Gaussian multiplicative chaos revisited

Robert, Raoul; Vargas, Vincent

doi:10.1214/09-aop490

Cited by 142 publications

(213 citation statements)

References 18 publications

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“…The inequality (7.1) and the classical corollary 6.2 in [19] imply the existence of some constant C q,r,δ > 0 such that for all x, y ∈ B(0, r) with |y − x| ≤ δ: We remind that the second inequality above results from a straightforward adaptation to the 2 dimensional case of the proof of Theorem 4.1 (in the log normal case). We then conclude by using the same argument than in the proof of Theorem 4.1.…”

Section: Then the Weak Limit (In The Sense Of Measures)mentioning

confidence: 94%

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KPZ formula for log-infinitely divisible multifractal random measures

Rhodes¹,

Vargas²

2011

ESAIM: PS

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123

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Abstract.We consider the continuous model of log-infinitely divisible multifractal random measures y]) and structure function ζ, we show that we have the following relation between the (Euclidian) Hausdorff dimension dimH of a measurable set K and the Hausdorff dimension dim ρ H with respect to ρ of the same set:Our results can be extended to all dimensions: inspired by quantum gravity in dimension 2, we focus on the log normal case in dimension 2.Mathematics Subject Classification. 60G57, 28A78, 28A80.

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Section: Then the Weak Limit (In The Sense Of Measures)mentioning

confidence: 94%

“…In this framework, the measure M is the multiplicative chaos associated with the function ln + R |x| and it can be defined almost surely (see Ex. 2.3 in [19]) as the limit (in the space of Radon measures) as l goes to 0 of the random measures M l (dx) defined by:…”

Section: The Log Normal Mrm Measure In Dimensionmentioning

confidence: 99%

KPZ formula for log-infinitely divisible multifractal random measures

Rhodes¹,

Vargas²

2011

ESAIM: PS

Self Cite

123

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“…Various specific physical models of log-correlated fields have been considered in [30][31][32][33][34]. Log-correlated fields are also of great significance in the area of Gaussian multiplicative chaos, introduced by Kahane [36], which recently became the object of renewed interest [2,[46][47][48][49]. We also mention the 2-dimensional Gaussian free field which plays an important role in statistical physics, the theory of random surfaces, and quantum field theory [17,26,29,50].…”

Section: Freezing and Log-correlated Fieldsmentioning

confidence: 99%

Freezing and Decorated Poisson Point Processes

Subag

Zeitouni

2015

Commun. Math. Phys.

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The limiting extremal processes of the branching Brownian motion (BBM), the two-speed BBM, and the branching random walk are known to be randomly shifted decorated Poisson point processes (SDPPP). In the proofs of those results, the Laplace functional of the limiting extremal process is shown to satisfy L θ y f = g y − τ f for any nonzero, nonnegative, compactly supported, continuous function f , where θ y is the shift operator, τ f is a real number that depends on f , and g is a real function that is independent of f . We show that, under some assumptions, this property characterizes the structure of SDPPP. Moreover, when it holds, we show that g has to be a convolution of the Gumbel distribution with some measure.The above property of the Laplace functional is closely related to a 'freezing phenomenon' that is expected to occur in a wide class of log-correlated fields, and which has played an important role in the analysis of various models. Our results shed light on this intriguing phenomenon and provide a natural tool for proving an SDPPP structure in these and other models.

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“…where φ is a bounded positive definite function: e.g., φ(x) = (1 − |x|) + , see [12] and the discussion in Example 2.3 in [13]). Define also the normalized field to bē X ε (x) = γX ε (x) − (γ 2 /2)σ ε , with σ ε = c ε (0, 0) = − log ε + 1, so that E(eX ε(x) ) = 1.…”

Section: Auxiliary Fieldsmentioning

confidence: 99%

“…But the integral is not necessarily a martingale and so we need a different argument which is based on comparison with another auxiliary field for which the martingale property does hold, using an idea of Robert and Vargas [13]. We summarise the argument below.…”

Section: Auxiliary Fieldsmentioning

confidence: 99%

Diffusion in planar Liouville quantum gravity

Berestycki¹

2015

Ann. Inst. H. Poincaré Probab. Statist.

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We construct the natural diffusion in the random geometry of planar Liouville quantum gravity. Formally, this is the Brownian motion in a domain D of the complex plane for which the Riemannian metric tensor at a point z ∈ D is given by exp(γh(z) − 1 2 γ 2 E(h(z) 2 )). Here h is an instance of the Gaussian Free Field on D and γ ∈ (0, 2) is a parameter. We show that the process is almost surely continuous and enjoys certain conformal invariance properties. We also estimate the Hausdorff dimension of times that the diffusion spends in the thick points of the Gaussian Free Field, and show that it spends Lebesgue-almost all its time in the set of γ-thick points, almost surely.The diffusion is constructed by a limiting procedure after regularisation of the Gaussian Free Field. The proof is inspired by arguments of Duplantier-Sheffield for the convergence of the Liouville quantum gravity measure, previous work on multifractal random measures, and relies also on estimates on the occupation measure of planar Brownian motion by Dembo, Peres, Rosen and Zeitouni.A similar but deeper result has been independently and simultaneously proved

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Gaussian multiplicative chaos revisited

Cited by 142 publications

References 18 publications

KPZ formula for log-infinitely divisible multifractal random measures

KPZ formula for log-infinitely divisible multifractal random measures

Freezing and Decorated Poisson Point Processes

Diffusion in planar Liouville quantum gravity

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