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Critical Gaussian multiplicative chaos: convergence of the derivative martingale. (English) Zbl 1306.60055
Summary: 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 atoms. In connection with the derivative martingale, we state 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.

MSC:
60G57 Random measures
60G15 Gaussian processes
60D05 Geometric probability and stochastic geometry
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