Liouville measure as a multiplicative cascade via level sets of the Gaussian free field
Annales de l'Institut Fourier, Volume 70 (2020) no. 1, pp. 205-245.

We provide new constructions of the subcritical and critical Gaussian multiplicative chaos (GMC) measures corresponding to the 2D Gaussian free field (GFF). As a special case we recover E. Aidekon’s construction of random measures using nested conformally invariant loop ensembles, and thereby prove his conjecture that certain CLE 4 based limiting measures are equal in law to the GMC measures for the GFF. The constructions are based on the theory of local sets of the GFF and build a strong link between multiplicative cascades and GMC measures. This link allows us to directly adapt techniques used for multiplicative cascades to the study of GMC measures of the GFF. As a proof of principle we do this for the so-called Seneta–Heyde rescaling of the critical GMC measure.

On propose de nouvelles constructions des mesures du chaos multiplicatif gaussien (GMC) sous-critique et critique correspondant au champ libre gaussien 2D (GFF). Comme cas particulier, on retrouve la construction des mesures aléatoires par E. Aidekon, qui utilise des ensembles de boucles emboîtées invariantes par transformations conformes. Ainsi, on prouve sa conjecture selon laquelle certaines mesures basées sur le CLE 4 emboîté sont égal en loi aux mesures de GMC pour le GFF. Nos constructions sont basées sur la théorie des ensembles locaux du GFF et permettent d’établir un lien fort entre les cascades multiplicatives et les mesures GMC. Ce lien nous permet d’adapter directement les techniques utilisées pour les cascades multiplicatives à l’étude des mesures de GMC pour le GFF. Comme exemple de ce principe on adapte l’argument de Seneta–Heyde pour construire la mesure critique de la GMC.

Received: 2017-05-03
Revised: 2017-12-27
Accepted: 2019-01-17
Published online: 2020-05-28
DOI: https://doi.org/10.5802/aif.3312
Classification: 60G57,  60G58,  60D05,  60J80
Keywords: Liouville measure, Gaussian free field, Gaussian multiplicative chaos, critical Gaussian multiplicative chaos, multiplicative cascades
@article{AIF_2020__70_1_205_0,
     author = {Aru, Juhan and Powell, Ellen and Sep\'ulveda, Avelio},
     title = {Liouville measure as a multiplicative cascade via level sets of the Gaussian free field},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {1},
     year = {2020},
     pages = {205-245},
     doi = {10.5802/aif.3312},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2020__70_1_205_0/}
}
Aru, Juhan; Powell, Ellen; Sepúlveda, Avelio. Liouville measure as a multiplicative cascade via level sets of the Gaussian free field. Annales de l'Institut Fourier, Volume 70 (2020) no. 1, pp. 205-245. doi : 10.5802/aif.3312. https://aif.centre-mersenne.org/item/AIF_2020__70_1_205_0/

[1] Aïdékon, Elie The extremal process in nested conformal loops (2015) (preprint available on the webpage of the author)

[2] Aïdékon, Elie; Jaffuel, Bruno Survival of branching random walks with absorption, Stochastic Processes Appl., Volume 121 (2011) no. 9, pp. 1901-1937 | Article | MR 2819234 | Zbl 1236.60080

[3] Aïdékon, Elie; Shi, Zhan Seneta–Heyde rescaling for the branching random walk, Ann. Probab., Volume 42 (2014) no. 3, pp. 959-993 | Article | Zbl 1304.60092

[4] Aru, Juhan; Lupu, Titus; Sepúlveda, Avelio First passage sets of the 2D continuum Gaussian free field, Probab. Theory Relat. Fields (2019) (online-first) | Article | Zbl 07191244

[5] Aru, Juhan; Lupu, Titus; Sepúlveda, Avelio The first passage sets of the 2D Gaussian free field: convergence and isomorphisms (2018) (https://arxiv.org/abs/1805.09204) | Zbl 07202500

[6] Aru, Juhan; Powell, Ellen; Sepúlveda, Avelio Critical Liouville measure as a limit of subcritical measures, Electron. Commun. Probab., Volume 24 (2019), 18, 16 pages | MR 3933042 | Zbl 07055622

[7] Aru, Juhan; Sepúlveda, Avelio; Werner, Wendelin On bounded-type thin local sets of the two-dimensional Gaussian free field, J. Inst. Math. Jussieu, Volume 18 (2019) no. 3, pp. 591-618 | Article | MR 3936643 | Zbl 07051731

[8] Berestycki, Nathanaël An elementary approach to Gaussian multiplicative chaos, Electron. Commun. Probab., Volume 22 (2017), 27, 12 pages | MR 3652040 | Zbl 1365.60035

[9] Bolthausen, Erwin On a functional central limit theorem for random walks conditioned to stay positive, Ann. Probab. (1976), pp. 480-485 | Article | MR 415702 | Zbl 0336.60024

[10] David, François; Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent Liouville quantum gravity on the Riemann sphere, Commun. Math. Phys., Volume 342 (2016) no. 3, pp. 869-907 | Article | MR 3465434 | Zbl 1336.83042

[11] Duplantier, Bertrand; Rhodes, Rémi; Sheffield, Scott; Vargas, Vincent Critical Gaussian multiplicative chaos: convergence of the derivative martingale, Ann. Probab., Volume 42 (2014) no. 5, pp. 1769-1808 | Article | MR 3262492 | Zbl 1306.60055

