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Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 18))

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Abstract

Airy processes arise through a scaling limit of our system of reflected Brownian motions. Their detailed structure depends on the initial conditions. For the various Airy processes appearing in the text we list here the definitions and, in particular, discuss their interrelation.

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References

  • M. Prähofer, H. Spohn, Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071–1106 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • I. Corwin, A. Hammond, Brownian Gibbs property for Airy line ensembles. Invent. Math. 195, 441–508 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • T. Sasamoto, Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38, L549–L556 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  • P.L. Ferrari, H. Spohn, A determinantal formula for the GOE Tracy-Widom distribution. J. Phys. A 38, L557–L561 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  • J. Quastel, D. Remenik, Local behavior and hitting probabilities of the Airy\(_1\) process. Prob. Theory Relat. Fields 157, 605–634 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • A. Borodin, P.L. Ferrari, M. Prähofer, T. Sasamoto, Fluctuation Properties of the TASEP with Periodic Initial Configuration. J. Stat. Phys. 129, 1055–1080 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • J. Baik, E.M. Rains, Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100, 523–542 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • J. Baik, P.L. Ferrari, S. Péché, Limit process of stationary TASEP near the characteristic line. Comm. Pure Appl. Math. 63, 1017–1070 (2010)

    MathSciNet  MATH  Google Scholar 

  • A. Borodin, I. Corwin, D. Remenik, Multiplicative functionals on ensembles of non-intersecting paths. Ann. Inst. H. Poincar Probab. Statist. 51, 28–58 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • J. Baik, R. Buckingham, J. DiFranco, Asymptotics of Tracy-Widom distributions and the total integral of a Painleve II function. Comm. Math. Phys. 280, 463–497 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • A. Borodin, P.L. Ferrari, T. Sasamoto, Transition between Airy\(_1\) and Airy\(_2\) processes and TASEP fluctuations. Comm. Pure Appl. Math. 61, 1603–1629 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • T. Imamura, T. Sasamoto, Fluctuations of the one-dimensional polynuclear growth model with external sources. Nucl. Phys. B 699, 503–544 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • A. Borodin, P.L. Ferrari, T. Sasamoto, Two speed TASEP. J. Stat. Phys. 137, 936–977 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • P.L. Ferrari, H. Spohn, Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265, 1–44 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • K. Johansson, Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242, 277–329 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • I. Corwin, J. Quastel, D. Remenik, Continuum statistics of the Airy\(_2\) process. Comm. Math. Phys. 317, 347–362 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • J. Quastel, D. Remenik, Airy processes and variational problems, Springer Proceedings in Mathematics & Statistics vol. 69, (2014), pp.121–171

    Google Scholar 

  • I. Corwin, Z. Liu, D. Wang, Fluctuations of TASEP and LPP with general initial data. Ann. Appl. Probab. (2014). arXiv:1412.5087

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Correspondence to Herbert Spohn .

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Weiss, T., Ferrari, P., Spohn, H. (2017). Airy Processes. In: Reflected Brownian Motions in the KPZ Universality Class. SpringerBriefs in Mathematical Physics, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-319-49499-9_4

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