathematical Reviews (MathSciNet): MR2930377
    Digital Object Identifier: doi:10.1142/S2010326311300014
  • [14] Corwin, I., Ferrari, P. L. and Péché, S. (2010). Limit processes for TASEP with shocks and rarefaction fans. J. Stat. Phys. 140 232–267.
    Mathematical Reviews (MathSciNet): MR2659279
    Digital Object Identifier: doi:10.1007/s10955-010-9995-7
  • [15] Corwin, I., Ferrari, P. L. and Péché, S. (2012). Universality of slow decorrelation in KPZ growth. Ann. Inst. Henri Poincaré Probab. Stat. 48 134–150.
    Mathematical Reviews (MathSciNet): MR2919201
    Digital Object Identifier: doi:10.1214/11-AIHP440
    Project Euclid: euclid.aihp/1327328017
  • [16] Corwin, I. and Hammond, A. (2014). Brownian Gibbs property for Airy line ensembles. Invent. Math. 195 441–508.
    Mathematical Reviews (MathSciNet): MR3152753
    Digital Object Identifier: doi:10.1007/s00222-013-0462-3
  • [17] Corwin, I., Quastel, J. and Remenik, D. (2013). Continuum statistics of the $\mathrm{Airy}_{2}$ process. Comm. Math. Phys. 317 347–362.
    Mathematical Reviews (MathSciNet): MR3010187
    Digital Object Identifier: doi:10.1007/s00220-012-1582-0
  • [18] Corwin, I., Quastel, J. and Remenik, D. (2015). Renormalization fixed point of the KPZ universality class. J. Stat. Phys. 160 815–834.
    Mathematical Reviews (MathSciNet): MR3373642
    Digital Object Identifier: doi:10.1007/s10955-015-1243-8
  • [19] Ferrari, P. L. (2008). Slow decorrelations in Kardar–Parisi–Zhang growth. J. Stat. Mech. Theory Exp. 2008 P07022.
  • [20] Ferrari, P. L. and Spohn, H. (2005). A determinantal formula for the GOE Tracy–Widom distribution. J. Phys. A 38 L557–L561.
    Mathematical Reviews (MathSciNet): MR2165698
    Digital Object Identifier: doi:10.1088/0305-4470/38/33/L02
  • [21] Ferrari, P. L. and Spohn, H. (2006). Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265 1–44.
    Mathematical Reviews (MathSciNet): MR2217295
    Digital Object Identifier: doi:10.1007/s00220-006-1549-0
  • [22] Geronimo, J. S. and Case, K. M. (1979). Scattering theory and polynomials orthogonal on the unit circle. J. Math. Phys. 20 299–310.
    Mathematical Reviews (MathSciNet): MR519213
    Digital Object Identifier: doi:10.1063/1.524077
  • [23] Imamura, T. and Sasamoto, T. (2004). Fluctuations of the one-dimensional polynuclear growth model with external sources. Nuclear Phys. B 699 503–544.
    Mathematical Reviews (MathSciNet): MR2098552
    Digital Object Identifier: doi:10.1016/j.nuclphysb.2004.07.030
  • [24] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476.
    Mathematical Reviews (MathSciNet): MR1737991
    Digital Object Identifier: doi:10.1007/s002200050027
  • [25] Johansson, K. (2003). Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 277–329.
    Mathematical Reviews (MathSciNet): MR2018275
    Digital Object Identifier: doi:10.1007/s00220-003-0945-y
  • [26] Kriecherbauer, T. and Krug, J. (2010). A pedestrian’s view on interacting particle systems, KPZ universality and random matrices. J. Phys. A 43 403001, 41.
    Mathematical Reviews (MathSciNet): MR2725545
    Digital Object Identifier: doi:10.1088/1751-8113/43/40/403001
  • [27] Krug, J., Meakin, P. and Halpin-Healy, T. (1992). Amplitude universality for driven interfaces and directed polymers in random media. Phys. Rev. A 45 638–653.
  • [28] Prähofer, M. and Spohn, H. (2002). Current fluctuations for the totally asymmetric simple exclusion process. In In and Out of Equilibrium (Mambucaba, 2000). Progress in Probability 51 185–204. Birkhäuser, Boston, MA.
    Mathematical Reviews (MathSciNet): MR1901953
  • [29] Prähofer, M. and Spohn, H. (2002). Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108 1071–1106.
    Mathematical Reviews (MathSciNet): MR1933446
    Digital Object Identifier: doi:10.1023/A:1019791415147
  • [30] Quastel, J. (2012). Introduction to KPZ. In Current Developments in Mathematics, 2011 125–194. Int. Press, Somerville, MA.
    Mathematical Reviews (MathSciNet): MR3098078
  • [31] Quastel, J. and Remenik, D. (2013). Supremum of the $\mathrm{Airy}_{2}$ process minus a parabola on a half line. J. Stat. Phys. 150 442–456.
    Mathematical Reviews (MathSciNet): MR3024136
    Digital Object Identifier: doi:10.1007/s10955-012-0633-4
  • [32] Quastel, J. and Remenik, D. (2014). Airy processes and variational problems. In Topics in Percolative and Disordered Systems (A. F. Ramírez, G. B. Arous, P. A. Ferrari, C. M. Newman, V. Sidoravicius and M. E. Vares, eds.). Springer Proc. Math. Stat. 69 121–171. Springer, New York.
    Mathematical Reviews (MathSciNet): MR3229288
  • [33] Sasamoto, T. (2005). Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38 L549–L556.
    Mathematical Reviews (MathSciNet): MR2165697
    Digital Object Identifier: doi:10.1088/0305-4470/38/33/L01
  • [34] Spohn, H. (2012). KPZ scaling theory and the semi-discrete directed polymer model. Preprint. Available at arXiv:1201.0645.
    arXiv: 1201.0645
  • [35] Tracy, C. A. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 727–754.
    Mathematical Reviews (MathSciNet): MR1385083
    Digital Object Identifier: doi:10.1007/BF02099545
    Project Euclid: euclid.cmp/1104286442

The Institute of Mathematical Statistics

The Annals of Applied Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more