On considère le modèle du -TASEP, qui est une modification du processus d’exclusion totalement asymétrique (TASEP) sur . Le taux des sauts d’une particule dépend de la distance avec la particule à sa droite et d’un paramètre . Dans cet article on considère la condition initiale où est completement occupé par les particules. On montre que les fluctuations du courant au temps sont d’ordre et la distribution asymptotique est la distribution de Tracy–Widom GUE. Ce résultat confirme la conjecture KPZ.
We consider the -TASEP that is a -deformation of the totally asymmetric simple exclusion process (TASEP) on for where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the current fluctuation of -TASEP at time is of order and asymptotically distributed as the GUE Tracy–Widom distribution, which confirms the KPZ scaling theory conjecture.
@article{AIHPB_2015__51_4_1465_0, 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}, volume = {51}, number = {4}, year = {2015}, doi = {10.1214/14-AIHP614}, mrnumber = {3414454}, zbl = {1376.60080}, language = {en}, url = {http://www.numdam.org/articles/10.1214/14-AIHP614/} }
TY - JOUR AU - Ferrari, Patrik L. AU - Vető, Bálint TI - Tracy–Widom asymptotics for $q$-TASEP JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 1465 EP - 1485 VL - 51 IS - 4 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/14-AIHP614/ DO - 10.1214/14-AIHP614 LA - en ID - AIHPB_2015__51_4_1465_0 ER -
%0 Journal Article %A Ferrari, Patrik L. %A Vető, Bálint %T Tracy–Widom asymptotics for $q$-TASEP %J Annales de l'I.H.P. Probabilités et statistiques %D 2015 %P 1465-1485 %V 51 %N 4 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/14-AIHP614/ %R 10.1214/14-AIHP614 %G en %F AIHPB_2015__51_4_1465_0
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] A phase transition for -TASEP with a few slower particles. Stochastic Process. Appl. 125 (2015) 2674–2699. | DOI | MR | Zbl
.[2] Discrete time -TASEPs. Int. Math. Res. Not. IMRN 2015 (2015) 499–537. | DOI | MR | Zbl
and .[3] Macdonald processes. Probab. Theory Related Fields 158 (2014) 225–400. | DOI | MR | Zbl
and .[4] Free energy fluctuations for directed polymers in random media in dimension. Comm. Pure Appl. Math. 67 (2014) 1129–1214. | DOI | MR | Zbl
, and .[5] Spectral theory for the -Boson particle system. Compos. Math. 151 (2015) 1–67. | DOI | MR | Zbl
, , and .[6] From duality to determinants for -TASEP and ASEP. Ann. Probab. 42 (2014) 2314–2382. | DOI | MR | Zbl
, and .[7] Large time asymptotics of growth models on space-like paths I: PushASEP. Electron. J. Probab. 13 (2008) 1380–1418. | DOI | MR | Zbl
and .[8] Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 (2007) 1055–1080. | DOI | MR | Zbl
, , and .[9] Universality of slow decorrelation in KPZ models. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 134–150. | DOI | Numdam | MR | Zbl
, and .[10] The -PushASEP: A new integrable model for traffic in dimension. J. Stat. Phys. 160 (2015) 1005–1026. | DOI | MR | Zbl
and .[11] From interacting particle systems to random matrices. J. Stat. Mech. 2010 (2010) P10016. | MR | Zbl
.[12] 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
and .[13] Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 (2003) 277–329. | DOI | MR | Zbl
.[14] The transition probability and the probability for the left-most particle’s position of the -TAZRP. J. Math. Phys. 55 (2014) 013301. | MR | Zbl
and .[15] Directed polymers and the quantum Toda lattice. Ann. Probab. 40 (2012) 437–458. | DOI | MR | Zbl
.[16] Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38 (2005) L549–L556. | MR
.[17] Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A 31 (1998) 6057–6071. | DOI | MR | Zbl
and .[18] KPZ scaling theory and the semi-discrete directed polymer model. MSRI Proceedings, 2012. Available at arXiv:1201.0645. | MR
.[19] Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1994) 151–174. | DOI | MR | Zbl
and .Cité par Sources :