Tracy–Widom asymptotics for q-TASEP
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1465-1485.

On considère le modèle du q-TASEP, qui est une modification du processus d’exclusion totalement asymétrique (TASEP) sur ℤ. Le taux des sauts d’une particule dépend de la distance avec la particule à sa droite et d’un paramètre q∈[0,1). Dans cet article on considère la condition initiale où ℤ - est completement occupé par les particules. On montre que les fluctuations du courant au temps τ sont d’ordre τ 1/3 et la distribution asymptotique est la distribution de Tracy–Widom GUE. Ce résultat confirme la conjecture KPZ.

We consider the q-TASEP that is a q-deformation of the totally asymmetric simple exclusion process (TASEP) on ℤ for q∈[0,1) where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the current fluctuation of q-TASEP at time τ is of order τ 1/3 and asymptotically distributed as the GUE Tracy–Widom distribution, which confirms the KPZ scaling theory conjecture.

DOI : 10.1214/14-AIHP614
Mots-clés : interacting particle systems, kpz universality class, $q$-TASEP, current fluctuation, Tracy–Widom distribution 
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     author = {Ferrari, Patrik L. and Vet\H{o}, B\'alint},
     title = {Tracy{\textendash}Widom asymptotics for $q${-TASEP}},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1465--1485},
     publisher = {Gauthier-Villars},
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Ferrari, Patrik L.; Vető, Bálint. Tracy–Widom asymptotics for $q$-TASEP. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1465-1485. doi : 10.1214/14-AIHP614. http://www.numdam.org/articles/10.1214/14-AIHP614/

[1] G. Barraquand. A phase transition for q-TASEP with a few slower particles. Stochastic Process. Appl. 125 (2015) 2674–2699. | DOI | MR | Zbl

[2] A. Borodin and I. Corwin. Discrete time q-TASEPs. Int. Math. Res. Not. IMRN 2015 (2015) 499–537. | DOI | MR | Zbl

[3] A. Borodin and I. Corwin. Macdonald processes. Probab. Theory Related Fields 158 (2014) 225–400. | DOI | MR | Zbl

[4] A. Borodin, I. Corwin and P. L. Ferrari. Free energy fluctuations for directed polymers in random media in 1+1 dimension. Comm. Pure Appl. Math. 67 (2014) 1129–1214. | DOI | MR | Zbl

[5] A. Borodin, I. Corwin, L. Petrov and T. Sasamoto. Spectral theory for the q-Boson particle system. Compos. Math. 151 (2015) 1–67. | DOI | MR | Zbl

[6] A. Borodin, I. Corwin and T. Sasamoto. From duality to determinants for q-TASEP and ASEP. Ann. Probab. 42 (2014) 2314–2382. | DOI | MR | Zbl

[7] A. Borodin and P. L. Ferrari. Large time asymptotics of growth models on space-like paths I: PushASEP. Electron. J. Probab. 13 (2008) 1380–1418. | DOI | MR | Zbl

[8] A. Borodin, P. L. Ferrari, M. Prähofer and T. Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 (2007) 1055–1080. | DOI | MR | Zbl

[9] I. Corwin, P. L. Ferrari and S. Péché. Universality of slow decorrelation in KPZ models. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 134–150. | DOI | Numdam | MR | Zbl

[10] I. Corwin and L. Petrov. The q-PushASEP: A new integrable model for traffic in 1+1 dimension. J. Stat. Phys. 160 (2015) 1005–1026. | DOI | MR | Zbl

[11] P. L. Ferrari. From interacting particle systems to random matrices. J. Stat. Mech. 2010 (2010) P10016. | MR | Zbl

[12] T. Imamura and T. Sasamoto. Dynamical properties of a tagged particle in the totally asymmetric simple exclusion process with the step-type initial condition. J. Stat. Phys. 128 (2007) 799–846. | DOI | MR | Zbl

[13] K. Johansson. Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 (2003) 277–329. | DOI | MR | Zbl

[14] M. Korhonen and E. Lee. The transition probability and the probability for the left-most particle’s position of the q-TAZRP. J. Math. Phys. 55 (2014) 013301. | MR | Zbl

[15] N. O’Connell. Directed polymers and the quantum Toda lattice. Ann. Probab. 40 (2012) 437–458. | DOI | MR | Zbl

[16] T. Sasamoto. Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38 (2005) L549–L556. | MR

[17] T. Sasamoto and M. Wadati. Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A 31 (1998) 6057–6071. | DOI | MR | Zbl

[18] H. Spohn. KPZ scaling theory and the semi-discrete directed polymer model. MSRI Proceedings, 2012. Available at arXiv:1201.0645. | MR

[19] C. A. Tracy and H. Widom. Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1994) 151–174. | DOI | MR | Zbl

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