An expository overview of the definition of Liouville quantum gravity surfaces, the three senses in which random planar maps converge to them, and the major open problems.
Critical Gaussian multiplicative chaos: Convergence of the derivative martingale
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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.
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2019 1verdicts
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Random surfaces and Liouville quantum gravity
An expository overview of the definition of Liouville quantum gravity surfaces, the three senses in which random planar maps converge to them, and the major open problems.