Localization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorder.

*(English)*Zbl 1432.60088Summary: We consider a Gaussian multiplicative chaos (GMC) measure on the classical Wiener space driven by a smoothened (Gaussian) space-time white noise. For \(d\geq 3\), it was shown in [the second author et al., Electron. Commun. Probab. 21, Paper No. 61, 12 p. (2016; Zbl 1348.60094)] that for small noise intensity, the total mass of the GMC converges to a strictly positive random variable, while larger disorder strength (i.e., low temperature) forces the total mass to lose uniform integrability, eventually producing a vanishing limit. Inspired by strong localization phenomena for log-correlated Gaussian fields and Gaussian multiplicative chaos in the finite dimensional Euclidean spaces [T. Madaule et al., Ann. Appl. Probab. 26, No. 2, 643–690 (2016; Zbl 1341.60094); M. Biskup and O. Louidor, Adv. Math. 330, 589–687 (2018; Zbl 1409.60053)], and related results for discrete directed polymers [V. Vargas, Probab. Theory Relat. Fields 138, No. 3–4, 391–410 (2007; Zbl 1113.60097); E. Bates and the second author, “The endpoint distribution of directed polymers”, Preprint, arXiv:1612.03443], we study the endpoint distribution of a Brownian path under the renormalized GMC measure in this setting. We show that in the low temperature regime, the energy landscape of the system freezes and enters the so-called glassy phase as the entire mass of the Cesàro average of the endpoint GMC distribution stays localized in few spatial islands, forcing the endpoint GMC to be asymptotically purely atomic [Vargas, loc. cit.]. The method of our proof is based on the translation-invariant compactification introduced in [the second author and S. R. S. Varadhan, Ann. Probab. 44, No. 6, 3934–3964 (2016; Zbl 1364.60037)] and a fixed point approach related to the cavity method from spin glasses recently used in [Bates and the second author, loc. cit.] in the context of the directed polymer model in the lattice.

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60J65 | Brownian motion |

60J55 | Local time and additive functionals |

60F10 | Large deviations |

35R60 | PDEs with randomness, stochastic partial differential equations |

35Q82 | PDEs in connection with statistical mechanics |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

82D60 | Statistical mechanics of polymers |

##### Keywords:

Gaussian multiplicative chaos; supercritical; renormalization; glassy phase; freezing; stochastic heat equation; strong disorder; asymptotic pure atomicity; translation-invariant compactification
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\textit{Y. Bröker} and \textit{C. Mukherjee}, Ann. Appl. Probab. 29, No. 6, 3745--3785 (2019; Zbl 1432.60088)

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##### References:

[1] | Aïdékon, E. (2013). Convergence in law of the minimum of a branching random walk. Ann. Probab. 41 1362-1426. · Zbl 1285.60086 |

[2] | Alberts, T., Khanin, K. and Quastel, J. (2014). The continuum directed random polymer. J. Stat. Phys. 154 305-326. · Zbl 1291.82143 |

[3] | 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 |

[4] | Barbato, D. (2005). FKG inequality for Brownian motion and stochastic differential equations. Electron. Commun. Probab. 10 7-16. · Zbl 1060.60015 |

[5] | Bates, E. (2018). Localization of directed polymers with general reference walk. Electron. J. Probab. 23 Paper No. 30, 45. · Zbl 1390.60359 |

[6] | Bates, E. and Chatterjee, S. (2016). The endpoint distribution of directed polymers. Preprint. Available at arXiv:1612.03443. |

[7] | Berestycki, N. (2017). An elementary approach to Gaussian multiplicative chaos. Electron. Commun. Probab. 22 Paper No. 27, 12. · Zbl 1365.60035 |

[8] | Bertini, L. and Cancrini, N. (1995). The stochastic heat equation: Feynman-Kac formula and intermittence. J. Stat. Phys. 78 1377-1401. · Zbl 1080.60508 |

[9] | Bertini, L. and Cancrini, N. (1998). The two-dimensional stochastic heat equation: Renormalizing a multiplicative noise. J. Phys. A 31 615-622. · Zbl 0976.82035 |

[10] | Bertini, L. and Giacomin, G. (1997). Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys. 183 571-607. · Zbl 0874.60059 |

[11] | Biskup, M. and Louidor, O. (2018). Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian free field. Adv. Math. 330 589-687. · Zbl 1409.60053 |

[12] | Bröker, Y. and Mukherjee, C. (2019). Quenched central limit theorem for the stochastic heat equation in weak disorder. In Probability and Analysis in Interacting Physical Systems 173-189. Springer, Cham. · Zbl 1430.60063 |

[13] | Caravenna, F., Sun, R. and Zygouras, N. (2017). Universality in marginally relevant disordered systems. Ann. Appl. Probab. 27 3050-3112. · Zbl 1387.82032 |

[14] | 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 |

[15] | Carpentier, D. and Le Doussal, P. (2001). Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and Sinh-Gordon models. Phys. Rev. E 63 026110. |

[16] | Chaterjee, S. (2018). Proof of the path localization conjecture for directed polymers. Preprint. Available at arXiv:1806.04220. |

[17] | Comets, F. and Cosco, C. (2018). Brownian polymers in Poissonian environment: A survey. Available at arXiv:1805.10899. |

