Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

By Bertrand Duplantier, Rémi Rhodes, Scott Sheffield, Vincent Vargas. 2012

166 citations recorded by the source corpus.

In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.

Subfields: Finance, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d872d2a6c7b148f927bcb58bd7e4a8973f7a8d9b

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Citations in the corpus

The Riemann zeta function and Gaussian multiplicative chaos: Statistics on the critical line (2016, background): “We refer to the nice survey [75] for the basic properties of these measures, to [12] for an elegant proof of the existence of subcritical chaos measures, and to [33, 34, 7] for the existence and basic properties of critical Gaussian chaos.”
Freezing and Decorated Poisson Point Processes (2014, background): “Wemention the class of logarithmically-correlated (log-correlated, for short)Gaussian fields [7,18,27,31,42], which are of great importance in a variety of fields (see Sect.”
Uniqueness of critical Gaussian chaos (2017, background): “More general results developing the method of [26] are contained in the review [25] due to Rhodes and Vargas, whose conditions are very similar to ours.”
Liouville Quantum Gravity on the Riemann Sphere (2014, background): “In [26,28] it was proven that the measure3 √ 2/π(− ln )1/2 2e2Xĝ, +2 ln ĝ dλ(x) (4.”
Convergence of the centered maximum of log-correlated Gaussian fields (2015, background): “It also occurs in the case of BRW, see [1], and plays a similar role in the study of critical Gaussian multiplicative chaos [12].”
The extremal process of branching Brownian motion (2011, background): “8) (after this paper has been submitted, partial progress towards this conjecture has been achieved in [27], and [28]).”
On Gaussian multiplicative chaos (2014, background): “inally we would like to remark there exist GMCs that are not subcritical. So far the only examples known to the author are the critical GMCs over logarithmic ﬁelds and their hierarchical counterparts [8, 6, 5]. For these critical GMCs we have for any measurable set A⊂ T EM[A] = ( 0, µ[A] = 0 ∞, µ[A]”
Maximum of a log-correlated Gaussian field (2013, methodology): “5) is to give a rigorous construction of a standard Gaussian multiplicative chaos (GMC) in the subcritical case which is formally defined as (see [19]) a random measure such that for any set A ∈ Bb(R),”
Gaussian multiplicative chaos and applications: A review (2013, methodology): “Based on heuristics on star scale invariance, it is conjectured in [17, 58] that this renormalized sequence actually admits only one possible limit if the kernel of X is star scale invariant: Conjecture 6.”
Convergence in Law of the Maximum of the Two‐Dimensional Discrete Gaussian Free Field (2013, background): “On the mathematical side, numerous results and conjectures have been formulated for such models; see [16] for recent progress.”