Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 102, 7 pages      arXiv:1608.01557      https://doi.org/10.3842/SIGMA.2016.102
Contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy

Moments Match between the KPZ Equation and the Airy Point Process

Alexei Borodin ab and Vadim Gorin ab
a) Department of Mathematics, Massachusetts Institute of Technology, USA
b) Institute for Information Transmission Problems of Russian Academy of Sciences, Russia

Received August 09, 2016, in final form October 21, 2016; Published online October 26, 2016

Abstract
The results of Amir-Corwin-Quastel, Calabrese-Le Doussal-Rosso, Dotsenko, and Sasamoto-Spohn imply that the one-point distribution of the solution of the KPZ equation with the narrow wedge initial condition coincides with that for a multiplicative statistics of the Airy determinantal random point process. Taking Taylor coefficients of the two sides yields moment identities. We provide a simple direct proof of those via a combinatorial match of their multivariate integral representations.

Key words: KPZ equation; Airy point process.

pdf (322 kb)   tex (14 kb)

References

  1. Amir G., Corwin I., Quastel J., Probability distribution of the free energy of the continuum directed random polymer in $1+1$ dimensions, Comm. Pure Appl. Math. 64 (2011), 466-537, arXiv:1003.0443.
  2. Anderson G.W., Guionnet A., Zeitouni O., An introduction to random matrices, Cambridge Studies in Advanced Mathematics, Vol. 118, Cambridge University Press, Cambridge, 2010.
  3. Borodin A., Determinantal point processes, in The Oxford Handbook of Random Matrix Theory, Oxford University Press, Oxford, 2011, 231-249, arXiv:0911.1153.
  4. Borodin A., Stochastic higher spin six vertex model and Macdonald measures, arXiv:1608.01553.
  5. Borodin A., Bufetov A., Corwin I., Directed random polymers via nested contour integrals, Ann. Physics 368 (2016), 191-247, arXiv:1511.07324.
  6. Borodin A., Corwin I., Macdonald processes, Probab. Theory Related Fields 158 (2014), 225-400, arXiv:1111.4408.
  7. Borodin A., Olshanski G., The ASEP and determinantal point processes, arXiv:1608.01564.
  8. Calabrese P., Le Doussal P., Rosso A., Free-energy distribution of the directed polymer at high temperature, Europhys. Lett. 90 (2010), 20002, 6 pages, arXiv:1002.4560.
  9. Corwin I., The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012), 1130001, 76 pages, arXiv:1106.1596.
  10. Dotsenko V., Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers, Europhys. Lett. 90 (2010), 20003, 5 pages, arXiv:1003.4899.
  11. Forrester P.J., Log-gases and random matrices, London Mathematical Society Monographs Series, Vol. 34, Princeton University Press, Princeton, NJ, 2010.
  12. Imamura T., Sasamoto T., Determinantal structures in the O'Connell-Yor directed random polymer model, J. Stat. Phys. 163 (2016), 675-713, arXiv:1506.05548.
  13. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995.
  14. O'Connell N., Directed polymers and the quantum Toda lattice, Ann. Probab. 40 (2012), 437-458, arXiv:0910.0069.
  15. Okounkov A., Generating functions for intersection numbers on moduli spaces of curves, Int. Math. Res. Not. 2002 (2002), 933-957, math.AG/0101201.
  16. Quastel J., Introduction to KPZ, in Current Developments in Mathematics, 2011, Int. Press, Somerville, MA, 2012, 125-194, available at http://math.arizona.edu/~mathphys/school_2012/IntroKPZ-Arizona.pdf.
  17. Sasamoto T., Spohn H., One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality, Phys. Rev. Lett. 104 (2010), 230602, 4 pages, arXiv:1002.1883.
  18. Tracy C.A., Widom H., A Fredholm determinant representation in ASEP, J. Stat. Phys. 132 (2008), 291-300, arXiv:0804.1379.
  19. Tracy C.A., Widom H., Integral formulas for the asymmetric simple exclusion process, Comm. Math. Phys. 279 (2008), 815-844, Erratum, Comm. Math. Phys. 304 (2011), 875-878, arXiv:0704.2633.
  20. Tracy C.A., Widom H., Asymptotics in ASEP with step initial condition, Comm. Math. Phys. 290 (2009), 129-154, arXiv:0807.1713.

Previous article  Next article   Contents of Volume 12 (2016)