Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

Abstract

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.1. Introduction. 1.1.Overview. In the 1980s, Kahane [45] developed a continuous parameter theory of multifractal random measures, called Gaussian multiplicative chaos; this theory emerged from the need to define rigorously the limit lognormal model introduced by Mandelbrot [59] in the context of turbulence. His efforts were followed by several authors [3,7,11,35,[67][68][69] [13] in the context of Mandelbrot's multiplicative cascades. This was done in the standard case of Liouville quantum gravity, namely strictly below the critical value of the GFF coupling constant γ in the Liouville conformal factor, that is, for γ < 2 (in a chosen normalization). Beyond this threshold, the standard construction yields vanishing random measures [29,45]. The issue of mathematically constructing singular Liouville measures beyond the phase transition (i.e., for γ > 2) and deriving the corresponding (nonstandard dual) KPZ formula has been investigated in [9,28,29], giving the first mathematical understanding of the so-called duality in Liouville quantum gravity; see [4,5,21,27,32,44,[48][49][50]54] for an account of physical motivations. However, the rigorous construction of random measures at criticality, that is, for γ = 2, does not seem to ever have been carried out.As stated above, once the Gaussian randomness is fixed, the standard Gaussian multiplicative chaos describes a random positive measure for each γ < 2 but yields 0 when γ = 2. Naively, one might therefore guess that −1 times the derivative at γ = 2 would be a random positive measure. This intuition leads one to consider the so-called derivative martingale, formally obtained by differentiating the standard measure w.r.t. γ at γ = 2, as explained below. In the case of branching Brownian motions [62], or of branching random walks [15,56] (see also [2] for a recent different but equivalent construction), the construction of such an object has already been carried out mathematically. In the context of branching random walks, the derivative martingale was introduced in the study of the fixed points of the smoothing transform at criticality (the smoothing transform is a generalization of Mandelbrot's ⋆-equation for discrete multiplicative cascades; see also [16]). Our construction will therefore appear as a continuous analogue of those works in the context of Gaussian multiplicative chaos.Besides the 2D-Liouville Quantum Gravity framework (and the KPZ formula), many other important models or qu...

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