×

Law of iterated logarithms and fractal properties of the KPZ equation. (English) Zbl 1516.35573

Summary: We consider the Cole-Hopf solution of the \((1+1)\)-dimensional KPZ equation started from the narrow wedge initial condition. In this article we ask how the peaks and valleys of the KPZ height function (centered by time/24) at any spatial point grow as time increases. Our first main result is about the law of iterated logarithms for the KPZ equation. As time variable \(t\) goes to \(\infty\), we show that the limsup of the KPZ height function with the scaling by \(t^{1/3}(\log \log t)^{2/3}\) is almost surely equal to \((3/4\sqrt{2})^{2/3}\), whereas the liminf of the height function with the scaling by \(t^{1/3}(\log \log t)^{1/3}\) is almost surely equal to \(-6^{1/3}\). Our second main result concerns with the macroscopic fractal properties of the KPZ equation. Under exponential transformation of the time variable, we show that the peaks of KPZ height function mutate from being monofractal to multifractal, a property reminiscent of a similar phenomenon in Brownian motion (Theorem 1.4 in [D. Khoshnevisan et al., Ann. Probab. 45, No. 6A, 3697–3751 (2017; Zbl 1418.60081)]).
The proofs of our main results hinge on the following three key tools: (1) a multipoint composition law of the KPZ equation which can be regarded as a generalization of the two point composition law from (Proposition 2.9 in [I. Corwin et al., Ann. Probab. 49, No. 2, 832–876 (2021; Zbl 1467.60045)]), (2) the Gibbsian line ensemble techniques from [I. Corwin and A. Hammond, Invent. Math. 195, No. 2, 441–508 (2014; Zbl 1459.82117); Probab. Theory Relat. Fields 166, No. 1–2, 67–185 (2016; Zbl 1357.82040); Corwin et al., loc. cit.], and (3) the tail probabilities of the KPZ height function in short time and its spatiotemporal modulus of continuity. We advocate this last tool as one of our new and important contributions which might garner independent interest.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alberts, T., Khanin, K. and Quastel, J. (2014). The intermediate disorder regime for directed polymers in dimension \[1+1\]. Ann. Probab. 42 1212-1256. · Zbl 1292.82014 · doi:10.1214/13-AOP858
[2] 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
[3] BALAN, R. M. and CONUS, D. (2016). Intermittency for the wave and heat equations with fractional noise in time. Ann. Probab. 44 1488-1534. · Zbl 1343.60081 · doi:10.1214/15-AOP1005
[4] BARLOW, M. T. and TAYLOR, S. J. (1989). Fractional dimension of sets in discrete spaces. J. Phys. A 22 2621-2628. · Zbl 0687.60088
[5] BARLOW, M. T. and TAYLOR, S. J. (1992). Defining fractal subsets of \[{\mathbf{Z}^d} \]. Proc. Lond. Math. Soc. (3) 64 125-152. · Zbl 0753.28006 · doi:10.1112/plms/s3-64.1.125
[6] Barraquand, G. and Corwin, I. (2017). Random-walk in beta-distributed random environment. Probab. Theory Related Fields 167 1057-1116. · Zbl 1382.60125 · doi:10.1007/s00440-016-0699-z
[7] BASU, R., GANGULY, S. and HAMMOND, A. (2021). Fractal geometry of \[{\text{A}iry_2}\] processes coupled via the Airy sheet. Ann. Probab. 49 485-505. · Zbl 1457.82165 · doi:10.1214/20-AOP1444
[8] BASU, R., GANGULY, S., HEGDE, M. and KRISHNAPUR, M. (2021). Lower deviations in \(β\)-ensembles and law of iterated logarithm in last passage percolation. Israel J. Math. 242 291-324. · Zbl 1489.60152 · doi:10.1007/s11856-021-2135-z
[9] BATES, E., GANGULY, S. and HAMMOND, A. (2022). Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape. Electron. J. Probab. 27 Paper No. 1, 44 pp. · Zbl 1496.60115 · doi:10.1214/21-ejp706
[10] BERTINI, L. and CANCRINI, N. (1995). The stochastic heat equation: Feynman-Kac formula and intermittence. J. Stat. Phys. 78 1377-1401. · Zbl 1080.60508 · doi:10.1007/BF02180136
[11] Bertini, L. and Giacomin, G. (1997). Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys. 183 571-607. · Zbl 0874.60059 · doi:10.1007/s002200050044
[12] Borodin, A. and Corwin, I. (2014). Macdonald processes. Probab. Theory Related Fields 158 225-400. · Zbl 1291.82077 · doi:10.1007/s00440-013-0482-3
[13] CAFASSO, M. and CLAEYS, T. (2019). A Riemann-Hilbert approach to the lower tail of the KPZ equation. Preprint. Available at arXiv:1910.02493.
