Geodesic
Moments Match between the KPZ Equation and the Airy Point Process
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: KPZ equation; Airy point process. 
@article{SIGMA_2016_12_a101,
     author = {Alexei Borodin and Vadim Gorin},
     title = {Moments {Match} between the {KPZ} {Equation} and the {Airy} {Point} {Process}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2016},
     volume = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a101/}
}
TY  - JOUR
AU  - Alexei Borodin
AU  - Vadim Gorin
TI  - Moments Match between the KPZ Equation and the Airy Point Process
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2016
VL  - 12
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a101/
LA  - en
ID  - SIGMA_2016_12_a101
ER  - 
%0 Journal Article
%A Alexei Borodin
%A Vadim Gorin
%T Moments Match between the KPZ Equation and the Airy Point Process
%J Symmetry, integrability and geometry: methods and applications
%D 2016
%V 12
%U http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a101/
%G en
%F SIGMA_2016_12_a101
Alexei Borodin; Vadim Gorin. Moments Match between the KPZ Equation and the Airy Point Process. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a101/

[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 | DOI | MR | Zbl

[2] Anderson G. W., Guionnet A., Zeitouni O., An introduction to random matrices, Cambridge Studies in Advanced Mathematics, 118, Cambridge University Press, Cambridge, 2010 | MR | Zbl

[3] Borodin A., “Determinantal point processes”, The Oxford Handbook of Random Matrix Theory, Oxford University Press, Oxford, 2011, 231–249, arXiv: 0911.1153 | MR | Zbl

[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 | DOI | MR

[6] Borodin A., Corwin I., “Macdonald processes”, Probab. Theory Related Fields, 158 (2014), 225–400, arXiv: 1111.4408 | DOI | MR | Zbl

[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 pp., arXiv: 1002.4560 | DOI

[9] Corwin I., “The Kardar–Parisi–Zhang equation and universality class”, Random Matrices Theory Appl., 1 (2012), 1130001, 76 pp., arXiv: 1106.1596 | DOI | MR | Zbl

[10] Dotsenko V., “Bethe ansatz derivation of the Tracy–Widom distribution for one-dimensional directed polymers”, Europhys. Lett., 90 (2010), 20003, 5 pp., arXiv: 1003.4899 | DOI

[11] Forrester P. J., Log-gases and random matrices, London Mathematical Society Monographs Series, 34, Princeton University Press, Princeton, NJ, 2010 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[13] Macdonald I. G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1995 | MR

[14] O'Connell N., “Directed polymers and the quantum Toda lattice”, Ann. Probab., 40 (2012), 437–458, arXiv: 0910.0069 | DOI | MR | Zbl

[15] Okounkov A., “Generating functions for intersection numbers on moduli spaces of curves”, Int. Math. Res. Not., 2002 (2002), 933–957, arXiv: math.AG/0101201 | DOI | MR | Zbl

[16] Quastel J., “Introduction to KPZ”, Current Developments in Mathematics (2011), Int. Press, Somerville, MA, 2012, 125–194 http://math.arizona.edu/\allowbreakm̃athphys/school_2012/IntroKPZ-Arizona.pdf | MR | Zbl

[17] Sasamoto T., Spohn H., “One-dimensional Kardar–Parisi–Zhang equation: an exact solution and its universality”, Phys. Rev. Lett., 104 (2010), 230602, 4 pp., arXiv: 1002.1883 | DOI

[18] Tracy C. A., Widom H., “A Fredholm determinant representation in ASEP”, J. Stat. Phys., 132 (2008), 291–300, arXiv: 0804.1379 | DOI | MR | Zbl

[19] Tracy C. A., Widom H., “Integral formulas for the asymmetric simple exclusion process”, Comm. Math. Phys., 279 (2008), 815–844, arXiv: ; Erratum Comm. Math. Phys., 304 (2011), 875–878 0704.2633 | DOI | MR | Zbl | DOI | MR | Zbl

[20] Tracy C. A., Widom H., “Asymptotics in ASEP with step initial condition”, Comm. Math. Phys., 290 (2009), 129–154, arXiv: 0807.1713 | DOI | MR | Zbl