×

Integration by parts and the KPZ two-point function. (English) Zbl 1498.60135

Summary: In this article, we consider the KPZ fixed point starting from a two-sided Brownian motion with an arbitrary diffusion coefficient. We apply the integration by parts formula from Malliavin calculus to establish a key relation between the two-point (correlation) function of the spatial derivative process and the location of the maximum of an Airy process plus Brownian motion minus a parabola. Integration by parts also allows us to deduce the density of this location in terms of the second derivative of the variance of the KPZ fixed point. In the stationary regime, we find the same density related to limit fluctuations of a second-class particle. We further develop an adaptation of Stein’s method that implies asymptotic independence of the spatial derivative process from the initial data.

MSC:

60F99 Limit theorems in probability theory
60H07 Stochastic calculus of variations and the Malliavin calculus
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
82D60 Statistical mechanics of polymers
60J65 Brownian motion
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Amir, G., Corwin, I. and Quastel, J. (2011). Probability distribution of the free energy of the continuum directed random polymer in \[1+1\] dimensions. Comm. Pure Appl. Math. 64 466-537. · Zbl 1222.82070 · doi:10.1002/cpa.20347
[2] Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119-1178. · Zbl 0932.05001 · doi:10.1090/S0894-0347-99-00307-0
[3] BAIK, J., FERRARI, P. L. and PÉCHÉ, S. (2010). Limit process of stationary TASEP near the characteristic line. Comm. Pure Appl. Math. 63 1017-1070. · Zbl 1194.82067 · doi:10.1002/cpa.20316
[4] BAIK, J., FERRARI, P. L. and PÉCHÉ, S. (2014). Convergence of the two-point function of the stationary TASEP. In Singular Phenomena and Scaling in Mathematical Models 91-110. Springer, Cham. · Zbl 1355.82024 · doi:10.1007/978-3-319-00786-1_5
[5] BAIK, J., LIECHTY, K. and SCHEHR, G. (2012). On the joint distribution of the maximum and its position of the \[{\text{Airy}_2}\] process minus a parabola. J. Math. Phys. 53 083303, 13 pp. · Zbl 1278.82070 · doi:10.1063/1.4746694
[6] BAIK, J. and RAINS, E. M. (2000). Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100 523-541. · Zbl 0976.82043 · doi:10.1023/A:1018615306992
[7] BALÁZS, M., CATOR, E. and SEPPÄLÄINEN, T. (2006). Cube root fluctuations for the corner growth model associated to the exclusion process. Electron. J. Probab. 11 1094-1132. · Zbl 1139.60046 · doi:10.1214/EJP.v11-366
[8] BORODIN, A., CORWIN, I., FERRARI, P. and VETŐ, B. (2015). Height fluctuations for the stationary KPZ equation. Math. Phys. Anal. Geom. 18 Art. 20, 95 pp. · Zbl 1332.82068 · doi:10.1007/s11040-015-9189-2
[9] Borodin, A., Ferrari, P. L., Prähofer, M. and Sasamoto, T. (2007). Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 1055-1080. · Zbl 1136.82028 · doi:10.1007/s10955-007-9383-0
[10] CATOR, E. and GROENEBOOM, P. (2006). Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab. 34 1273-1295. · Zbl 1101.60076 · doi:10.1214/009117906000000089
[11] Chhita, S., Ferrari, P. L. and Spohn, H. (2018). Limit distributions for KPZ growth models with spatially homogeneous random initial conditions. Ann. Appl. Probab. 28 1573-1603. · Zbl 1397.82040 · doi:10.1214/17-AAP1338
[12] CORWIN, I. (2012). The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl. 1 1130001, 76 pp. · Zbl 1247.82040 · doi:10.1142/S2010326311300014
[13] Corwin, I. and Hammond, A. (2014). Brownian Gibbs property for Airy line ensembles. Invent. Math. 195 441-508. · Zbl 1459.82117 · doi:10.1007/s00222-013-0462-3
[14] CORWIN, I., HAMMOND, A., HEGDE, M. and MATETSKI, K. (2021). Exceptional times when the KPZ fixed point violates Johansson’s conjecture on maximizer uniqueness. Preprint. Available at arXiv:2101.04205.
[15] Corwin, I., Liu, Z. and Wang, D. (2016). Fluctuations of TASEP and LPP with general initial data. Ann. Appl. Probab. 26 2030-2082. · Zbl 1356.82013 · doi:10.1214/15-AAP1139
[16] Corwin, I., Quastel, J. and Remenik, D. (2013). Continuum statistics of the \[{\text{Airy}_2}\] process. Comm. Math. Phys. 317 347-362. · Zbl 1257.82112 · doi:10.1007/s00220-012-1582-0
[17] Corwin, I., Quastel, J. and Remenik, D. (2015). Renormalization fixed point of the KPZ universality class. J. Stat. Phys. 160 815-834. · Zbl 1327.82064 · doi:10.1007/s10955-015-1243-8
[18] DAUVERGNE, D., ORTMANN, J. and VIRÁG, B. The directed landscape. Preprint. Available at arXiv:1812.00309.
