Gaussian multiplicative chaos and applications: A review

By Rémi Rhodes, Vincent Vargas. 2013

480 citations recorded by the source corpus.

In this article, we review the theory of Gaussian multiplicative chaos initially introduced by Kahane’s seminal work in 1985. Though this beautiful paper faded from memory until recently, it already contains ideas and results that are nowadays under active investigation, like the construction of the Liouville measure in 2d-Liouville quantum gravity or thick points of the Gaussian Free Field. Also, we mention important extensions and generalizations of this theory that have emerged ever since and discuss a whole family of applications, ranging from nance, through the Kolmogorov-Obukhov model of turbulence to 2d-Liouville quantum gravity. This review also includes new results like the convergence of discretized Liouville measures on isoradial graphs (thus including the triangle and square lattices) towards the continuous Liouville measures (in the subcritical and critical case) or multifractal analysis of the measures in all dimensions.

View Paper: https://www.semanticscholar.org/paper/92ad9b2d6ebaae6603faf14e5bc8ecaca5354f5c

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4. Christian Webb
5. Rémi Rhodes
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8. Adam J Harper

Citations in the corpus

• Maximum of the Characteristic Polynomial of Random Unitary Matrices (2015, background): “Log-correlated models are closely related to Gaussian Multiplicative chaos, see [48] for a review.”
• On Gaussian multiplicative chaos (2014, methodology): “Mollifying operators [11, 3, 10] This technique restricts the generality to the case where T is a domain in R, μ is the Lebesgue measure, and the covaraince kernel K (t, s) := EX (t)X (s) has the special form K (t, s) = γ(2) log ‖t− s‖ + g (t, s) , (2) where the function g : T ×T → R is bounded and ”
• Uniqueness of critical Gaussian chaos (2017, methodology): “We refer to [25, 11] for a much more thorough discussion of chaos measures and their applications, as well as for further references on the topic.”
• Branching random walks (2015, background): “In parallel, in the continuous-time setting, following Kahane’s pioneer work in [151], the study of Gaussian multiplicative chaos has witnessed importance recent progress (Duplantier, Rhodes, Sheffield and Vargas [101], Garban, Rhodes and Vargas [116], Rhodes and Vargas [217]).”
• The Riemann zeta function and Gaussian multiplicative chaos: Statistics on the critical line (2016, methodology): “[64] T. Madaule, R. Rhodes, and V. Vargas: Glassy phase and freezing of log-correlated Gaussian potentials.”
• MOMENTS OF RANDOM MULTIPLICATIVE FUNCTIONS, I: LOW MOMENTS, BETTER THAN SQUAREROOT CANCELLATION, AND CRITICAL MULTIPLICATIVE CHAOS (2017, background): “certain events A related to the definition of L, and it requires quite a lot of work to set this up carefully. Now we make a few remarks about multiplicative chaos: see the survey of Rhodes and Vargas [22] for more about the field. The theory begins with a random function h(·) on some space, say the i”
• Liouville Quantum Gravity on the Riemann Sphere (2014, methodology): “Convergence in law of the circle average based regularization measure is carried out via the smooth Gaussian approximations introduced in [52].”
• Extrema of log-correlated random variables: Principles and Examples (2016, methodology): “I am grateful to Jean-Philippe Bouchaud, Pierluigi Contucci, Cristian Giardina, Pierre Nolin, Vincent Vargas, and Vladas Sidoravicius for the organization of the trimester Disordered systems, Random spatial processes and Applications at the the Institut HenriPoincaré in the spring 2015.”
• Continuity and estimates of the Liouville heat kernel with applications to spectral dimensions (2014, methodology): “n R2 is given by Mn(dx) := exp γXn(x) − γ 2 2 E Xn(x)2 dx (2.3) with a parameter γ≥ 0. By the classical theory of Gaussian multiplicative chaos established in Kahane’s seminal work [20] (see also [25]) we have the following: P-a.s. the family (Mn)n≥1 converges vaguely on R2 to a random Radon measu”