Liouville measure as a multiplicative cascade via level sets of the Gaussian free field

Abstract

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Critical Gaussian multiplicative chaos: a review

“…This result was conjectured in [DRSV14a], and an analogous version proven for the branching random walk in [Mad16]. However, it was not proven in the Gaussian multiplicative chaos setting until [APS20,APS19], in which the underlying field is assumed to be a planar Gaussian free field (or a GFF on the unit circle). The papers [APS20, APS19] make use of the special "local set" approximation to the free field described in (2.10) to transfer the result of [Mad16] to the GFF.…”

Critical Brownian multiplicative chaos

“…We mention that the construction of the critical measure as a limit of subcritical measures is only partially known in the GMC realm. Such a result has first been proved to hold in the specific case of the two-dimensional GFF [4] exploiting on the one hand the construction of Liouville measures as multiplicative cascades [3] and on the other hand the strategy of Madaule [37] who proves a result analogous to Theorem 1.2 in the case of multiplicative cascades/branching random walk. It has then been extended to a wide class of log-correlated Gaussian fields in dimension two by comparing them to the GFF [30].…”

“…We believe that the approach we use in this paper to prove Theorem 1.2 can be adapted in order to show that critical GMC measures can be built from their subcritical versions in any dimension. 3 Theorem 1.2 can be seen as exchanging the limit in ε and the derivative with respect to γ . Surprisingly, a factor of 2 pops up when one exchanges the two:…”

“…The theory was introduced by Kahane [32] and has expanded significantly in recent years. By now it is relatively well understood, at least in the subcritical case where γ < √ 2d [9,22,[45][46][47] and even in the critical case γ = √ 2d [3,4,18,19,29,30,39]. In this article, the log-correlated field we have in mind is the (square root of) the local time process of a planar Brownian motion, appropriately stopped.…”

Critical Brownian multiplicative chaos

“…We mention that the construction of the critical measure as a limit of subcritical measures is only partially known in the GMC realm. Such a result has first been proved to hold in the specific case of the two-dimensional GFF [APS19] exploiting on the one hand the construction of Liouville measures as multiplicative cascades [APS17] and on the other hand the strategy of Madaule [Mad16] who proves a result analogous to Theorem 1.2 in the case of multiplicative cascades/branching random walk. It has then been extended to a wide class of log-correlated Gaussian fields in dimension two by comparing them to the GFF [JSW19].…”