[19] DAUVERGNE, D. and VIRÁG, B. (2021). The scaling limit of the longest increasing subsequence. Preprint. Available at arXiv:2104.08210. · Zbl 1484.60107
[20] DEUSCHEL, J.-D., FLORES, G. R. M. and ORENSHTEIN, T. (2020). Aging for the stationary Kardar-Parisi-Zhang equation and related models. Preprint. Available at arXiv:2006.10485.
[21] DOUSSAL, P. L. (2017). Maximum of an Airy process plus Brownian motion and memory in Kardar-Parisi-Zhang growth. Phys. Rev. E (3) 96 060101. · doi:10.1103/PhysRevE.96.060101
[22] Ferrari, P. L. and Spohn, H. (2006). Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265 1-44. · Zbl 1118.82032 · doi:10.1007/s00220-006-1549-0
[23] FLORES, G. M., QUASTEL, J. and REMENIK, D. (2013). Endpoint distribution of directed polymers in \[1+1\] dimensions. Comm. Math. Phys. 317 363-380. · Zbl 1257.82117 · doi:10.1007/s00220-012-1583-z
[24] FUKAI, Y. T. and TAKEUCHI, K. A. (2020). Kardar-Parisi-Zhang interfaces with curved initial shapes and variational formula. Phys. Rev. Lett. 124 060601, 6 pp. · doi:10.1103/PhysRevLett.124.060601
[25] Groeneboom, P. and Wellner, J. A. (2001). Computing Chernoff’s distribution. J. Comput. Graph. Statist. 10 388-400. · doi:10.1198/10618600152627997
[26] HUANG, J., NUALART, D., VIITASAARI, L. and ZHENG, G. (2020). Gaussian fluctuations for the stochastic heat equation with colored noise. Stoch. Partial Differ. Equ. Anal. Comput. 8 402-421. · Zbl 1451.60066 · doi:10.1007/s40072-019-00149-3
[27] IMAMURA, T. and SASAMOTO, T. (2013). Stationary correlations for the 1D KPZ equation. J. Stat. Phys. 150 908-939. · Zbl 1266.82045 · doi:10.1007/s10955-013-0710-3
[28] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437-476. · Zbl 0969.15008 · doi:10.1007/s002200050027
[29] Johansson, K. (2003). Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 277-329. · Zbl 1031.60084 · doi:10.1007/s00220-003-0945-y
[30] Kardar, M., Parisi, G. and Zhang, Y.-C. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 889-892. · Zbl 1101.82329
[31] MAES, C. and THIERY, T. (2017). Midpoint distribution of directed polymers in the stationary regime: Exact result through linear response. J. Stat. Phys. 168 937-963. · Zbl 1374.82044 · doi:10.1007/s10955-017-1839-2
[32] MATETSKI, K., QUASTEL, J. and REMENIK, D. (2021). The KPZ fixed point. Acta Math. 227 115-203. · Zbl 1505.82041 · doi:10.4310/acta.2021.v227.n1.a3
[33] NICA, M., QUASTEL, J. and REMENIK, D. (2020). One-sided reflected Brownian motions and the KPZ fixed point. Forum Math. Sigma 8 e63, 16 pp. · Zbl 1455.60131 · doi:10.1017/fms.2020.56
[34] NUALART, D. (2019). Malliavin calculus and normal approximations. Ensaios Mat. 34 1-74. · Zbl 1464.60055
[35] Pimentel, L. P. R. (2016). Duality between coalescence times and exit points in last-passage percolation models. Ann. Probab. 44 3187-3206. · Zbl 1361.60095 · doi:10.1214/15-AOP1044
[36] PIMENTEL, L. P. R. (2021). Brownian aspects of the KPZ fixed point. In In and Out of Equilibrium 3. Celebrating Vladas Sidoravicius. Progress in Probability 77 711-739. Birkhäuser/Springer, Cham. · Zbl 1469.60343 · doi:10.1007/978-3-030-60754-8_29
[37] Prähofer, M. and Spohn, H. (2002). Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108 1071-1106. · Zbl 1025.82010 · doi:10.1023/A:1019791415147
[38] PRÄHOFER, M. and SPOHN, H. (2004). Exact scaling functions for one-dimensional stationary KPZ growth. J. Stat. Phys. 115 255-279. · Zbl 1157.82363 · doi:10.1023/B:JOSS.0000019810.21828.fc
[39] QUASTEL, J. and REMENIK, D. (2019). KP governs random growth of a one dimensional substrate. Preprint. Available at arXiv:1908.10353. · Zbl 1502.60161
[40] Ross, N. (2011). Fundamentals of Stein’s method. Probab. Surv. 8 210-293. · Zbl 1245.60033 · doi:10.1214/11-PS182
[41] SARKAR, S. and VIRÁG, B. (2021). Brownian absolute continuity of the KPZ fixed point with arbitrary initial condition. Ann. Probab. 49 1718-1737. · Zbl 1473.60150 · doi:10.1214/20-aop1491
[42] SASAMOTO, T. (2005). Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38 L549-L556. · doi:10.1088/0305-4470/38/33/L01
[43] SCHEHR, G. (2012). Extremes of \(N\) vicious walkers for large \(N\): Application to the directed polymer and KPZ interfaces. J. Stat. Phys. 149 385-410. · Zbl 1259.82146 · doi:10.1007/s10955-012-0593-8
[44] TAKEUCHI, K. A., SANO, M., SASAMOTO, T. and SPOHN, H. (2011). Growing interfaces uncover universal fluctuations behind scale invariance. Sci. Rep. 1 34. · doi:10.1038/srep00034
[45] Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151-174. · Zbl 0789.35152
[46] Tracy, C. A. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 727-754 · Zbl 0851.60101
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.