[12] Duplantier, Bertrand; Rhodes, Rémi; Sheffield, Scott; Vargas, Vincent Renormalization of critical Gaussian multiplicative chaos and KPZ relation, Commun. Math. Phys., Volume 330 (2014) no. 1, pp. 283-330 | Article | MR 3215583 | Zbl 1297.60033

[13] Duplantier, Bertrand; Sheffield, Scott Liouville quantum gravity and KPZ, Invent. Math., Volume 185 (2011) no. 2, pp. 333-393 | Article | MR 2819163 | Zbl 1226.81241

[14] Durrett, Rick Probability: theory and examples, Cambridge Series in Statistical and Probabilistic Mathematics, Volume 31, Cambridge University Press, 2010, x+428 pages | Article | MR 2722836 | Zbl 1202.60001

[15] Høegh-Krohn, Raphael A general class of quantum fields without cut-offs in two space-time dimensions, Commun. Math. Phys., Volume 21 (1971) no. 3, pp. 244-255 | Article | MR 292433

[16] Huang, Yichao; Rhodes, Rémi; Vargas, Vincent Liouville Quantum Gravity on the unit disk, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 54 (2018) no. 3, pp. 1694-1730 | Article | MR 3825895 | Zbl 1404.60071

[17] Junnila, Janne; Saksman, Eero Uniqueness of critical Gaussian chaos, Electron. J. Probab., Volume 22 (2017), 11, 31 pages | MR 3613704 | Zbl 1357.60040

[18] Kahane, Jean-Pierre Sur le chaos multiplicatif, Ann. Sci. Math. Qué., Volume 9 (1985) no. 2, pp. 105-150 | MR 829798 | Zbl 0596.60041

[19] Kozlov, Mykyta V. The asymptotic behavior of the probability of non-extinction of critical branching processes in a random environment, Theory Probab. Appl., Volume 21 (1976), pp. 791-804 | Article | MR 428492 | Zbl 0384.60058

[20] Kyprianou, Andreas E. Martingale convergence and the stopped branching random walk, Probab. Theory Relat. Fields, Volume 116 (2000) no. 3, pp. 405-419 | Article | MR 1749282 | Zbl 0955.60078

[21] Lyons, Russell A simple path to Biggins’ martingale convergence for branching random walk, Classical and modern branching processes (The IMA Volumes in Mathematics and its Applications), Springer, 1997 no. 84, pp. 217-221 | Article | Zbl 0897.60086

[22] Madaule, Thomas First order transition for the branching random walk at the critical parameter, Stochastic Processes Appl., Volume 126 (2016), pp. 470-502 | Article | MR 3434991 | Zbl 1384.60093

[23] Miller, J.; Sheffield, Scott The GFF and CLE(4) (2011) (slides and private communications)

[24] Nakayama, Yu Liouville field theory: a decade after the revolution, Int. J. Mod. Phys. A, Volume 19 (2004) no. 17-18, pp. 2771-2930 | Article | MR 2073993 | Zbl 1080.81056

[25] Powell, Ellen Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation, Electron. J. Probab., Volume 23 (2018), 31, 26 pages | MR 3785401 | Zbl 1390.60182

[26] Qian, Wei; Werner, Wendelin Coupling the Gaussian free fields with free and with zero boundary conditions via common level lines, Commun. Math. Phys. (2018), pp. 1-28 | MR 3825935 | Zbl 1430.60064

[27] Rhodes, Rémi; Vargas, Vincent Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity (2016) (https://arxiv.org/abs/1602.07323) | Zbl 1386.60139

[28] Robert, Raoul; Vargas, Vincent Gaussian multiplicative chaos revisited, Ann. Probab., Volume 38 (2010) no. 2, pp. 605-631 | Article | MR 2642887 | Zbl 1191.60066

[29] Schramm, Oded; Sheffield, Scott A contour line of the continuum Gaussian free field, Probab. Theory Relat. Fields, Volume 157 (2013) no. 1-2, pp. 47-80 | Article | MR 3101840 | Zbl 1331.60090

[30] Schramm, Oded; Sheffield, Scott; Wilson, David B. Conformal radii for conformal loop ensembles, Commun. Math. Phys., Volume 288 (2009) no. 1, pp. 43-53 | Article | MR 2491617 | Zbl 1187.82044

[31] Sepúlveda, Avelio On thin local sets of the Gaussian free field, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 55 (2019) no. 3, pp. 1797-1813 | Article | MR 4010952 | Zbl 07133738

[32] Shamov, Alexander On Gaussian Multiplicative Chaos, J. Funct. Anal., Volume 270 (2016) no. 9, pp. 3224-3261 | Article | MR 3475456 | Zbl 1337.60054

[33] Sheffield, Scott Conformal weldings of random surfaces: SLE and the quantum gravity zipper, Ann. Probab., Volume 44 (2016) no. 5, pp. 3474-3545 | Article | MR 3551203 | Zbl 1388.60144

[34] Webb, Christian The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos - the L 2 phase, Electron. J. Probab., Volume 20 (2015), 104, 21 pages | MR 3407221 | Zbl 1328.15052

[35] Werner, Wendelin Topics on the GFF and CLE(4) (2015) (lecture notes available on his webpage)