[14] CAFASSO, M., CLAEYS, T. and RUZZA, G. (2021). Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations. Comm. Math. Phys. 386 1107-1153. · Zbl 1477.37080 · doi:10.1007/s00220-021-04108-9
[15] CALABRESE, P., DOUSSAL, P. L. and ROSSO, A. (2010). Free-energy distribution of the directed polymer at high temperature. Europhys. Lett. 90 20002. · doi:10.1209/0295-5075/90/20002
[16] CARMONA, P. and HU, Y. (2002). On the partition function of a directed polymer in a Gaussian random environment. Probab. Theory Related Fields 124 431-457. · Zbl 1015.60100 · doi:10.1007/s004400200213
[17] CARMONA, R. A. and MOLCHANOV, S. A. (1994). Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 viii+125. · Zbl 0925.35074 · doi:10.1090/memo/0518
[18] CHEN, L. (2017). Nonlinear stochastic time-fractional diffusion equations on \[\mathbb{R} \]: Moments, Hölder regularity and intermittency. Trans. Amer. Math. Soc. 369 8497-8535. · Zbl 1406.60093 · doi:10.1090/tran/6951
[19] CHEN, L. and DALANG, R. C. (2015). Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions. Ann. Probab. 43 3006-3051. · Zbl 1338.60155 · doi:10.1214/14-AOP954
[20] CHEN, L., HU, Y. and NUALART, D. (2019). Nonlinear stochastic time-fractional slow and fast diffusion equations on \[{\mathbb{R}^d} \]. Stochastic Process. Appl. 129 5073-5112. · Zbl 1427.60119 · doi:10.1016/j.spa.2019.01.003
[21] CHEN, X. (2015). Precise intermittency for the parabolic Anderson equation with an \[(1+1)\]-dimensional time-space white noise. Ann. Inst. Henri Poincaré Probab. Stat. 51 1486-1499. · Zbl 1333.60136 · doi:10.1214/15-AIHP673
[22] Conus, D., Joseph, M., Khoshnevisan, D. and Shiu, S.-Y. (2013). On the chaotic character of the stochastic heat equation, II. Probab. Theory Related Fields 156 483-533. · Zbl 1286.60061 · doi:10.1007/s00440-012-0434-3
[23] 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
[24] Corwin, I. (2018). Exactly solving the KPZ equation. In Random Growth Models. Proc. Sympos. Appl. Math. 75 203-254. Amer. Math. Soc., Providence, RI. · Zbl 1423.60153
[25] CORWIN, I. and GHOSAL, P. (2020). KPZ equation tails for general initial data. Electron. J. Probab. 25 Paper No. 66, 38 pp. · Zbl 1456.60253 · doi:10.1214/20-ejp467
[26] Corwin, I. and Ghosal, P. (2020). Lower tail of the KPZ equation. Duke Math. J. 169 1329-1395. · Zbl 1457.35096 · doi:10.1215/00127094-2019-0079
[27] CORWIN, I., GHOSAL, P. and HAMMOND, A. (2021). KPZ equation correlations in time. Ann. Probab. 49 832-876. · Zbl 1467.60045 · doi:10.1214/20-aop1461
[28] CORWIN, I., GHOSAL, P., KRAJENBRINK, A., DOUSSAL, P. L. and TSAI, L.-C. (2018). Coulomb-gas electrostatics controls large fluctuations of the Kardar-Parisi-Zhang equation. Phys. Rev. Lett. 121 060201. · doi:10.1103/PhysRevLett.121.060201
[29] Corwin, I., Ghosal, P., Shen, H. and Tsai, L.-C. (2020). Stochastic PDE limit of the six vertex model. Comm. Math. Phys. 375 1945-2038. · Zbl 1439.82024 · doi:10.1007/s00220-019-03678-z
[30] CORWIN, I. and GU, Y. (2017). Kardar-Parisi-Zhang equation and large deviations for random walks in weak random environments. J. Stat. Phys. 166 150-168. · Zbl 1364.35365 · doi:10.1007/s10955-016-1693-7
[31] 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
[32] Corwin, I. and Hammond, A. (2016). KPZ line ensemble. Probab. Theory Related Fields 166 67-185. · Zbl 1357.82040 · doi:10.1007/s00440-015-0651-7
[33] Corwin, I. and Quastel, J. (2013). Crossover distributions at the edge of the rarefaction fan. Ann. Probab. 41 1243-1314. · Zbl 1285.82034 · doi:10.1214/11-AOP725
[34] CORWIN, I., SHEN, H. and TSAI, L.-C. (2018). \[ \operatorname{ASEP}(q,j)\] converges to the KPZ equation. Ann. Inst. Henri Poincaré Probab. Stat. 54 995-1012. · Zbl 1391.60155 · doi:10.1214/17-AIHP829
[35] CORWIN, I. and TSAI, L.-C. (2017). KPZ equation limit of higher-spin exclusion processes. Ann. Probab. 45 1771-1798. · Zbl 1407.60120 · doi:10.1214/16-AOP1101
[36] DAS, S. and GHOSAL, P. (2023). Supplement to “Law of iterated logarithms and fractal properties of the KPZ equation.” https://doi.org/10.1214/22-AOP1603SUPP
[37] DAS, S. and TSAI, L.-C. (2021). Fractional moments of the stochastic heat equation. Ann. Inst. Henri Poincaré Probab. Stat. 57 778-799. · Zbl 1472.60051 · doi:10.1214/20-aihp1095
[38] Dauvergne, D., Ortmann, J. and Virág, B. (2018). The directed landscape. Preprint. Available at arXiv:1812.00309.
