## Annals of Applied Probability

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

#### Abstract

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 (Electron. Commun. Probab. 21 (2016) 61) 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 (Ann. Appl. Probab. 26 (2016) 643–690; Adv. Math. 330 (2018) 589–687), and related results for discrete directed polymers (Probab. Theory Related Fields 138 (2007) 391–410; Bates and Chatterjee (2016)), 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 (Probab. Theory Related Fields 138 (2007) 391–410). The method of our proof is based on the translation-invariant compactification introduced in (Ann. Probab. 44 (2016) 3934–3964) and a fixed point approach related to the cavity method from spin glasses recently used in (Bates and Chatterjee (2016)) in the context of the directed polymer model in the lattice.

#### Article information

Source
Ann. Appl. Probab., Volume 29, Number 6 (2019), 3745-3785.

Dates
First available in Project Euclid: 7 January 2020

https://projecteuclid.org/euclid.aoap/1578366325

Digital Object Identifier
doi:10.1214/19-AAP1491

Mathematical Reviews number (MathSciNet)
MR4047991

Zentralblatt MATH identifier
07172345

#### Citation

Bröker, Yannic; Mukherjee, Chiranjib. Localization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorder. Ann. Appl. Probab. 29 (2019), no. 6, 3745--3785. doi:10.1214/19-AAP1491. https://projecteuclid.org/euclid.aoap/1578366325

#### References

• [1] Aïdékon, E. (2013). Convergence in law of the minimum of a branching random walk. Ann. Probab. 41 1362–1426.
• [2] Alberts, T., Khanin, K. and Quastel, J. (2014). The continuum directed random polymer. J. Stat. Phys. 154 305–326.
• [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.
• [4] Barbato, D. (2005). FKG inequality for Brownian motion and stochastic differential equations. Electron. Commun. Probab. 10 7–16.
• [5] Bates, E. (2018). Localization of directed polymers with general reference walk. Electron. J. Probab. 23 Paper No. 30, 45.
• [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.
• [8] Bertini, L. and Cancrini, N. (1995). The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys. 78 1377–1401.
• [9] Bertini, L. and Cancrini, N. (1998). The two-dimensional stochastic heat equation: Renormalizing a multiplicative noise. J. Phys. A 31 615–622.
• [10] Bertini, L. and Giacomin, G. (1997). Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys. 183 571–607.
• [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.
• [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.
• [13] Caravenna, F., Sun, R. and Zygouras, N. (2017). Universality in marginally relevant disordered systems. Ann. Appl. Probab. 27 3050–3112.
• [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.
• [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.
• [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.
• [23] Comets, F., Shiga, T. and Yoshida, N. (2003). Directed polymers in a random environment: Path localization and strong disorder. Bernoulli 9 705–723.
• [24] Comets, F. and Yoshida, N. (2005). Brownian directed polymers in random environment. Comm. Math. Phys. 254 257–287.
• [25] Comets, F. and Yoshida, N. (2006). Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 1746–1770.
• [26] Derrida, B. and Spohn, H. (1988). Polymers on disordered trees, spin glasses, and traveling waves. J. Stat. Phys. 51 817–840.
• [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.
• [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.
• [29] Duplantier, B. and Sheffield, S. (2011). Liouville quantum gravity and KPZ. Invent. Math. 185 333–393.
• [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.
• [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.
• [33] Hairer, M. (2013). Solving the KPZ equation. Ann. of Math. (2) 178 559–664.
• [34] Kahane, J.-P. (1985). Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9 105–150.
• [35] Madaule, T. (2015). Maximum of a log-correlated Gaussian field. Ann. Inst. Henri Poincaré Probab. Stat. 51 1369–1431.
• [36] Madaule, T., Rhodes, R. and Vargas, V. (2016). Glassy phase and freezing of log-correlated Gaussian potentials. Ann. Appl. Probab. 26 643–690.
• [37] Mandelbrot, B. (1974). Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire. C. R. Acad. Sci. Paris Sér. A 278 289–292.
• [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.
• [40] Mukherjee, C. and Varadhan, S. R. S. (2016). Brownian occupation measures, compactness and large deviations. Ann. Probab. 44 3934–3964.
• [41] Robert, R. and Vargas, V. (2010). Gaussian multiplicative chaos revisited. Ann. Probab. 38 605–631.
• [42] Sasamoto, T. and Spohn, H. (2010). The one-dimensional KPZ equation: An exact solution and its universality. Phys. Rev. Lett. 104 230602.
• [43] Shamov, A. (2016). On Gaussian multiplicative chaos. J. Funct. Anal. 270 3224–3261.
• [44] Vargas, V. (2007). Strong localization and macroscopic atoms for directed polymers. Probab. Theory Related Fields 138 391–410.