[18] | Comets, F., Cosco, C. and Mukherjee, C. (2018). Fluctuation and rate of convergence of the stochastic heat equation in weak disorder. Preprint. Available at arXiv:1807.03902. |

[19] | Comets, F., Cosco, C. and Mukherjee, C. (2019). Renormalizing the Kardar-Parisi-Zhang equation in \(d\geq 3\) in weak disorder. Preprint. Available at arXiv:1902.04104. |

[20] | Comets, F., Cosco, C. and Mukherjee, C. (2019). Space-time fluctuation of the Kardar-Parisi-Zhang equation in \(d\geq 3\) and the Gaussian free field. Preprint. Available at 1905.03200. |

[21] | Comets, F. and Cranston, M. (2013). Overlaps and pathwise localization in the Anderson polymer model. Stochastic Process. Appl. 123 2446-2471. · Zbl 1290.60102 |

[22] | Comets, F. and Neveu, J. (1995). The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: The high temperature case. Comm. Math. Phys. 166 549-564. · Zbl 0811.60098 |

[23] | Comets, F., Shiga, T. and Yoshida, N. (2003). Directed polymers in a random environment: Path localization and strong disorder. Bernoulli 9 705-723. · Zbl 1042.60069 |

[24] | Comets, F. and Yoshida, N. (2005). Brownian directed polymers in random environment. Comm. Math. Phys. 254 257-287. · Zbl 1128.60089 |

[25] | Comets, F. and Yoshida, N. (2006). Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 1746-1770. · Zbl 1104.60061 |

[26] | Derrida, B. and Spohn, H. (1988). Polymers on disordered trees, spin glasses, and traveling waves. J. Stat. Phys. 51 817-840. · Zbl 1036.82522 |

[27] | Duplantier, B., Rhodes, R., Sheffield, S. and Vargas, V. (2014). Critical Gaussian multiplicative chaos: Convergence of the derivative martingale. Ann. Probab. 42 1769-1808. · Zbl 1306.60055 |

[28] | Duplantier, B., Rhodes, R., Sheffield, S. and Vargas, V. (2014). Renormalization of critical Gaussian multiplicative chaos and KPZ relation. Comm. Math. Phys. 330 283-330. · Zbl 1297.60033 |

[29] | Duplantier, B. and Sheffield, S. (2011). Liouville quantum gravity and KPZ. Invent. Math. 185 333-393. · Zbl 1226.81241 |

[30] | Fyodorov, Y. V. and Bouchaud, J.-P. (2008). Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential. J. Phys. A 41 372001, 12. · Zbl 1214.82016 |

[31] | Fyodorov, Y. V., Le Doussal, P. and Rosso, A. (2009). Statistical mechanics of logarithmic REM: Duality, freezing and extreme value statistics of \(1/f\) noises generated by Gaussian free fields. J. Stat. Mech. Theory Exp. 2009 P10005, 32. |

[32] | Gubinelli, M. and Perkowski, N. (2017). KPZ reloaded. Comm. Math. Phys. 349 165-269. · Zbl 1388.60110 |

[33] | Hairer, M. (2013). Solving the KPZ equation. Ann. of Math. (2) 178 559-664. · Zbl 1281.60060 |

[34] | Kahane, J.-P. (1985). Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9 105-150. · Zbl 0596.60041 |

[35] | Madaule, T. (2015). Maximum of a log-correlated Gaussian field. Ann. Inst. Henri Poincaré Probab. Stat. 51 1369-1431. · Zbl 1329.60138 |

[36] | Madaule, T., Rhodes, R. and Vargas, V. (2016). Glassy phase and freezing of log-correlated Gaussian potentials. Ann. Appl. Probab. 26 643-690. · Zbl 1341.60094 |

[37] | Mandelbrot, B. (1974). Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire. C. R. Acad. Sci. Paris Sér. A 278 289-292. · Zbl 0276.60096 |

[38] | Mukherjee, C. (2017). Central limit theorem for Gibbs measures on path spaces including long range and singular interactions and homogenization of the stochastic heat equation. Preprint. Available at arXiv:1706.09345. |

[39] | Mukherjee, C., Shamov, A. and Zeitouni, O. (2016). Weak and strong disorder for the stochastic heat equation and continuous directed polymers in \(d\geq 3\). Electron. Commun. Probab. 21 Paper No. 61, 12. · Zbl 1348.60094 |

[40] | Mukherjee, C. and Varadhan, S. R. S. (2016). Brownian occupation measures, compactness and large deviations. Ann. Probab. 44 3934-3964. · Zbl 1364.60037 |

[41] | Robert, R. and Vargas, V. (2010). Gaussian multiplicative chaos revisited. Ann. Probab. 38 605-631. · Zbl 1191.60066 |

[42] | Sasamoto, T. and Spohn, H. (2010). The one-dimensional KPZ equation: An exact solution and its universality. Phys. Rev. Lett. 104 230602. · Zbl 1204.35137 |

[43] | Shamov, A. (2016). On Gaussian multiplicative chaos. J. Funct. Anal. 270 3224-3261. · Zbl 1337.60054 |

[44] | Vargas, V. (2007). Strong localization and macroscopic atoms for directed polymers. Probab. Theory Related Fields 138 391-410. · Zbl 1113.60097 |

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