[39] Dimitrov, E. (2020). Two-point convergence of the stochastic six-vertex model to the Airy process. Preprint. Available at arXiv:2006.15934.
[40] FERRARI, P. L. and SPOHN, H. (2011). Random growth models. In The Oxford Handbook of Random Matrix Theory 782-801. Oxford Univ. Press, Oxford. · Zbl 1234.60010
[41] FOONDUN, M. and KHOSHNEVISAN, D. (2009). Intermittence and nonlinear parabolic stochastic partial differential equations. Electron. J. Probab. 14 548-568. · Zbl 1190.60051 · doi:10.1214/EJP.v14-614
[42] Gärtner, J. and Molchanov, S. A. (1990). Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm. Math. Phys. 132 613-655. · Zbl 0711.60055
[43] GHOSAL, P. (2017). Hall-Littlewood-PushTASEP and its KPZ limit. Preprint. Available at arXiv:1701.07308.
[44] GHOSAL, P. (2018). Moments of the SHE under delta initial measure. Preprint. Available at arXiv:1808.04353.
[45] GHOSAL, P. and LIN, Y. (2020). Lyapunov exponents of the SHE for general initial data. Preprint. Available at arXiv:2007.06505.
[46] GIBBON, J. D. and DOERING, C. R. (2005). Intermittency and regularity issues in 3D Navier-Stokes turbulence. Arch. Ration. Mech. Anal. 177 115-150. · Zbl 1129.76014 · doi:10.1007/s00205-005-0382-5
[47] GIBBON, J. D. and TITI, E. S. (2005). Cluster formation in complex multi-scale systems. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 3089-3097. · Zbl 1206.82123 · doi:10.1098/rspa.2005.1548
[48] Gonçalves, P. and Jara, M. (2014). Nonlinear fluctuations of weakly asymmetric interacting particle systems. Arch. Ration. Mech. Anal. 212 597-644. · Zbl 1293.35336 · doi:10.1007/s00205-013-0693-x
[49] GUBINELLI, M., IMKELLER, P. and PERKOWSKI, N. (2015). Paracontrolled distributions and singular PDEs. Forum Math. Pi 3 e6, 75 pp. · Zbl 1333.60149 · doi:10.1017/fmp.2015.2
[50] Gubinelli, M. and Perkowski, N. (2017). KPZ reloaded. Comm. Math. Phys. 349 165-269. · Zbl 1388.60110 · doi:10.1007/s00220-016-2788-3
[51] Hairer, M. (2013). Solving the KPZ equation. Ann. of Math. (2) 178 559-664. · Zbl 1281.60060 · doi:10.4007/annals.2013.178.2.4
[52] HU, Y., HUANG, J., NUALART, D. and TINDEL, S. (2015). Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electron. J. Probab. 20 no. 55, 50 pp. · Zbl 1322.60113 · doi:10.1214/EJP.v20-3316
[53] KARDAR, M. (1987). Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities. Nuclear Phys. B 290 582-602. · doi:10.1016/0550-3213(87)90203-3
[54] KARDAR, M., PARISI, G. and ZHANG, Y.-C. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 889. · Zbl 1101.82329
[55] Khoshnevisan, D. (2014). Analysis of Stochastic Partial Differential Equations. CBMS Regional Conference Series in Mathematics 119. Amer. Math. Soc., Providence, RI. · Zbl 1304.60005 · doi:10.1090/cbms/119
[56] KHOSHNEVISAN, D., KIM, K. and XIAO, Y. (2017). Intermittency and multifractality: A case study via parabolic stochastic PDEs. Ann. Probab. 45 3697-3751. · Zbl 1418.60081 · doi:10.1214/16-AOP1147
[57] KHOSHNEVISAN, D., KIM, K. and XIAO, Y. (2018). A macroscopic multifractal analysis of parabolic stochastic PDEs. Comm. Math. Phys. 360 307-346. · Zbl 1454.60095 · doi:10.1007/s00220-018-3136-6
[58] KRAJENBRINK, A. and LE DOUSSAL, P. (2019). Linear statistics and pushed Coulomb gas at the edge of \(β\)-random matrices: Four paths to large deviations. Europhys. Lett. 125 20009. · doi:10.1209/0295-5075/125/20009
[59] Kupiainen, A. (2016). Renormalization group and stochastic PDEs. Ann. Henri Poincaré 17 497-535. · Zbl 1347.81063 · doi:10.1007/s00023-015-0408-y
