Maximum of a log-correlated gaussian field
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 4, pp. 1369-1431.

We study the maximum of a Gaussian field on [0,1] 𝚍 (𝚍1) whose correlations decay logarithmically with the distance. Kahane (Ann. Sci. Math. Québec 9 (1985) 105–150) introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint, Renormalization of critical Gaussian multiplicative chaos and KPZ formula (2012) Preprint) extended Kahane’s construction to the critical case and established the KPZ formula at criticality. Moreover, they made in (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint) several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint): we establish the convergence in law of the maximum and show that the limit law is the Gumbel distribution convoluted by the limit of the derivative martingale.

Nous étudions le maximum d’un champ Gaussien sur [0,1] 𝚍 (𝚍1) dont les corrélations décroissent logarithmiquement avec la distance. Kahane (Ann. Sci. Math. Québec 9 (1985) 105–150) a introduit ce modèle pour construire mathématiquement le chaos Gaussien multiplicatif dans le cas sous-critique. Duplantier, Rhodes, Sheffield et Vargas (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint, Renormalization of critical Gaussian multiplicative chaos and KPZ formula (2012) Preprint) ont étendu cette construction au cas critique et ont établi la formule KPZ. De plus, dans (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint), ils fournissent plusieurs conjectures sur le cas sur-critique ainsi que sur le maximum de ce champ Gaussien. Dans ce papier nous établissons la convergence en loi du maximum et montrons que loi limite est une variable aléatoire de Gumbel convoluée avec la limite de la martingale dérivée, résolvant ainsi la Conjecture 12 de (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint).

DOI: 10.1214/14-AIHP633
Keywords: gaussian multiplicative chaos, log-correlated gaussian field, minimal position, Gumbel distribution
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Madaule, Thomas. Maximum of a log-correlated gaussian field. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 4, pp. 1369-1431. doi : 10.1214/14-AIHP633. http://www.numdam.org/articles/10.1214/14-AIHP633/

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

[2] E. Aïdékon and Z. Shi. The Seneta–Heyde scaling for the branching random walk. Ann. Probab. 42 (2014) 959–993. | MR | Zbl

[3] E. Aïdékon, J. Berestycki, É. Brunet and Z. Shi. Branching Brownian motion seen from its tip. Probab. Theory Related Fields 157 (2013) 405–451. | MR | Zbl

[4] E. Aïdékon and Z. Shi. Weak convergence for the minimal position in a branching random walk: A simple proof. Period. Math. Hungar. 61 (1–2) (2010) 43–54. | MR | Zbl

[5] R. Allez, R. Rhodes and V. Vargas. Lognormal -scale invariant random measures. Probab. Theory Related Fields 155 (2013) 751–788. | MR | Zbl

[6] L.-P. Arguin, A. Bovier and N. Kistler. Poissonian statistics in the extremal process of branching Brownian motion. Ann. Appl. Probab. 22 (2012) 1693–1711. | MR | Zbl

[7] L.-P. Arguin, A. Bovier and N. Kistler. Genealogy of extremal particles of branching Brownian motion. Comm. Pure Appl. Math. 64 (2011) 1647–1676. | MR | Zbl

[8] L.-P. Arguin, A. Bovier and N. Kistler. The extremal process of branching Brownian motion. Probab. Theory Related Fields 157 (2013) 535–574. | MR | Zbl

[9] L.-P. Arguin and O. Zindy. Poisson–Dirichlet statistics for the extremes of a log-correlated Gaussian field. Ann. Appl. Probab. 24 (2014) 1446–1481. | MR | Zbl

[10] J. Barral, A. Kupiainen, M. Nikula, E. Saksman and C. Webb Basic properties of critical lognormal multiplicative chaos. Preprint, 2013. Available at arXiv:1303.4548. | MR

[11] J. Barral, R. Rhodes and V. Vargas. Limiting laws of supercritical branching random walks. C. R. Math. Acad. Sci. Paris 350 (2012) 535–538. | MR | Zbl

[12] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York, 1999. | MR | Zbl

[13] C. M. Bingham, N. H. Goldie and J. L. Teugels. Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge, 1989. | MR | Zbl

[14] E. Bolthausen, J.-D. Deuschel and O. Zeitouni. Recursions and tightness for the maximum of the discrete, two dimensional Gaussian free field. Electron. Commun. Probab. 16 (2011) 114–119. | MR | Zbl

[15] J.-D. Bolthausen, J.-D. Deuschel and G. Giacomin. Entropic repulsion and the maximum of the two-dimensional harmonic crystal. Ann. Probab. 29 (4) (2001) 1670–1692. | MR | Zbl

[16] M. Bramson, J. Ding and O. Zeitouni. Convergence in law of the maximum of the two-dimensional discrete Gaussian free field. Preprint, 2013. Available at arXiv:1301.6669.

[17] M. Bramson and O. Zeitouni. Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field. Comm. Pure Appl. Math. 65 (2012) 1–20. | MR | Zbl

[18] D. Carpentier and P. Le Doussal. Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon models. Phys. Rev. E (3) 63 (2001) 026110.

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

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

[21] X. Fernique. Regularité des trajectoires des fonctions aléatoires gaussiennes. In École d’Été de Probabilités de Saint-Flour IV – 1974 1–96. Lecture Notes in Math. 480. Springer, Berlin, 1975. | MR | Zbl

[22] J.-P. Kahane. Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9 (2) (1985) 105–150. | MR | Zbl

[23] T. Madaule. Convergence in law for the branching random walk seen from its tip. Preprint, 2011. Available at arXiv:1107.2543v4.

[24] L. D. Pitt and L. T. Tran. Local sample path properties of Gaussian fields. Ann. Probab. 7 (3) (1979) 477–493. | MR | Zbl

[25] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin, 1999. | MR | Zbl

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