Local behaviour of Airy processes.(English)Zbl 1405.82029
Summary: The Airy processes describe limit fluctuations in a wide range of growth models, where each particular Airy process depends on the geometry of the initial profile. We show how the coupling method, developed in the last-passage percolation context, can be used to prove existence of a continuous version and local convergence to Brownian motion. By using similar arguments, we further extend these results to a two parameter limit fluctuation process (Airy sheet).
MSC:
| 82C43 | Time-dependent percolation in statistical mechanics |
| 60J65 | Brownian motion |
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 82C41 | Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics |
| 35Q82 | PDEs in connection with statistical mechanics |
| 60K37 | Processes in random environments |
Cite
References:
| [1] | Amir, G.; Corwin, I.; Quastel, J., Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions, Commun. Pure Appl. Math., 64, 466-537, (2011) ·Zbl 1222.82070 ·doi:10.1002/cpa.20347 |
| [2] | Balázs, M.; Cator, EA; Seppäläinen, T., Cube root fluctuations for the corner growth model associated to the exclusion process, Electron. J. Probab., 11, 1094-1132, (2006) ·Zbl 1139.60046 ·doi:10.1214/EJP.v11-366 |
| [3] | Borodin, A.; Ferrari, PL, Large time asymptotics of growth models on space-like paths I: PushASEP, Electron. J. Probab., 13, 1380-1418, (2008) ·Zbl 1187.82084 ·doi:10.1214/EJP.v13-541 |
| [4] | Borodin, A.; Ferrari, PL; Prähofer, M.; Sasamoto, T., Fluctuation properties of the TASEP with periodic initial configuration, J. Stat. Phys., 129, 1055-1080, (2007) ·Zbl 1136.82028 ·doi:10.1007/s10955-007-9383-0 |
| [5] | Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968) ·Zbl 0172.21201 |
| [6] | Cator, EA; Groeneboom, P., Second class particles and cube root asymptotics for Hammersley’s process, Ann. Probab., 34, 1273-1295, (2006) ·Zbl 1101.60076 ·doi:10.1214/009117906000000089 |
| [7] | Cator, EA; Pimentel, LPR, Busemann functions and the speed of a second class particle in the rarefaction fan, Ann. Probab., 41, 2401-2425, (2013) ·Zbl 1276.60108 ·doi:10.1214/11-AOP709 |
| [8] | Cator, EA; Pimentel, LPR, On the local fluctuations of last-passage percolation models, Stoch. Process. Appl., 125, 538-551, (2016) ·Zbl 1326.60134 ·doi:10.1016/j.spa.2014.08.009 |
| [9] | Chhita, S., Ferrari, P.L., Spohn, H.: Limit distributions for KPZ growth models with spatially homogeneous random initial conditions. arXiv:1611.06690 (2016) ·Zbl 1397.82040 |
| [10] | Corwin, I.; Hammond, A., Brownian Gibbs property for Airy line ensembles, Invent. Math., 195, 441-508, (2014) ·Zbl 1459.82117 ·doi:10.1007/s00222-013-0462-3 |
| [11] | Corwin, I.; Quastel, J.; Remenik, D., Renormalization fixed point of the KPZ universality class, J. Stat. Phys., 160, 815-834, (2015) ·Zbl 1327.82064 ·doi:10.1007/s10955-015-1243-8 |
| [12] | Corwin, I.; Liu, Z.; Wang, D., Fluctuations of TASEP and LPP with general initial data, Ann. Appl. Probab., 26, 2030-2082, (2016) ·Zbl 1356.82013 ·doi:10.1214/15-AAP1139 |
| [13] | Hägg, J., Local fluctuations in the Airy and discrete PNG process, Ann. Probab., 36, 1059-1092, (2008) ·Zbl 1142.60025 ·doi:10.1214/07-AOP353 |
| [14] | Johansson, K., Shape fluctuations and random matrices, Commun. Math. Phys., 209, 437-476, (2000) ·Zbl 0969.15008 ·doi:10.1007/s002200050027 |
| [15] | Johansson, K., Discrete polynuclear growth and determinantal processes, Commun. Math. Phys., 242, 277-329, (2003) ·Zbl 1031.60084 ·doi:10.1007/s00220-003-0945-y |
| [16] | Kardar, M.; Parisi, G.; Zhang, Y-C, Dynamic scaling of growing interfaces, Phys. Rev. Lett., 56, 889-892, (1986) ·Zbl 1101.82329 ·doi:10.1103/PhysRevLett.56.889 |
| [17] | Matetski, K., Quastel, J., Remenik, D.: The KPZ fixed point. arXiv:1701.00018 (2016) |
| [18] | Neuhaus, G., On weak convergence of stochastic processes with multidimensional time parameter, Ann. Math. Stat., 42, 1285-1295, (1971) ·Zbl 0222.60013 ·doi:10.1214/aoms/1177693241 |
| [19] | Pimentel, LPR, On the location of the maximum of a continuous stochastic process, J. Appl. Probab., 51, 152-161, (2014) ·Zbl 1305.60029 ·doi:10.1239/jap/1395771420 |
| [20] | Pimentel, LPR, Duality between coalescence times and exit points in last-passage percolation models, Ann. Probab., 44, 3187-3206, (2016) ·Zbl 1361.60095 ·doi:10.1214/15-AOP1044 |
| [21] | Pimentel, L.P.R.: Ergodicity of the KPZ fixed point. arXiv:1708.06006 (2017) |
| [22] | Prähofer, M.; Spohn, H., Scale invariance of the PNG droplet and the Airy process, J. Stat. Phys., 108, 1076-1106, (2002) ·Zbl 1025.82010 ·doi:10.1023/A:1019791415147 |
| [23] | Quastel, J.; Remenik, D., Local behavior and hitting probabilities of the Airy1 process, Probab. Theory. Relat. Fields, 157, 605-634, (2013) ·Zbl 1285.60095 ·doi:10.1007/s00440-012-0466-8 |
| [24] | Resnick, S.: Adventures in Stochastic Processes. Birkhäuser, Basel (1992) ·Zbl 0762.60002 |
| [25] | Sasamoto, T., Spatial correlations of the 1D KPZ surface on a flat substrate, J. Phys. A, 38, l549, (2007) ·doi:10.1088/0305-4470/38/33/L01 |
| [26] | Straf, M.: Weak convergence of stochastic processes with several parameters. In: Proc. Sixth Berkeley Symp. on Math. Statist. and Prob. (vol. 2, pp. 187-221). Univ. of Calif. Press, Berkeley, CA (1972) ·Zbl 0255.60019 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