[60] LEDOUX, M. (2018). A law of the iterated logarithm for directed last passage percolation. J. Theoret. Probab. 31 2366-2375. · Zbl 1428.60049 · doi:10.1007/s10959-017-0775-z
[61] LIN, Y. (2020). KPZ equation limit of stochastic higher spin six vertex model. Math. Phys. Anal. Geom. 23 Paper No. 1, 118 pp. · Zbl 1434.82058 · doi:10.1007/s11040-019-9325-5
[62] LIN, Y. and TSAI, L.-C. (2021). Short time large deviations of the KPZ equation. Comm. Math. Phys. 386 359-393. · Zbl 1469.82024 · doi:10.1007/s00220-021-04050-w
[63] 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
[64] MOLCHANOV, S. (1996). Reaction-diffusion equations in the random media: Localization and intermittency. In Nonlinear Stochastic PDEs (Minneapolis, MN, 1994). IMA Vol. Math. Appl. 77 81-109. Springer, New York. · Zbl 0851.35143 · doi:10.1007/978-1-4613-8468-7_5
[65] MORENO FLORES, G. R. (2014). On the (strict) positivity of solutions of the stochastic heat equation. Ann. Probab. 42 1635-1643. · Zbl 1306.60088 · doi:10.1214/14-AOP911
[66] MOTOO, M. (1958). Proof of the law of iterated logarithm through diffusion equation. Ann. Inst. Statist. Math. 10 21-28. · Zbl 0084.35801 · doi:10.1007/BF02883984
[67] MUELLER, C. and NUALART, D. (2008). Regularity of the density for the stochastic heat equation. Electron. J. Probab. 13 2248-2258. · Zbl 1191.60077 · doi:10.1214/EJP.v13-589
[68] PALEY, R. and ZYGMUND, A. (1932). A note on analytic functions in the unit circle. In Mathematical Proceedings of the Cambridge Philosophical Society 28 266-272. Cambridge University Press, Cambridge. · JFM 58.1076.03
[69] PAQUETTE, E. and ZEITOUNI, O. (2017). Extremal eigenvalue correlations in the GUE minor process and a law of fractional logarithm. Ann. Probab. 45 4112-4166. · Zbl 1390.60036 · doi:10.1214/16-AOP1161
[70] 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
[71] QUASTEL, J. (2012). Introduction to KPZ. In Current Developments in Mathematics, 2011 125-194. Int. Press, Somerville, MA. · Zbl 1316.60019
[72] QUASTEL, J. and SARKAR, S. (2023). Convergence of exclusion processes and the KPZ equation to the KPZ fixed point. J. Amer. Math. Soc. 36 251-289. · Zbl 1520.60063 · doi:10.1090/jams/999
[73] Quastel, J. and Spohn, H. (2015). The one-dimensional KPZ equation and its universality class. J. Stat. Phys. 160 965-984. · Zbl 1327.82069 · doi:10.1007/s10955-015-1250-9
[74] STRASSEN, V. (1964). An invariance principle for the law of the iterated logarithm. Z. Wahrsch. Verw. Gebiete 3 211-226. · Zbl 0132.12903 · doi:10.1007/BF00534910
[75] Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151-174. · Zbl 0789.35152
[76] TSAI, L.-C. (2022). Exact lower-tail large deviations of the KPZ equation. Duke Math. J. 171 1879-1922. · Zbl 1492.60069 · doi:10.1215/00127094-2022-0008
[77] VIRAG, B. (2020). The heat and the landscape I. Preprint. Available at arXiv:2008.07241.
[78] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 265-439. Springer, Berlin. · Zbl 0608.60060 · doi:10.1007/BFb0074920
[79] ZHONG, C. (2019). Large deviation bounds for the Airy point process. Preprint. Available at arXiv:1910.00797.
[80] ZIMMERMANN, M. G., TORAL, R., PIRO, O. and SAN MIGUEL, M. (2000). Stochastic spatiotemporal intermittency and noise-induced transition to an absorbing phase. Phys. Rev. Lett. 85 3612-3615 · doi:10.1103/PhysRevLett.85.3612
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.